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Renewable Energy 145 (2020) 1543e1571

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Renewable Energy

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Review

Uncertainty models for stochastic optimization in renewable energyapplications

A. Zakaria a, Firas B. Ismail a, M.S. Hossain Lipu b, M.A. Hannan c, *

a Power Generation Unit, Institute of Power Engineering, Universiti Tenaga Nasional, 43000 Kajang, Malaysiab Centre for Integrated Systems Engineering and Advanced Technologies (Integra), Faculty of Engineering and Built Environment, Universiti KebangsaanMalaysia, 43600 Bangi, Malaysiac Department of Electrical Power Engineering, College of Engineering, Universiti Tenaga Nasional, 43000 Kajang, Malaysia

a r t i c l e i n f o

Article history:Received 8 March 2019Received in revised form7 July 2019Accepted 15 July 2019Available online 16 July 2019

Keywords:Stochastic optimizationsUncertainty modelScenario generationsRenewable energy applications

* Corresponding author.E-mail address: [email protected] (M.A. Han

https://doi.org/10.1016/j.renene.2019.07.0810960-1481/© 2019 Elsevier Ltd. All rights reserved.

a b s t r a c t

With the rapid surge of renewable energy integrations into the electrical grid, the main questionsremain; how do we manage and operate optimally these surges of fluctuating resources? However, vastoptimization approaches in renewable energy applications have been widely used hitherto to aiddecision-makings in mitigating the limitations of computations. This paper comprehensively reviews thegeneric steps of stochastic optimizations in renewable energy applications, from the modelling of theuncertainties and sampling of relevant information, respectively. Furthermore, the benefits and draw-backs of the stochastic optimization methods are highlighted. Moreover, notable optimization methodspertaining to the steps of stochastic optimizations are highlighted. The aim of the paper is to introducethe recent advancements and notable stochastic methods and trending of the methods going into thefuture of renewable energy applications. Relevant future research areas are identified to support thetransition of stochastic optimizations from the traditional deterministic approaches. We concluded basedon the surveyed literatures that the stochastic optimization methods almost always outperform thedeterministic optimization methods in terms of social, technical, and economic aspects of renewableenergy systems. Thus, this review will catalyse the effort in advancing the research of stochastic opti-mization methods within the scopes of renewable energy applications.

© 2019 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15442. Overview of stochastic optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15443. Uncertainty modelling in stochastic optimization approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545

3.1. Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15453.1.1. Types of Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15453.1.2. Recent MCS applications in renewable energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546

3.2. Notable uncertainty modelling method: Generative Adversarial Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15494. Sampling methods in scenario generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1550

4.1. Importance Sampling (IS) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15514.1.1. Type of IS method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15514.1.2. IS method implementations in renewable energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1551

4.2. Notable sampling method: Markov Chain Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15544.2.1. Overview of MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15544.2.2. MCMC sampling procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1554

5. Stochastic optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15565.1. Stochastic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556

nan).

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A. Zakaria et al. / Renewable Energy 145 (2020) 1543e15711544

5.1.1. Two e stage models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15565.1.2. Multi e stage models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558

5.2. Approximate stochastic dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15605.2.1. Model predictive control (MPC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15615.2.2. Notable overview of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15625.2.3. Stochastic MPC implementations in renewable energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15625.2.4. MPC's comparison and future trending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1568References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1568

1. Introduction

Renewable energy sectors have seen tremendous growth in thelast decade throughout the world especially in Northern America,Western Europe, and China accounting for almost half of theexpansion [1]. The recent rapid energy shift in these parts of theworld are mainly due to the reduction of production costs of therenewable energy generators, the drive to reduce carbon emissions,and attractive tariffs offered [2]. Wind energy and solar energyaccounts for the most rapid growth in renewable energy genera-tions with an approximate 77% of new capacity, with hydropowerdominating the rest [3]. Despite being a clean and abundantlyavailable energy (in some parts of the world), renewable energyresources still suffer from its lack of energy density and its inter-mittency [4]. The latter presents the most challenge to researchersin terms of successfully predicting and utilizing the usage andcontrol of renewable energy resources. In contrast to the conven-tional generators (i.e. coal or steam turbine power plants), renew-able energy generators can only generate energy, when there arerenewable resources available. Therefore, appropriate prediction,control, and precise representation of renewable energy systemsplay an important role to ensure stable and uninterrupted energysupply. Optimization of renewable energy systems can be accu-rately solved if uncertainties, probabilities, and fluctuating behav-iours of renewable energy systems are being properly represented[5].

The current wave of optimization approach in renewable energyapplications are shifting. The first wave was in the form of deter-ministic approaches [6]. During this wave, mixed integer pro-gramming has stood out from the earlier modelling approachesnamely; dynamic programming, priority list, Lagrangian Relaxa-tion, etc. However, deterministic methods with an assumption ofperfect information produced idealistic results which contradictedwith the core value of renewable energy systems. With the fluc-tuations of renewable resources, varying demands, and intermit-tent economic parameters, deterministic approaches alone couldnot fully capture the dynamics of the whole renewable energysystems [7].

Studies are now moving towards stochastic optimization inwhich the optimization considers uncertainties and probabilities asinputs, then evaluate its influence on the output of the system [8]. Astochastic optimization then utilizes these scenario uncertainties inits objective function's formulations. Hitherto, vast amount of lit-eratures has been found regarding the stochastic optimizationtechniques [9]. Stochastic optimization provides a range of possiblesolutions whichmodels closer to realeworld situations that wouldbenefit operators/consumers in assessing the risks involved in theuncertainties of renewable energy generations. Therefore, thecharacteristics of stochastic optimization methods are more suit-able in handling renewable energy system's fluctuating and inter-mittent nature. Stochastic optimization however, generally suffers

from huge computational expenses due to large number of sce-narios that needs to be considered in its calculations [10].Numerous techniques have been developed by many authors inincreasing the stochastic optimizations' efficacy to reduce compu-tational expenses [11]. Nonetheless, it is still computationallydemanding and suffers from the ‘curse of dimensionality’ in casesof assessing the problems over multivariate andmultiple periods oftime intervals. Despite the advantages of the stochastic optimiza-tions, its implementations in renewable energy applications arestill relatively new. The problems in its transparencies, computa-tional efficacies, and their full practical implementations are stillbeing addressed by system operators and other interested parties.

Based on vast relevant surveys conducted, the paper's motiva-tion is to analyse recent and notable stochastic optimizationmethods in the lights of renewable energy applications whileidentifying its current and future research directions. We alsopointed out various advantages and disadvantages of highlightedstochastic optimization methods. Given the vastness of stochasticoptimization methods that exist hitherto, the focus of the paper isto provide a basic introduction to the highlighted methods whiledirecting interested readers towards notable works of other au-thors mainly in the field of renewable energy applications.

2. Overview of stochastic optimization

The general stochastic optimization in renewable energy ap-plications is broken down into several steps as summarized in Fig.1.The paper highlights these steps and focuses on the notable sto-chastic methods in recent renewable energy applications. As stated,the idea of the paper is to provide new researchers as well asadvanced readers in the optimization field with insights on therecent and notable stochastic optimization methods in renewableenergy applications. The paper is focused on the intuitive part ofthe stochastic optimization methods rather than the mathematicaldiscourses of the field. Readers are also exposed to the recenttrending of the stochastic optimizations in renewable energy ap-plications as well as future works and relevant research themes inthese areas. From these recent works, gaps and future works fromthe literatures are analysed. The trending from the surveyed recentliteratures is highlighted and the efficacy of the stochastic optimi-zation approaches is presented from the main results of theliteratures.

We consider only the most recent literatures in stochasticoptimization methods in the field of renewable energy applica-tions. Key aspects pertaining to the stochastic optimizations arefeatured such as its main results and contributions, its researchgaps, and its uncertainty parameters. The notable mentionedmethods are chosen based on its contributions in the field as well asits future implementation prospects. Past authoritative works arealso highlighted for interested readers to research further. In thispaper, within the scope of renewable energy applications,

StochasticOptimization

Parallel Scenario / ScenarioTree generations / Chance

Constraints Scenario /Forecasts of renewable data

with varying PDF(Uncertainty Modelling)

Sampling ofrelevant scenario

generations

Optimization/Decomposition ofsamples generated

using stochasticmodels

Max iteration?

Yes

Output of possiblebest solutions

No

Fig. 1. Stochastic optimization flowchart.

A. Zakaria et al. / Renewable Energy 145 (2020) 1543e1571 1545

uncertainty modelling/scenario generation approaches are initiallyaddressed, followed by notable sampling methods to capturerelevant scenarios in stochastic optimizations. Next, the stochasticoptimization approach is highlighted mainly in the lights ofapproximate stochastic dynamic programming. The paper con-cludes with main issues and challenges of stochastic optimizationapproaches in renewable energy applications, followed by its crit-ical remarks and future relevant themes in renewable energyapplications.

3. Uncertainty modelling in stochastic optimization approach

In stochastic optimizations, representing the correct un-certainties are critical. Each uncertainty modelling techniquewould yield a different representation of the systems. Therefore,appropriate selection of uncertainty modelling methods is crucial.Uncertainty modelling is a typical way to represent the stochas-ticity of renewables' systems. Instead of assuming perfect knowl-edge of the parameters (i.e. wind speed, solar irradiation and loaddemand) as opposed to a deterministic approach, random distri-butions are added as inputs to a stochastic optimization approachto mimic the probabilistic characteristics of a renewable energysystem. In representing the uncertainties, it is critical that thedistribution dynamics of the scenarios arewell captured. One of theways to do that is by generating large number of scenarios, whereeach scenario would capture the possible realization of the un-derlying uncertainties. The idea is to find the close approximationof the uncertainties' true distributions. In other words, the maingoal is to infer a probability distribution(s) of an output(s) based ona given probability distribution function(s) (PDF) of an assumedknown input(s). PDF distributions of inputs varies from parameters

Fig. 2. Uncertainty mo

or variables involved. Fig. 2 shows an overview of the uncertainties’modelling approaches. The scope of the paper is only limited to thenumerical method of the uncertainty modelling approaches,mainly in the recent Monte Carlo Simulation (MCS) approach inrenewable energy applications. Interested readers are encouragedto read the works made by Refs. [11e13] for further informationregarding other uncertainty modelling approaches.

3.1. Monte Carlo Simulation

MCS is one of the most used methods in the probabilistic un-certainty modelling approach [15,16]. Historical probability distri-bution function, forecasting errors, and market variability are theparameters that can be utilized by the MCS method to learn andpopulate the scenarios’ generations. The MCS method is favoureddue to its ability to systematically sample from random processes[16]. Furthermore, a transfer function is not necessarily needed inMCS. The problem can be treated as a black box system which canyield related output with given samples of inputs. MCS is alsointuitive and relatively easy to implement. MCS can also beimplemented in non e differentiable as well as non e convexproblems. Apart from that, it supports all probabilistic distributionfunction (PDF) types. Regardless, MCS has some deficiencies issuesuch as expensive computation due to its iterative behaviour,especially when the degrees of freedom and the space search ex-pands [17]. The general MCS method in renewable energy appli-cations can be described in Fig. 3.

3.1.1. Types of Monte Carlo SimulationAccording to Ref. [18], MCS method is typically divided in three

different types. The first one is called the SequentialeMCSmethod.

delling overview.

Start

Input PDF model ofrenewable and load

Constraintssatisfied?

Record results

Stop

Initialize MCSparameters

Generate statistic datausing MCS

Output PDFparameterscalculations

Yes

Maxiteration?

YesObtain PDF of output

parameters’ results

No No

Fig. 3. MCS Flowchart in renewable energy applications.


This method represents the uncertainties in chronological orderand is valued for its flexibility for assessing the reliability of thesystem's posterior distribution. This method is suitable in the ap-plications of time series of variable energy sources and variableload. Nonetheless, it requires a significant amount of computationaleffort as the dimensions of the uncertainties grow due to sequentialand iterative characteristics of this MCS. This method also may beinfeasible for some applications which are non e sequential. Thesecond type of theMCS is the Pseudoe Sequential MCS. It is nameddue to its ability of non e sequential sampling of system states andchronological simulation of only the sub e sequences tied with thefailed states. This method has a faster convergence rate than thesequentialeMCSmethod. However, this method is still demandingin terms of computational expenses as the degree of the problemincreases. The third and the last type is called the NoneSequentialMCS method. It is known for its high computational efficiency butlacks the ability to simulate the chronological aspects of a renew-able system operation. The summary of the three types of the MCSmethod mentioned with respect to the literatures mentioned isshown in Tables 1e3.

3.1.2. Recent MCS applications in renewable energy

3.1.2.1. Sequential MCS applications in renewable energy. Thesequential MCS method has been implemented as follows inrenewable energy applications [19]. implemented the MCS in sce-nario generations of irradiation, wind speed, load, and temperatureas inputs to optimize for the control of PVewinde diesele batterystand e alone systems. Authors suggested that a reduction insearch space is recommended to optimize the problem withinacceptable time frames. Genetic Algorithm (GA) has been usedinitially to reduce the possible scenario generations’ space of theMCS. A novel hybrid GA and MCS approach was proposed byRef. [20] to predict hourly energy consumption and generation by acluster of Net Zero Energy Buildings. The MCS aspect considers thevariability and the modelling aspects of the random energy con-sumption in a building at a given specific hour. An analyticalconvolution process combined with MCS in the works of [21] to

determine the optimal amount of power generation to becommitted by incorporating renewable power forecasting errorsand system reliability. A study by Ref. [22] proposed a sliding timewindow optimization approach to find an optimal design anddispatch scheduling strategy in a hybrid renewable energy systemconsisting of biomass, wind, solar, gas e fired boiler, battery, andthermal energy storage. Different scenarios were created to find outwhich combinations of renewable generators would yield the op-timum design and dispatch strategy. MCS was then tasked togenerate the cost of energy (COE) distribution as an output toprovide a risk indication of the chosen design. MCS has been uti-lized by Ref. [23] to sample different system states from the self eadapted evolutionary strategy (SAES) combined with Fischer-Burmeister algorithm in optimizing the one e time investmentand the operational costs of hybrid wind e energy storage powersystem. Lopes et al. (2015) have addressed the impact and systemreliability on the combination of wind generation and small hydropower plants [24].

The uncertainties of power generations are modelled using theMCS. A novel risk management method was investigated in thework of [25] based onmanaged charging of pluge in hybrid electricvehicle (PHEV) and vehicle e to e grid (V2G) using MCS. MCSanalysis of wind farm lightning surge transients aided by a light-ning detection network data is implemented by Ref. [26] to producea statistical depiction of over e voltages distribution within thewind farm electrical network. The statistic depiction can be used toassist wind farm lightning risk management and surge protectionoptimization. MCS has been implemented by Ref. [27] to considerthe uncertainties of load and irradiation in the economic optimi-zation of energy supply at off e grid healthcare facilities. In thework of [28], the authors have utilized MCS by performing Tem-perature e Augmented Probabilistic Load Flow (TPLF) to charac-terize the aspects of overe limit probabilities of events such as overand undere loading of loads and voltages in a 39e bus test system.

3.1.2.2. Pseudo e sequential MCS applications in renewable energy.The MCS method handled the uncertainties which are; the solarirradiance which is modelled using kernel density estimation, theload demand using a Gaussian distribution, the wind speed using aWeibull distribution. In the work of [29], the author coupled MCSwith quantile estimation techniques, and an efficient stochasticoptimizer, Adaptive Global Local Search (AGLS) in sizing hybridrenewable energy systems while considering the renewables un-certainty as inputs to MCS. Authors found that the approach hasenabled the control of the upside risk, consequently enhancing thedecision quality regarding the hybrid renewable energy systems.Implementation of MCS in Ref. [30] has been represented to showthe possible distribution of thermal energy collected at a solarthermal power plant. Applying a pseudo e MCS reduce the searchspace of the non e convex stochastic optimizer which is the PSO, tofind an optimal design that leads to an improvement of yearlythermal energy collected between 3.34% and 23.5%. MCS has beenapplied in Ref. [31] to consider the intrinsic variability of electricpower consumption in the probabilistic assessment simulation ofDG penetration in medium voltage distribution networks. Thefluctuations and uncertainties of load demands and generations ofsolar PVs are represented using MCS in the work of [32]. A multi elinear MCS method is proposed by Ref. [17] to analyse the steadystate operating conditions of an active electrical distribution sys-tem with Wind and PV generation plants. The uncertainties ofpower load demand and power production from renewable gen-erations are considered using the MCS combined with multi e

linearized power flow equations. In the work of [33] MCS isimplemented to model the uncertainties of energy demand, solarenergy availability, and electricity prices followed by a space search

Table 1Sequential MCS in renewable energy applications.

References Method Objective function Type of MCS Uncertain input Parameters Main results/contributions Future work/Gaps

[19] MCS e GA Minimize investment andoperational costs

Sequential Solar irradiation,Temperature, Wind speed,Annual fuel price interestrate, Average daily load

More information availableon expected performanceand costs of the systemwith respect to thedeterministic optimization

Optimize MCS samples andcomputational time

[20] GA e MCS Minimize instantaneousand cumulative energybalance

Sequential Buildings' energyconsumptions

Reduction in net energybalance in buildings

Extending the method'speriod to more than oneyear

[21] MCS e AnalyticalConvolutional Process

Optimize cost/benefitrelationship of REgenerations

Sequential Wind speed forecast error,generation unit reliability

Considerable improvementof computational efficiencywith reasonable cost/benefit

Increment of MCScomputational efficiency

[22] Receding HorizonOptimization e MCS

Minimize Cost of Energy,minimize risks

Sequential Wind power, Solar power,Battery storage, Biomasscombined Heat Power,thermal energy storage, gasproducer

Lowest cost option mayhave a higher risk of failing.The model provides rangesof possible microgriddesigns to determine majorrisk factors

Comparison of short-termperformance with whileconsidering demand sideuncertainties

[23] SAES e ARM eMCS Minimize investment andoperational costs

Sequential Load demand, Wind speed Reduces iteration in acomplex search space;Investigate discharge cycleefficiency of differentenergy storage on thesystem

Investigate impact ofenergy management onplanning decisions

[24] Risks based e MCS Minimize loss loadprobability (LOLP), EPNS,LOLD and LOLF

Sequential Wind speed, River inflows Precise estimation ofenergy delivered at a giventime and reducing loadshedding risks

Integration of variousintermittent RESs

[25] LOEE e MCS Minimize loss of energyexpected and expectedenergy not supplied

Sequential PHEV owner's behaviour,Solar and Wind power

LOEE in novel chargingapplications reduced by75% in comparison tounmanaged charging

N/A

[26] LINET - MCS Mitigation of lightning risks Sequential Lightning transients/activity at wind turbines

Cost e effectiveovervoltage protectionselection

N/A

[27] NPC e MCS Minimize net present cost(NPC)

Sequential Load, Solar irradiation Realistic stochastic batterylifetime prediction usingweighted Ah Schiffermethod

Analysing cost reductionand fossil fuel consumption,Improving supply reliability

[28] TPLF - MCS Minimize risk of systemover e voltage and risk ofsystem over e load

Sequential Load, Solar irradiation,Solar PV output,Temperature

Accurate uncertaintymodelling of Solar PVoutput, load, andtemperature at multi etime instants

Considering multi e timespatial and temporalcorrelations in powergeneration dispatchstrategy

Table 2Pseudo - Sequential MCS in renewable energy applications.

References Method Objective function Type of MCS Uncertain inputParameters

Main results/contributions

Future work/Gaps

[29] MCS e AGLS Minimize risk Pseudo - Sequential Possible sizing of HRES Sizing of HRES withminimal risk

An efficient quantileestimation method to solvelargeescale problems

[30] Ray Tracing MCS e PSO Maximize yearlythermal energycollection

Pseudo e Sequential Sun ray's position, daysof the year

Increment in yearlythermal energycollected

Integration of electricaloutput in the system,optimization of levelizedcost of energy

[33] Multi objective e RouletteWheel ee MCS

Minimize energy costsand environmentalimpacts

Pseudo e Sequential Supply side; DemandSide, domestic hotwater, space heatingand cooling)

Models and proposedmethods providedaccurate optimizationresults in identifyingthe economic/environmental paretofronts

N/A

[36] Cholesky Decomposition e

MCSMinimize economicrisks and maximizefinancial returns

Pseudo e Sequential water inflow, windspeed, solar irradiance,temperature of PVpanels, and averagegeneration capacity

Main characteristics ofthe random variablesare accurately modelledfor energy applications

Test proposed method withplant's installation site data(real data); Adaptation ofthe method in othermarkets

[34] Various techniques e MCS Minimize LOLP,Expected UnservedEnergy (EUE)

Pseudo - Sequential Load demands,Conventionalgeneration resources,Wind resources

Quantify the impacts ofintegrated renewableresources on reliability,power economics, andemissions

Integration of otherstochastic parameters suchas solar, demands, andstorages


Table

3Non

-Se

quen

tial

MCSin

renew

able

energy

applic

ations.

Referen

ces

Method

Objective

function

Typeof

MCS

UncertaininputPa

rameters

Mainresu

lts/co

ntribution

sFu

ture

work/Gap

s

[35]

NPV

eMCS

OptimizeNPV

Non

-Se

quen

tial

Risks

andfinan

ces

param

eters(refer

tothe

pap

erforfurther

clarification

s)

Optimizationof

conceptual

designwithresp

ectto

inve

stmen

ts,security,a

nd

return

s

Gen

eralizationof

analysis

method

whereco

rrelations

betw

eenparam

etersare

permissible

[37]

LCAe

MCS

Evaluaterandom

variab

les

environmen

talim

pact

Non

-Se

quen

tial

Wellfluid

composition,

Drillingtime,

geothermal

welllife

Mainen

vironmen

tal

impacts

ofge

othermal

plants

aresh

own

Com

parison

ofen

han

ced

geothermal

plants

LCAwith

trad

itional

plants

[38]

Ran

dom

MCS(RMCS)

with

annual

bran

chfault&

Sequ

ential

MCS

Minim

izesystem

interruption

;SA

IFI,CAID

Ian

dSA

IDI(refer

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Outage

per

year,d

uration

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osed

method

outperform

ssequ

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MCSin

term

sof

risks

analysis

N/A


reduction technique (RouletteeWheelMechanism) to reduce thecomputational expenses. Pinheiro et al. (2017) has implementedMCS associated with Cholesky Decomposition as inputs togenerate synthetic time series of water inflow, wind speed, solarirradiance, temperature of PV panels, and average generationcapacity [34]. TheMCS's goal is to perform risk analysis with 2000scenarios that spans over a period of 300 months. Authors inRef. [34] deployed the MCS to systematically sample randomprocesses of intermittent renewable resources and simulated thepower system and transmission constrained day e ahead marketoperations. MCS approach to investigate the economic risk anal-ysis of decentralized renewable energy infrastructures has beenused in the work of [35]. The MCS method considers the netpresent value (NPV) estimation and its ranges for each scenarioinvolved.

3.1.2.3. Non e sequential MCS applications in renewable energy.Few recent works that have implemented the non e sequentialMCS in renewable energy applications are mentioned as follows.The MCS approach to investigate the economic risk analysis ofdecentralized renewable energy infrastructures has been imple-mented in the work of [35]. The MCS method considers the NPVestimation and its ranges for each scenario involved. Hanbury andVasquez, 2018 employed the usage of MCS in geothermal plant'sconstruction to stochastically capture the environmental impactin terms of complete Life Cycle Analysis (LCA) relative to othermethods of energy production [37]. A systematic approach ofMCSto address the system distribution reliability considering inten-tional islanding was implemented in the work of [38].

3.1.2.4. Trending of MCS applications in renewable energy.Trending of MCS applications in renewable energy applications ishybridized with either a (meta) e heuristic method, strategicsampling methods, or other optimization methods. The (meta) eheuristic method typically acted as a space search reducer for theMCS method in performing the stochastic optimization as shownin Ref. [19], thus decreasing the overall computational expenses.Other method such as sampling methods (Typically related to

Generator DNNs

Noise Input

Generated SamplesHistorical Samples

DiscriminatorDNNs

Output PredictionReal Fake

Fig. 4. Example of GANs structure for wind scenario generation.


Pseudo e Sequential MCS) has strengthened the weakness of MCSmethod that requires large amount of sampling data to be accurate[38,39]. With strategic sampling of the scenario generations onlythe most important components are considered that contributesthe most to the objective function's stochastic optimization. Due tothe time dependent nature of renewable energy problems, most ofthe applications of MCS methods in this field is either Sequential orPseudo e Sequential. Section 4 describes further methods involvedin stochastic optimization sampling processes.

3.2. Notable uncertainty modelling method: Generative AdversarialNetworks

It is important to note that as computational advancementshave been growing rapidly throughout the years, model e drivenuncertainty modelling or scenario generations' methods have beenbecoming less viable, difficult to apply, and hard to scale [15,41].These are caused by the complex dynamics of renewable energysystems, time e varying nature of weather, and complex temporaland spatial connections. Studies are now converging towards thedata e driven methods in generating new sets of unique anddistinctive scenarios in renewable energy application [41]. As one

Table 4Characteristic and benefits of different types of GANs.

References Type of GANs Characteristic

[45] Wasserstein GANs Using the Earth e Moverdistance to evaluate thedistribution gap between reaand generated data

[46] Loss e Sensitive GANs Limiting the modelling abilityof the discriminator to betterdistinguish the real andgenerated samples regardlessof their complexity

[47] Semi - GANs Adding labels of real data to ttraining of discriminator

[48] Bidirectional GANs Mapping the real data to thelatent variable space in anunsupervised learningenvironment

[49] Info GANs Capturing mutual informatiobetween a small subset of latevariables and observations

[50] Auxiliary e Classifier GANs Incorporating label informatiinto the generator andadjusting objective function fthe discriminator

[51] Sequence GANs Generating sequences ofdiscrete tokens

[52] Boundary e Equilibrium GANs Equilibrium enforcing methopaired with a loss derived frothe Wasserstein distance

day might not be the same as another due to erratic weatherchanges and global warming, a new method can't only rely ongenerating/projecting scenarios based on historical data but mustalso correctly capture the rapid variations and strong diurnal cyclesof renewable resources in generating authentic new scenarios.Numerous amounts of literatures exist in scenario generations ofrenewable resources such as wind and solar as well as demands[42]. However, most of them were model e driven and it iscumbersome to pin point themost efficient usage of an exact modelto an exact situation. A recent study derived from Artificial NeuralNetworks (ANNs), namely Generative Adversarial Networks (GANs)by Ref. [43] has been gaining a lot of attention due to its ability tosynthesize artificial images from trainings of real ones. Only fewworks have been identified in literatures that implemented data e

driven GANs in renewable energy applications. The method suc-cessfully synthesizes renewable system's scenarios in Ref. [41] us-ing Wasserstein GANs. The generated scenarios are successful insynthesizing new and distinct scenarios by capturing the intrinsicfeatures of the historical data.

Fig. 4 depicts an overview of GANs system. The intuition behindGANs is to exploit the capacity of Deep Neural Networks (DNNs) inboth classifying complex signals (Discriminator) and expressing

Main advantage(s) Future work(s)

l

� Stable training of GANs� improves the learning

parameter and optimizationmethod of conventionalGANs

� Developing new algorithmsfor calculating Wassersteindistance between differentdistributions

� Reduces over fitting ofgenerated samples

� improves the learningparameter and optimizationmethod of conventionalGANs

� N/A

he � Generates a higher qualitysample than conventionalGANs

� Reduces training times forthe generator

� Weighting of discriminationand classification

� Generating examples withclass labels

� No assumptions ofunderlying structure of dataare needed

� Outperforms manyunsupervised featurelearning approaches

� Testing of the BidirectionalGANs under other space ofarchitecture models

nnt

� Learns interpretable anddisjointed representationson challenging datasetscompletely unsupervised

� Negligible increment incomputational expensescompared to conventionalGANs and easy to train

� Applying mutual informationand induce representation toother methods such asvariational autoencoder

on

or

� Generation anddiscrimination capability ofGANs are enhanced

� Produces a more diversifiedsamples of data

� Improving the reliability ofthe proposed GANs

� Improving visualdiscriminability

� Excellent performance insynthesizing speech, poem,and music generation

� Monte Carlo tree search inimproving the actiondecision making for largescale data in cases of long e

term planningdm

� Balances the discriminatorand generator in training

� Provides trade e offsbetween samples' diversityand quality

� Determining the best latentspace size for a given dataset

� Determining when and hownoises should be added tothe input

Variance Reduction Techniques(VRTs)

AntitheticVariates

(AV)

ImportanceSampling

(IS)

StratifiedSampling (SS)

CommonRandom

Numbers (CRN)

Latin HypercubeSampling (LHS)

Control Variates(CV)

Sequential ImportanceSampling (SIS)

DaggerSampling (DS)

Cross Entropy(CE)

Adaptive ImportanceSampling (AIS)

Fig. 5. Overview of variance reduction techniques (VRTs).


complex non e linear interactions (Generator). The idea behindGANs is to set up a minimax game of two DNNs which are in anadversarial relationship with each other. The Generator's DNNsupdates its weights during each training epochs to “trick” thediscriminator by generating “fake” samples of scenarios, while theDiscriminator's DNNs attempts to distinguish between true his-torical scenarios and the “fake” ones. Theoretically, after reachingequilibrium, the optimal solution of GANs will yield scenario dis-tributions from Generators which are hardly distinguishable froman authentic real historical data.

Hence, the Discriminator can no longer differentiate the originof the data, whether it is from the generator or the real historicaltraining data distributions. It is easier to imagine the GANs as acounterfeiter and a police game where the counterfeiter (Gener-ator) keeps on improving its technique to deceive the police, whilethe police (Discriminator) are also getting better at catching thecounterfeiter. The counterfeiter in the end would produce a “fakeproduct” that resembles the authentic product successfully, whichno longer can be identified by the police.

To summarize, the GANs method in scenario generation canleverage the power of DNNs and vast sets of historical data inperforming the tasks for directly generating scenarios conformingto the same distribution of historical data, without the need ofexplicit modelling of the distribution [43]. However, it is importantto note that the architecture of DNNs is complex in nature andrequires high computational efficacy in solving GANs problems.Yize et al. in his work [41] has suggested the usage of efficientGPU(s) to accelerate the DNNs training procedures. Future works inrenewable energy systems using GANs would be in the decision e

making strategy for unit commitments with high penetration ofrenewable energy generations and incorporation of the method inprobabilistic forecasting problems. Interested readers are directedto thework of [44] for a comprehensive overview of GANs as well asits future trending. Hitherto, several main variants of GANs havebeen identified and are summarized in Table 4. Characteristics,main advantages, and identified future works of the GANs' variantsare highlighted for the perusal of interested readers. It is to be notedthat the future works identified are mostly in the realms ofcomputational and mathematical sciences. However, implementa-tions of these GANs’ variants in renewable energy systems are yetto be tested.

4. Sampling methods in scenario generations

Increasing scenario generations would intuitively mean a closerand more comprehensive representation of possible futures.Nonetheless, increment of scenario generations (samples taken)

might only marginally increase the quality of the solution and theobjective function until a certain threshold [6]. One need to care-fully evaluate the trade e offs between the accuracy and the rate ofconvergence of a given algorithm. One popular technique to in-crease the sampling precision is called Variance Reduction Tech-niques (VRTs) [12]. VRTs can be broken down into several main sube categories as shown in Fig. 5. The estimates of scenario genera-tions’ precision depend on standard deviation between the sam-ples. The standard deviation can be expressed in equation (1)below:

s ¼ffiffiffiffiffiffiffiffiffiffiVðzÞpffiffiffiffiN

p (1)

where VðzÞthe unbiased sample variance and N is the samplenumber.

According to equation (1), the precision of the estimates can beintuitively increased by increasing the number of samples, N.However, increasing the samples’ size would mean reducing theefficacy of computation. In cases of sequential sampling processthroughout a year, with 8760 h steps, each hour containing its ownmultivariate properties, a sample increase of 1 would mean arepetition of 8760 h of sampling process. Therefore, another way tokeep the sample size small yet still maintaining a desired precisionis to reduce the variance between the samples. The main ideabehind VRTs is to decrease the amount of sampling needed to thedesired level of accuracy or increasing the accuracy of the expectedvalue for a given number of samples. There are various VRTs whichhave been reported in literature in renewable energy applications,as depicted in Fig. 5.

The authors in Ref. [53] used a range of randomvariables (RV) todevelop an improved stochastic model for power system sched-uling in the presence of uncertain renewables. A work in Ref. [54]focused on reliability evaluation through sequential Monte Carlosimulation to address cascading failure in power systems operation.The Weibull distribution together with antithetic variates (AV) isimplemented in order to reduce the large computational burden insimulations. Kardooni et al. [55] conducted a survey on climatechange and renewable energy in Peninsular Malaysia based onstratified sampling (SS). The authors in Ref. [56] identified thefactors shaping public opinion based on random stratified samplingto examine willingness to pay for expansion of renewable energysources in the electricity mix. A novel modified Latin hypercube-important (LHS) sampling method is suggested in Ref. [57] toenhance the accuracy and speed of correlation processing underlow sampling times. A LHS method is proposed in Ref. [58] toanalyse the reliability of power systems considering the


intermittent behaviour of renewable generations such as wind,solar power and fluctuation of bus loads. Dahlblom [59] appliedcontrol variates (CV) for Monte Carlo-pricing on three-asset spreadoptions with a view towards energy markets. A control variablebased dagger sampling (DS) technique is proposed in Ref. [60] todecrease the computational effort in Monte-Carlo reliability eval-uation for composite systems. Apart from the methods mentionedabove, Importance Sampling (IS) Method has become popular inrenewable energy applications. IS method and sub-division areexplained in the following section.

4.1. Importance Sampling (IS) method

Recent surveys from the literatures have shown that the ISmethod boost the sampling efficiency [61]. Typically, in MCS, thesampling representations would be excellent, if and only if samplescan be drawn from the target distributions. However, certain rarecases in renewable energy applications such as extreme wind cy-cles, sudden power outages, and rare occurrences of device failuresare difficult to account for. IS method focuses on sampling theimportant region (usually named “proposal distribution”) in whichthe important region have greater occurrence probabilities incomparison to the original distribution. The intuition is to constructa proposal distribution that “boosts” the sampling of importantregions. The method can bring enormous advantage, making anotherwise seemingly impossible problem for typical MCS,amenable. Nonetheless, applying IS method requires experience insampling due to its doublee edged characteristics. One could easilygo wrong by yielding an estimate with infinite variance, when asimple sampling method could have yielded a finite one. Therefore,a well e chosen proposal distribution is the key to maximizecomputation efficiency.

Fig. 6. Framework of CE based dispatch model to handle u

4.1.1. Type of IS method“Trainings” are encouraged with a trial distribution to capture

the appropriate estimate distributions using MCS. With repetitionsof MCS simulations, a better trial distribution can be drawn outbased on the weighted MCS samples. The process is repeated untiltermination criteria are met. The “Trainings” and the trial distri-bution procedure is called the Adaptive Importance Sampling (AIS)method, as the proposal distribution is updated adaptively. Anothertypical form of IS is called the Sequential Importance Sampling(SIS). As the name suggests, SIS constructs the proposal distributionsequentially and typically requires a decomposition procedure. SISis normally implemented in high e dimensional problems inbuilding up proposal distributions sequentially. Cross e Entropy(CE) was proposed by Ref. [62] to enable the inclusion of very un-likely events in computations. CE is a popular sub e category of ISmethod in VRTs to account for the optimizations of rare events [63].Based on repeated sampling, the method utilizes each iteration intwo steps; random data generation using a specific randommethodand updating the specific method's parameters to yield animproved sample in the next iteration. According to Ref. [64], ISmethod is the hardest variance reduction method to use, thereforeexpertise in the field is a necessity. Readers are advised to read theworks of [46] for the detailed mathematical representations andimplementations of IS methods. The following paragraph brieflypresents the recent literatures in IS, CE and SIS applications inrenewable energy systems. From our extensive literature searches,only a few recent literatures existed in the implementation of IS inrenewable energy applications.

4.1.2. IS method implementations in renewable energyIS in reducing the computational time of MCS has been imple-

mented by Ref. [65] in a probabilistic security management forpower system operations with large amounts of wind power.

ncertainties in PHEVs and renewable generation [66].

Fig. 7. Flowchart of SIS based hybrid probabilistic method for electricity market [71].


Author has found out that the IS method significantly reduced thecomputational time needed in sampling by two to three orders andhas shifted the original distribution to the desired proposal distri-bution. In the work of [66], CE method is utilized in hybridrenewable generation's optimal dispatch strategy of Plug e inHybrid Electric Vehicles (PHEVs) to improve the voltage profile andthe power flow with a 33 e nodes distribution systems. Authorshave found that the proposedmethod has managed to decrease thepower flows in heavy loaded lines and renewable generation fluc-tuations. The objective function is developed using two parts. Thefirst part presents the expectation of population variance ofrenewable generation outputs, while the second part denotes theexpected operation cost including battery degradation, PHEVowner benefits and control of the fleet of the vehicles. The objectivefunction is formulated as follows:

Mina1E�DP�Pe;t��þ a2E

"CBDP

XTt¼1

ð1þ u%ÞPdch;tDt#

(2)

Where, E denotes the outputs of PV system/wind generator, CBDPis

the per MWh cost of the battery degradation, Pdch;t is the averagepower consumption of a single PHEV in time interval t, Pe;t is thetotal charging/discharging power of PHEVs in time interval t, Dtisthe time interval, a1and a2 are the probability density function(PDF) parameters.

The comprehensive framework is developed with the multiplecases such as typical situations of seasonal renewable generationand vehicle usage, as shown in Fig. 6. Different renewable marketshare and peak generation or demand circumstances are alsodiscussed.

An efficient sampling method for MCS in Ref. [67] has beeninvestigated using CE and Copula theory to analyse generationadequacy of multi e area power systems with high penetrations ofwind power. Results have shown that the sampling methodsignificantly reduced the number of samples required to estimatereliability parameters of interest. A robust Multi e Objective CE(MOCE) algorithm is proposed by Ref. [68] in integrated schedulingapproach to solve for microgrid supply and demand schedulingproblem under uncertainties. A multi-objective function is devel-oped using fuel price, maintenance cost, buying and selling elec-tricity price, depreciation cost of battery and penalty cost which canbe presented in the following equation,

F1 ¼XjT jt¼1

8<:cfuelf mt

t þXdg2A

cdgpdgt þ�zpgt cbuyt ppgt þ �1� zpgt

�

þ csellt pdgt�þ sbtpbtt þ

�sespest þ schs

pchst

�9=; (3)

where cfuelis the natural gas fuel price,f mtt is waste heat by burning

natural gas, A is DG unit set, cdg presents the DG unit maintenancecost, pdgt is DG power output, cbuyt and csellt denote the buying andselling electricity price, respectively, ppgt is power grid poweroutput, sbt is depreciation cost of battery, ses and schs are the penaltycost of shortage/excess electricity and cooling/heating respectively,and pbtt is the battery power.

Another objective function F2 containing coal and natural gascombustion emissions, can be expressed as follows

F2 ¼XjT jt¼1

�εpgppgt þ ε

fuelf mtt

�(4)

where εpg and ε

fuel denote the conversion factor of carbon

emissions generated from electricity and natural gas, respectively.Authors have shown that the proposed algorithm has managed

to simultaneously minimize operation costs and emissions underthe worst e case scenario of fluctuating renewable generations anduncertain loads.

Leite and Castro [69] has presented a new probabilistic methodin evaluating spinning reserve margins using CE in renewable en-ergy systems with transmission restrictions. CE is utilized intreating the rare events and identifying necessity equipment foroperation in such events. Authors have shown that the CE methodwas successful in managing higher penetration of renewablesources and ensuring a reliable operation. Graf. et al. [70] has uti-lized the Adaptive Stratified Importance Sampling (ASIS) method inhybrid extrapolation and MCS method for estimating wind turbineextreme loads. Authors have shown that the variance of the hybridmethod are reduced swiftly with the implementation of the ASIS.The minimal variance importance distribution can be derived asfollows,

q*ðxÞ ¼ YðxÞf ðxÞEf ½YðxÞ�

(5)

Ef ½YðxÞ� �1

MtotPMtot

iYðxiÞ

;with xi drawn form f (6)

Table 5IS, CE and SIS implementations in recent renewable energy applications.

References Method Objective IS/CE distribution parameters Main Results Future work/Gaps

[70] Adaptive stratified e IS To estimate wind turbineextreme loads

Extreme loads, wind speed The proposed methodoutperforms sample e based ISe MCS method

Root causes of extremeresponse variation in windturbine loads

[65] Risk assessment e IS To estimate very low operatingrisks in power systems

Load, Wind power Decrease in computationalexpenses of two to three ordersof magnitude with respect tocrude MCS

Robustness tests with differentvalues of controllable activepower outputs and wind powerforecast distributions

[66] Normal DistributionParameterized e CE

To provide an optimal dispatchstrategy for PHEV

PHEV's driving behaviour, windspeed, solar irradiance, system,and load data

With introduction of Vehicle 2Grid (V2G), PHEV could act asstorage devices and proposedCE models solved for multiplepatterns of seasonal profiles forPHEV dispatch cases

Consider future studiesintervals in seconds andminutes relevant to powermarkets like spinning reserves

[67] Copula Theory & CE To analyse generation adequacyof multi e area power systemswith high penetrations of windpower

Conventional generation, Load,Wind power generation

Proposed method outperformscrudeMCS in terms of efficiencyand accuracy by three to fourorders. Number of samplesrequired does not increase withthe decrease of probabilityinterests' level

N/A

[68] Multi e objective e CE To schedule energy supply anddemand in integratedscheduling under uncertainty

Load profiles, Solar PV power Total cost and carbon emissionsare significantly reduced usingproposed method

Large scale integration ofdistributive resource andrenewable energy in regionalintegrated energy systems

[69] MCS e CE To assess probabilistic spinningreserve considering renewableresources and transmissionrestrictions

Wind generation capacity,Equipment failures, capacitylimits of transmissionequipment

Using risk assessments andknowing the critical elementsof the system, planners canbetter manage the highpenetration of renewablesources, ensuring sustainableand reliable operation

The configuration of theBrazilian interconnectedsystem to demonstrate thepracticality of the proposedapproach

[63] MCS-SIS To assess the deviation of price,possible occurrence of pricespike in electricity market

System load, renewable energyoutput, generator biddingstrategy, and outage rate

Estimations for both expectednormal price and price spikeprobability are accurate and fastusing less than 3% of the MCsimulation time.

The proposed is promising to beimplemented on onlineapplications.


Then for any given load Yj, The estimation of ASIS can beexpressed as follows

P�Y <Yj

� ¼ Ef�I�Y <Yj

��(7)

¼ðI�YðxÞ<Yj

�f ðxÞdx (8)

¼ðI�YðxÞ<Yj

� f ðxÞgðxÞ gðxÞdx (9)

¼ðI�YðxÞ<Yj

� f ðxÞgðxÞ gðxÞdx;�

1Mtot

Xi

IYðxiÞ

f ðxiÞgðxiÞ

�with xi

drawn from g (10)

¼ 1Mtot

Xi

8>>>><>>>>:

f ðxiÞgðxiÞ

if i< f

0 otherwise

(11)

¼ 1Mtot

Xi< j

f ðxiÞgðxiÞ

(12)

where, q*ðxÞis the auxiliary importance variable, YðxÞis load, f ðxÞ isthe distribution of wind speed, Ef denotes expectation with regardto f, Mtotis the total number of samples and gðxÞ is the arbitrary

distribution.Huang et al. [71] established a hybrid probabilistic assessment

method based on SIS for electricity market risk management. Theproposed method has considered various uncertainties such assystem load, renewable energy output, generator bidding strategy,and outage rate. The performance is checked under AustralianNational Electricity Market consisting of 59 buses, with 38 con-ventional generation units and one wind farm. The authors havefound that the method has resonance accuracy similar to MCS re-sults and fast executionwith regard to normal price and price spikeprobability. The implementation flow is illustrated in Fig. 7 wheresystem load is classified into “STATE”, and reported price of eachunit into “ACTION”.

Vast amount of recent literatures pertaining to recent IS adap-tations and improvements have been found outside of the renew-able energy applications which has proven to be efficient androbust to implement, mainly in the fields of signal processing andcomputational sciences. Recent adaptations of various IS methodsin the renewable energy applications are still scarce. Readers areencouraged to read the work of [39] which provides a compre-hensive overview of IS methods. In this work, the IS methods'(mainly AIS) scopes are discoursed at great depths from the past,the present, and on to the future. Future works in IS involves theimplementations of proposed IS methods with different and wideranges of distribution parameters in high dimensional problems inwhich the characteristics of the problems are very similar to therenewable energy applications. IS method's promising new appli-cations involves utilization of the method in the deep learning fieldfor computing the weights of hidden layers.


Summary of the references mentioned above is specified inTable 5.

4.2. Notable sampling method: Markov Chain Monte Carlo method

The Markov Chain Monte Carlo (MCMC) is a popular and rapidlygrowing sampling method which combines the properties ofMarkov Chain andMCS [72]. The intuition behind theMarkov Chainis to generate random samples through a special sequential pro-cess. The next generated random sample depends only on its pre-vious random sample and not affected by any samples prior to theprevious ones, thus creating a chain of random generated samplesuntil the end of iterations. This is the well e known “Markov”property. MCMC proved to be advantageous especially in Bayesianinference due to the difficulty of predicting the posterior distribu-tions via analytic methods. MCMC grants the user the ability toapproximate the posterior distribution, with minimal number ofsamples [73].

4.2.1. Overview of MCMCA simple yet concise introduction to MCMC was written by

Ref. [72]. The goal of the authors was to demystify MCMC samplingmethod and provide a comprehensive example to encourage newresearchers/users in adopting the MCMC method for their ownresearch purposes. Interested readers are directed to the work of[74] for in depth analysis and advanced coverage of MCMC. A moretechnical approach of MCMC method can be found in the work of[75].

In recent renewable energy applications, the MCMCmethod hasbeen implemented as follows. MCMC simulation model has beenutilized by Ref. [76] to consider the uncertainties of renewableenergy generation outputs and plug-in electric vehicle (PEVs)charging demand in a combined resource allocation framework indistributed energy storage systems (DESS). The objective function isexpressed as follows:

minU1;U1

XsProðsÞ �

Xi

hCREðiÞ � RREði;sÞ þ CESðiÞ þ CEVði;sÞ

i

þhClossðsÞ þ CconsðsÞ

i!(13)

Where

CREðiÞ ¼CkWREðiÞ � PCapREðIÞ

.LV (14)

RREði;sÞ ¼Xd

NdaysAðdÞ

Xh

rkWhREði;h;d;sÞ � PREði;h;d;sÞ (15)

Measured windspeed data

Data partitionedinto large time step

periods

Embedded Mchain gene

Generatiotransition m

for large tim

Generatiotransition m

for short tim

Fig. 8. Markov Chain development and r

CESði;sÞ ¼�CkWESðiÞP

CapESðiÞ þ CkW

ESðiÞECapESðiÞ

�.LV þ COM

ESðiÞ (16)

CEVði;sÞ ¼ CCHcEVðiÞ.LV (17)

hClossðsÞ þCconsðsÞ

i¼Xi

Xd

NdaysAðdÞ

Xh

rkWhGridði;h;d;sÞ �PGridði;h;d;sÞ

!!

(18)

Where, CEV , CREand CESdenote the capital and operating costs forPEV chargers, renewable energy resource units and DESS, respec-tively, PREis the active power provided by renewable energy re-sources in kW,RRE is the return of selling energy, PESis the activepower consumed by the DESS, EES denotes capacity of DESS in KWh,Clossand Cconsrepresent the costs of energy losses and energyconsumed by PEVs, normal load, and DESS, respectively, rkWh

Gridði;h;d;sÞis the selling price of renewable energy resources, PGridði;h;d;sÞis theenergy cost distributed from the grid in $/kWh, Ndays

AðdÞ is the numberof actual day, PGrid is the generated active power from grid, LV is thelevelized cost factor,CCHis the capital cost of PEV chargers, and cEVis number of charging stations installed at bus i .

A work in Ref. [77] presented a review of the measurementuncertainty in energy monitoring, where the MCMC method'scontributions in this area are elucidated. The authors in Ref. [78]used MCMC method in simulating the wind speed data andimplemented an embedded Markov Chain to incorporate the longterm effects in modelling the turbulent wind flow, as depicted inFig. 8. Authors have discovered that the proposed embeddedMarkov chain outperform the conventional MCMC method.

A slice sampling in MCMC simulation in a case study of proba-bility assessment for power system voltage stability margin withrenewable energy source has been presented in Ref. [79]. The slicesampling method performs better than Gibbs sampling methodwith respect to smaller simulation size, and the calculation effi-ciency. Besides, the proposed slice sampling method is more effi-cient and simpler to implement in the power system probabilisticcase study. The execution process of the proposed algorithm forpower system voltage stability margin using slice sampling inMCMC is illustrated in Fig. 9.

4.2.2. MCMC sampling proceduresTypically, the MCMC sampling is broken down in three main

sampling procedures namely; the basic Metropolis e Hastings al-gorithm, Gibbs sampling algorithm, and Differential Evolution [72].Each has its own advantages and complexity as well as types ofapplications. The basic Metropolis e Hastings algorithm is known

arkovration

Monte CarloSimulation

Simulated outputdata

n ofatrixe step

Mean wind speedsimulation for large

time step

n ofatrixe step

Large time steppopulated usingshort time step

elated Monte Carlo simulation [78].

Step 1: Weibull PDF and the beta PDF are constructed by collecting the historical data from photovoltaic power plant(s) and the wind farm(s) locations for a period of one year.

Step 2: The slice sampling algorithm is utilized to obtain the wind speed and illumination intensity samples from their PDFs.

Step 3: The extended continuation power flow (CPF) with the sampling matrix is executed to obtain the voltage stability.

Step 4: The power flow equation in critical point is linearized to achieve the eigenvector related to zero eigenvalue and evaluate distribution of the load margin variable.

Step 5: The sensitivity matrix is calculated corresponding to stability margin of wind speed and illumination intensity

Step 6: The distribution of voltage stability margin is achieved by using statistical method

Fig. 9. The voltage stability margin estimation process for power system with renewable energy source using slice sampling.

Table 6Main advantages vs disadvantages of main MCMC variants.

References Type of MCMC Advantages Disadvantages

[80] Metropolis Hastings � Knowing the posterior distribution without knowing all themathematical properties through random sampling despiteonly knowing the density for different samples

� Particularly useful in representing posterior distributions thatare hard to determine using analytical means

� Simple implementations for highly correlated distributions

� The values calculated must be proportional to the posteriorlikelihood

� Only applicable to very strongly correlated parameters� Requires a suitable step size to avoid too many rejections

from the next sampling sequence or resulting in a poorexploration

� Struggles in multi e modal distributions[80] Gibbs Sampling � Produces posterior distribution with good accuracy

� Easy to evaluate the conditional distributions� Conditional distributions will be in lower dimensions and

rejection sampling or importance sampling can be appliedto these dimensions

� Suffers from Computational efficiency in a long run� Suffers from manoeuvrability in cases of strong variables'

dependencies

[81] Differential Evolution � Faster convergence rate in a higher dimension samplingproblem

� Reduction in rejection rate of proposal distributions due tomultiple chains of sampling learning from each other

� Requires simple tuning parameters

� Cross e over and exchange between parallel chains ofsampling needs to be addressed for better convergence

[82] Slice Sampling � Does not require much tweakable parameters such asproposal distributions

� No rejections of samples� Suitable when little is known about the sampling distribution

� Suffers from curse of dimensionality� Sampling is done for each variable in turn using one

dimensional sampling in a multi e dimensional distribution

[83] Annealing Methods � Suitable for sample transitioning from high probability regionto another high probability region

� Does not suffer greatly from curse of dimensionality� A heuristic method that is easy to implement

� May be developed by trial and error� Moving in small steps from one iteration to the next� Requires knowledge in tuning its parameters

Stochastic Optimizationmethods

(Approximate)Stochastic Dynamic

Programming

StochasticProgramming

Robust Optimization

Value FunctionApproximation

Policy Iteration /Model Predictive

Control

State – SpaceApproximation

Fig. 10. General overview of stochastic optimization [85].



for its simplicity but lacks the ability to converge properly inproblems where parameters are highly correlated. Therefore, amore complex approach would be suitable in a multivariate envi-ronment. The Gibbs sampling method by Ref. [80] separates themultivariate problem and treats them independently by samplingfrom conditional distributions of parameters. The Gibbs method isknown for its accuracy but suffers in computational efficiency in along run. The Differential Evolution sampling procedure is a heu-ristic method that generates random chained samples that “learn”from each other. Instead of relying on a single random sample andcreating a chain from that random sample, multiple random sam-ples with multiple chains are generated using this method. Bylearning from other chains of samples, the parameter's correlationsbetween the samples are respected. Hence, it causes the chains ofsampling to be formed within the parameter's correlations limits,leading to a greater efficiency of sampling within the true under-lying distribution. However, the Differential Evolution algorithmrequires a certain “tuning” parameter to sample efficiently. Moreinformation regarding the DE sampling procedure in MCMC can befound in Ref. [81]. Many other main variants of MCMC exist hithertoand is summarized in Table 6. The table highlights the main MCMCsampling variants' advantages and disadvantages.

5. Stochastic optimization methods

As opposed to the deterministic optimization method whichassumes a perfect knowledge of a system, the stochastic algorithmmodels include uncertainties either in predictions, in the decisione

making processes, or both. In optimizing the problem formulationsunder uncertainties in stochastic models, the main approaches aredivided in three categories, namely; stochastic programming,robust optimization, and (approximate) stochastic dynamic pro-gramming (ASDP) as shown in Fig. 6. The paper's scope is focusedon the renewable energy applications which are in the field ofASDP. Brief information on the stochastic programming methodswhich are still prevalent in renewable energy applications areshown in the next section. Robust optimization approach is notconsidered in the paper's scope. The robust optimization approachgenerally produced over e conservative results, needed expertiseas well as rationale in uncertainty set construction, and difficult toimplement in dynamic uncertainty cases [84]. Nonetheless, inter-ested readers are directed to the recent notable works of stochasticrobust optimization in renewable energy applications asmentioned in the works of following authors [85e87] (See. Fig. 10).

5.1. Stochastic programming

In dealing with power generation problems, stochastic unitcommitment in the form of stochastic programming has beenimplemented as a promising tool [88]. The utilization of scenario e

based/tree uncertainty's representation and probabilities in theoptimization is the main idea of the stochastic programming. Thestochastic programming models are divided into two e stagemodels as well as multistage models. The methods were mainlyused as stochastic mixed integer programming (SMIP, linear or none linear SMIP are denoted as SMILP or SMINLP) problems formu-lations in renewable energy applications.

5.1.1. Two e stage modelsThe former two e stage models separate the optimizations in

day e ahead (1st stage) versus real e time (2nd stage) decisions.Typically, in the 1st stage (day e ahead), decisions for conventionalgenerators such as coal power plants and nuclear generators aremade beforehand as the start e up and shutdown times for thesegenerators are not immediate. The commitment decision in

operating these conventional generators depends on up/down timerequirements of the generators and the various starts up andshutdown costs. Therefore, the uncertainties and quality in fore-casting plays a major role in stochastic optimization as it effects theprior decision that must be made.

In the second stage which is the real time operations (i.e. theexpected real time operations' costs), the input variables’ PDF mustbe known beforehand to generate large number of relevant sce-narios relating to the output PDFs. The 2nd stage normally involvesthe strategy in dispatching renewable resources and reserves (e.g.Pump e hydro storage) over multiple periods of time whileconsidering uncertainties involved. Despite the huge number ofscenarios generated in the 2nd stage, the scenarios are not inter-twined with each other, implicating that each scenarios outcome isindependent of each other. Once the decision has beenmade for the1st stage problems, decomposition method is generally used in twoe stage models to treat the 2nd stage scenarios independently,resulting in a cluster of much lesser scenarios needed to be opti-mized. Common decomposition methods used in two e stageprogramming models are the Benders Decomposition (BD) method[75,76], Lagrangian Relaxation (LR) method [77,78], Bundlemethods [92], and Sample Average Approximation (SAA) method[88]. A stochastic two-level model is offered in Ref. [93] to maxi-mize the profit of the EV aggregator in the upper level and mini-mize the cost paid by the EV owners in the lower level in thecompetitive electricity markets. The upper level problem relates tothe revenue obtained from selling energy to the EV owners andfrom reducing load based on negative imbalance prices. The upperlevel problem can be modelled as follows:

MaximizeEDAt;w;E

þ;Bt;w ;E�;B

t;w ;lcht;w;E

cht;w;z;tðwÞ

Xw2T

hEcht;wl

cht;w � EDAt;wl

DAt;w � Eþ;B

t;w ; Eþ;Bt;w

þ E�;Bt;w ; E�;B

t;w

iþ b

"z� 1

1� a

Xw2U

pðwÞlðwÞ#

(19)

Where Echt;w is EV demand provided by the aggregator, lcht;w is theaggregator selling price EDAt;w day ahead (DA) EV demand, lDAt;w is theDA price at time t, Eþ;B

t;w ; E�;Bt;w are the positive/negative energy bal-

ance, zis the rival aggregator scenario index, Uis the number ofscenarios with regard to price and demand, ais confidence level ofconditional value at risk (CVaR), bis the risk factor and pðwÞis theprobability of occurrence with respect to demand and price andlðwÞdenotes the auxiliary variable to control CVaR.

The lower-level problem narrates the decision-making of EVowners and their reaction to the prices which can be expressed asbelow:

Xs0ðzÞ2 argnMinimizeXSðzÞ;ZS;S0 ðzÞ

E_D

t

26666666664lcht;wXs0ðzÞ þ

Xs2S

ss0

ls;t;z

37777777775

þXs02S

Xs02S

ss0

E_D

t Ks;s

0ZS;S

0 ðzÞ (20)

Where Xs0 is the EV demand percentage delivered by the aggre-gator, E

_D

t is the total EV demand, ls;t;z is the electricity selling pricesoffered by each rival aggregator, Ks;s

0is the cost relates to the

reluctance of EV owners for shifting between the aggregators ZS;S0 ðzÞ

is the EVs percentage that are shifted between the aggregators.

Start

Collect load data obtained from customers'registration

Identify types of loads and their average powerconsumption

Identify demand density of end-users based onhistorical data

Determine total demand of all customers (shiftable,sheddable and non-sheddable loads)

Input historical data of wind andPV power

Obtain mean and standard deviation throughforecasted and historical data

Scenario generation for load, electricity price, windand PV power by using MCS and RWM

Scenario reduction to Ns Scenario by using K-meansmethod

Master Problem(Find the optimal solution for UC and economic

dispatch without network constraints)

t = 0

s = 0

Subproblem for scenario s(Hourly evaluation of AC constraints for scenario s )

Is it feasible?

S < NS

t < NT

Number of cuts > ()

s = s + 1

Create cut forscenario s, time t

Add cuts to masterproblem

Get the optimalsolution

Stag

e1

Det

erm

ine

the

dem

and

resp

onse

stru

ctur

e

Stag

e2

Scen

ario

gene

ratio

nan

dre

duct

ion

t = t + 1

NO

YES

YES

YES NO

Stag

e3

Sche

dulin

g

YES

NO

NO

Fig. 11. Implementation framework of stochastic model to solve the optimal scheduling problem in autonomous microgrids [94].


A risk-constrained two-stage stochastic programming is sug-gested in Ref. [94] to maximize the expected profit during micro-grid operator considering uncertainties such as renewableresources, demand load and electricity price. A three-stage efficientflow diagram is developed to represent the underlying the optimalscheduling problem, as shown in Fig. 11. In the first phase, thecustomers electrical devices and equipment demand are assessed.In the second phase, the scenario generation and reduction processare executed for stochastic parameters. In the third phase, theoptimization problem is solved by employing a risk-constraintstochastic programming approach.

The authors in Ref. [95] developed a stochastic model of ACsecurity-constrained unit commitment (AC-SCUC) problem relatedwith demand response (DR) considering uncertainties of wind, PVunits and demand-side participation for the day-ahead energy andreserve scheduling in an islanded residential microgrid. In additionto that, an economic model of responsive loads is established basedon real-time pricing (RTP) scheme in view of the price elasticity ofdemand and customers' utility function. The objective function ofthe proposed model is designed with two terms including theprofits associated with here-and-now (H&N) and wait-and-see(W&S) decisions. The objective function includes the purchasing


energy cost from renewable, dispatchable units and buying reservefrom DG.

Max�PH&N þ EPW&S

�(21)

PH&N ¼XNT

t¼1

XNJ

j¼1

rj;tDj;t �XNT

t¼1

XNG

i¼1

h�Ai:ui;t þ Bi:Pi;t

�þ SUCi:yi;t: þ SDCi:zi;t�CRD

i;t :RDi;t þ CRU

i;t :RUi;t þ CRNS

i;t :RNSi;t�i

þ�CRD

j;i :RDj;i þ CRU

j;t :RUj;t

�XNT

t¼1

"XNW

w¼1

rw;t:Pw;t þXNV

v¼1

rv;tPv;t

# (22)

EPW&S ¼ �XNS

s¼1

XNT

t¼1

XNG

i¼1

ps:

hSUCi

�yi;t;s � yi;t

�þ SDCi:

�zi;t;s � zi;t

�þ rDepi;t :�rUi;t;s þ rNSi;t;s � rDi;t;s

�i

�XNsS¼1

XTt¼1

24ps::

XnJ

j¼1

rDepj;t :�rUi;t;s � rDi;t;s

�35�XNsS¼1

XTt¼1

"ps::

XnW

w¼1

rDepw;t DPw;t;s þXNv

v¼1

rDepv;t DPv;t;s

#�XNJ

j¼1

EENSj

(23)

In the above equations, the profit of microgrid operator, PH&N isassessed using sum of 5 terms. The first one denotes the electricityconsumption revenue from customers. The second term denotesthe cost associated with distributed generations (DGs) and theirstart-up/shut-down. The third and the fourth term represents thescheduled reserve cost of generating units and loads, respectively.Finally, the last term represents the energy cost delivered by windand PV units. Similarly, expected profit of microgrid, PH&N is eval-uated based on the total of 5 terms. The first one denotes the unitcommitment (UC) cost. The second and the third terms representthe deploying reserves cost from DG units and loads, respectively.The fourth term stands for the power cost delivered fromwind andPV units in real-time and the day-ahead energy forecasted. Finally,the last term corresponds to the expected energy cost which is notserved (EENS). The detail parameter description of the in equations(22)-(23) can be found in Ref. [95].

A work in Ref. [96], proposed a risk constrained two-stage sto-chastic programming model to determine the optimal schedulingto maximize the expected profit of operator. The flow diagram ofthe propose framework is operated in two stages, as illustrated inFig. 12. As it can be observed that, the input data is categorized intotwo groups, deterministic data and stochastic data. After, a set ofscenarios is generated using MG uncertainties. Then, an appro-priate scenario-reduction algorithm is employed to reduce thegenerated scenarios into an optimal subset. In the next stage, thestochastic security and risk-constrained scheduling problems areaddressed. The optimal scheduling of the generating units is per-formed based on unit commitment (UC) algorithm and AC/DCoptimal power flow (OPF) procedure by taking into account ofobjective function and constraints.

Readers are encouraged to read the works specified for eachdecomposition algorithm, which highlights the past notableimplementations of the two-stage methods in power generationsand renewable energy applications. Table 7 presents the recentworks of two-stage stochastic programming in renewable energyapplications.

5.1.2. Multi e stage modelsIn multi e stage stochastic programming models, uncertainties

are captured dynamically as possible events branched out of ascenario tree. Each uncertainty in events at a later time tþ1,

depends on its previous states at time, t. Decision e making pro-cesses are adjusted and updated hourly (or multi e hourly or sub e

hourly). Therefore, the multi e stage models represent a more ac-curate and realistic interactions between decision e makings and

unfolding uncertainties as time goes by. Each scenario generatedtakes a unique path starting from its root node, x1 to correspondingend nodes (i.e., x6, x8, and x15), where each node along the pathrepresents the time at which decisions were made. For each cor-responding scenario, n (i.e. n1 taking the node from x1/ x2/ x3/ x5), the problem is treated as an individual deterministicproblem. The difficulty of the multi e stage models rises from thenon e anticipative constraints, which means that only one set ofdecision variables are permissible at each node. The advantages ofthemultie stagemodels comewith a huge computational expense.The number of scenarios grows exponentially as shown in Fig. 13.Hence, multi e stage models are harder to solve than the two e

stage models. Advanced decomposition models/algorithms aretypically introduced in these cases. Often, the techniques used arenested or multi e layered decompositions and are further dividedinto scenario e based decomposition and unit e based decompo-sition targets [6]. Common advanced decomposition algorithms inmulti e stage stochastic programming and its past notable worksare shown as follows; Augmented LR [105], Dantzig e Wolfedecomposition (Column Generation (CG)) [106], ProgressiveHedging [107], Nested CG [108], Stabilized LR or CG [109].

The algorithms summarized in Table 9 are used in the pastnotable works of multi e stage e stochastic programming. InTable 9, readers are also enlightened with the qualitative advan-tages and disadvantages of the highlighted algorithms in multi estage stochastic programming, while Table 8 presents the recentworks of multi e stage stochastic programming in renewable en-ergy applications. Quantitative comparisons of the two e stage andmulti e stage models can be found in the past works of [110,111].Qualitative advantages and disadvantages of these methods aresummarized in Table 9.

From the literatures surveyed based on Table 7 & Table 8 inrenewable energy applications, it is apparent that the two e stagestochastic models are preferably implemented due to its simplicityin implementations and a guaranteed convergence in obtaining thesolution. However, the multi e stage stochastic models arebecoming more reliable as it better represents the complexity ofrenewable systems with significant increase in renewable re-sources and storages. Various advanced decomposition in two e

stage and multi e stage models have proven to yield better resultsthan the deterministic as well as perfect foresight cases (i.e.[101,114]). The literatures in two e stage stochastic models pro-vided a rather conservative solution with respect to multi e stage

Input data (Wind and PV power, load and demandelasticity, electricity price and etc.)

Scenario generation (Generate scenario data based onnormal operation contingency - based uncertainties)

Scenario reduction (Reducing the number of scenariosto NS based on PDF algorithm)

Master problem(find network optimal solution for UC and economic

dispatch)

Sub problem (I)(Hourly network evaluation for DC constraints)

Sub problem (II)(Hourly network evaluation for AC constraints)

Sub optimization(Hourly network evaluation for AC constraints for all scenarios)

Stage I

Is it feasible?

Is it feasible?

Scenario S1 Scenario SN

Master problem(Evaluation of DCOPF)

Sub problem(Evaluation of ACOPF)

Master problem(Evaluation of DCOPF)

Sub problem(Evaluation of ACOPF)

Is it feasible? Is it feasible?

Is it feasible?

Final results

Stage II

YES YES

YES

YES

NO

NO

NO NO

Cut

Cut

Cut Cut

...

...

...

...

YES

NO

Fig. 12. Methodological framework of a stochastic model for energy and reserve scheduling considering risk management strategy [96].


stochastic models that may lead to inefficiencies in generating thebest solutions. With advancements of computational efficacies,multi e stage stochastic models are becoming more viable insolving stochastic renewable energy problems. Applicability ofmulti e stage stochastic models (short e term and long e term)especially in big e scaled renewable economic dispatch are yet tobe fully explored. Demand side uncertainty and considering de-mand side response has been gaining a lot of attentions in formu-lating the stochastic renewable systems' problems. Many recent

literatures on stochastic programming (i.e. [82,83,88,89]) havestarted to consider the demand side uncertainty and managing thedemand side in optimizing the renewable's system. Main advan-tages of a responsive demand side management are the reductionin costs and minimization of energy wastage. It is to be noted thatliteratures combining the stochastic programming methods withmetaeheuristic algorithms were not being considered in this sec-tion and only SMIP method variations were highlighted.

Table 7Two e stage stochastic programming methods in renewable energy applications.

References Methods Structures Objective(s) System's uncertainty Main Result(s) Future work(s)

[97] SMILP Two e stage Minimize daily operationalcosts

Wind power and Energystorage

The proposed stochasticmethods reduced the totaldaily costs and loadshedding

N/A

[98] Multi e objective SMILP Two e stage Minimize operational costand pollution

Demand side, supply side(renewable), and energystorage

Applying portablerenewable energyresources have decreasedthe cost and increasedprofits

N/A

[99] Novel decomposition e

SMILPTwo e stage Minimize NPV of total

expected costsSolar irradiance, wind, andload

The proposed methoddemonstrated theeffectiveness in finding theoptimal battery energystorage system (BESS)power and energy sizes

Envisaged to be used in MGplanners, Govt. and privateagencies

[100] BD e SMIP Two e stage Minimize day aheadpurchase cost and expectedresource cost

Demand side, supply side(RE), electricity prices

Day e ahead powerprocurement and theformulation as a two e

stage SMIP problem isproposed

Demandeside procurementby twoestage stochastic am

[101] BD e SMINLP Two e stage Minimize expected totaloperation costs includinggeneration, day e aheadmarket, and battery wearfor the next 24 h

Demand, EV availability andstorage capacity, renewableenergy resources

Using the energy from EVreduces the total operationcost of the microgrid. Theresults yielded better costsavings than adeterministic model

N/A

[102] BD e SMILP Two e stage Minimize environmentaland social impacts

Wind speed, solarirradiation, and demand

Including demand responseas a flexible load reducesload curtailment andreduces energy generationneeded

N/A

[103] ε e Constraint multi eobjective SMILP

Two e stage Maximize DG owners'profits and minimizeDistribution Company's(DisCo) costs

Wind speed, load,electricity price

Solving the reconfigurationof the network andallocation of DGsimultaneously produced amore desired schedulingbetween the stakeholders.The stochastic optimizationis compared to adeterministic optimizationwith an improved profit onbehalf of the DG owners

N/A

[104] Scenario e based SMINLP Two e stage Minimize active andreactive power purchasingcosts

Load demand, wind power Reduction of expected costsof energy and reactivepower as well as emissioncosts

N/A


5.2. Approximate stochastic dynamic programming

Stochastic dynamic programming is an optimization method insolving discrete multi e stage decision e making processes withunderlying uncertainties or probabilities. Decisions made to lowerthe objective function's costs at a current stage might

x2

x3 x4

x7x6x5

t

t + 1

t + 2

t + 3

Fig. 13. Scenario tree with multiple stages

unintentionally increase the total costs throughout the wholeperiod of optimizations. One need to evaluate decisions made at allstages carefully to obtain the most cost e efficient objective func-tion. Stochastic dynamic programming method can capture thetrade e off between decisions made in the present and the futurestages. Due to these properties, it is instinctive that stochastic

x1

x9

x10 x11

x8 x12 x13 x14 x15

(4 stages, 8 scenarios, and 15 nodes).

Table 9Qualitative advantages and disadvantages of the two e stage and multi e stage stochastic programming algorithms.

Stochastic OptimizationMethod

Type References and Algorithms Advantages Disadvantages

Stochastic Programming Two e stage [80,81] LagrangianRelaxation (LR) [88] SampleAverage Approximation(SAA) [78,79] BendersDecomposition[92]Bundle Methods

� Simple Implementations andeasier to understand

� Convergence and goodperformances are guaranteed asvarious decomposition methodshave been tested

� Robustness issues can beaddressed using riskmeasurements

� Value of stochastic solution andexpected value of perfectinformation can be provided

� Probabilities of scenariogenerated need to be known

� Computationally expensive forlarge number of scenariosgenerated

� Complexity in dealing withinteger variables during the 2ndstage (i.e. unit rescheduling inreal e time)

� Assumption of staticuncertainties

Multi e stage [105] Augmented LR[106] Column Generation(CG) [107] ProgressiveHedging[109] Stabilized LR or CG

� Considering over multiple timeintervals of uncertainties indecision e making processes

� Uncertainties and decisions canbe adjusted dynamically asscenarios unfold

� Advantageous for systems thatneeds quick rescheduling

� Providing value of perfectinformation and value ofstochastic solution

� Size of problems growsexponentially with increasingtime intervals and scenarios

� Requires explicit scenario treerepresentations

� Difficulties increase with integervariables present in all stages

Table 8Multi e stage stochastic programming methods in renewable energy applications.

References Methods Structures Objective(s) System's uncertainty Main Result(s) Future work(s)

[112] Dynamic Response e

SMILPMulti e stage Maximize net social benefit Demand side and Energy

StorageA responsive demand sideprovided a more flexibleand smarter power systems

Enhancing planningmethodologies using k e

means and system states[113] Decision dependent e

SMILPMulti e stage Maximize total profit Wind capacity penetrations

and demandThe proposed methodprovided effectiveoptimization informationon investment and long e

term expansion planning

Developing new modelswith uncertaintiesconstraints

[114] Two e period multi estage SMILP

Multi - stage Minimize NPV related tolosses, emission,maintenance, operation,and unserved energy

Generation sources,electricity demand,emission prices, demandgrowth

The proposed methodproduced significantlybetter results in terms ofobjectives and yieldedrobust decision - makingsin comparison todeterministic methods

N/A

[115] Piecewise multi e stagelinear stochasticoptimization

Multi - stage Minimize operational costsand computational time oflong e term generationscheduling of hydropower

Load and Water inflow Inclusion of piecewiselinear approximationboosted the computationalefficacy and minimized theoperational costs inoperating large storagecapacity hydro powerplants

N/A


dynamic programming is suitable in the applications of renewableenergy optimizations.

The usage of dynamic programming can be dated back to late1970s [90] in solving deterministic problem. The solving approachwas based on Bellman's Principle of Optimality [91] which uses thebackward induction method. The past works of dynamic pro-gramming suffer from heavy computational expenses due to thecurse of dimensionality. As the number of scenarios and states in-creases as stages unfolds, the time needed in yielding a solutiongrows exponentially. Hitherto, various methods and broad class ofalgorithms have been tested to overcome the computationalexpenses.

Approximate stochastic dynamic programming (ASDP) hasproven to lighten the burden of dimensionality's curse of dynamicprogramming and is well suited for models with uncertainties andstochasticity [116]. Generally, the ASDP method can be divided in

three categories as shown in Fig. 6. The scope of the paper is withinthe policy function approximation in the form of stochastic ModelPredictive Control (MPC) in renewable energy applications. Readersare directed to the recent renewable energy applicationsmentioned which highlights the usage of value function approxi-mation [117e119] and state e space approximation [93,94]methods. A comparison of approximate dynamic programmingtechniques was carried out by Ref. [122]. Authors have comparedvarious policy iteration and value function approximation tech-niques. Authors have argued that new theory and methodology areneeded for these techniques in order to solve real e world prob-lems, which are becoming more difficult.

5.2.1. Model predictive control (MPC)MPC, also known as receding control horizon approximates

policies by iteratively solving a finite horizon optimal control

Fig. 14. Framework of the MPC optimization-based heating, ventilation, and air conditioning (HVAC) systems. The boxes highlighted with blue denote the factors that have impactto the optimization problem directly; the boxes highlighted with green indicate the optimization problems results [123].


problem. The horizon recedes once the optimal control for a currentstage, t has been found moving on to another finite horizon at alater stage, t þ 1. The process is repeated until the optimal controlhas been found for all stages; t (initial stage) until tmax (final stage).Serale et al. [123] have suggested several parameters which have adirect impact to the MPC optimization problems namely; Objectivefunction, receding horizon, control model, constraints, and distur-bances, where the optimization problems of MPC can be furtherdivided into Linear and Non e Linear problem formulations. Theframework of the proposed method is illustrated in Fig. 14.

Many works in the scope of MPC have been found in literaturesand are highlighted in the next paragraph. Recent literatures instochastic MPC are mentioned in the later paragraphs of this sec-tion and summarized in Table 10, Table 11, and Table 12.

5.2.2. Notable overview of MPCSeveral recent reviews in MPC have been published in the lights

of power generations, building and environments, and renewableenergy applications. Interested readers are encouraged to read theworks of following authors where theoretical modelling and ap-plications of MPC are further discoursed. The state e of e the e artdevelopment of MPC has been reviewed by Ref. [124] for renewableenergy applications. The authors have presented a systematic re-view of MPC applications in the field of solar PV and wind energyrenewable systems. The authors aimed to help researchers infurther exploring the flexibility of MPC for design, implementation,and analysis in renewable energy applications. In Ref. [125], hier-archical energy management strategy based on model predictivecontrol is proposed for microgrid management operation consid-ering different endogenous and exogenous sources of uncertainties.In Ref. [123], MPC in the themes of enhancing building and HVACsystem energy efficiency have been systematically reviewed. Thepotential benefits and future of MPC are discoursed at a great depthin authors’work. ANNs based MPC has been reviewed by Ref. [126]in a case study of a residential HVAC system. The authors haveutilized new ANNs algorithm to design a supervisory MPC whichsuccessfully reduced operating costs of equipment while con-straints are not violated with a cost reduction percentage range of6%e73% depending on the season. A similar review was made by

the main author in his past work [127] with regards to theory andapplications of HVAC control systems using MPC and was regardedas the most remarkable review on MPC due to clear classificationsand comprehensive scheme of MPC implementations.

5.2.3. Stochastic MPC implementations in renewable energyStochasticity of MPC in recent renewable energy applications

are typically represented as probability e constrained scenarios orforecasts, uncertainty modelling of scenario generations, andrandom disturbances. Stochastic MPC can be further derived intothree main categories which are tree e based, chance e con-strained, and multi e scenario MPC. The summary of recent liter-atures pertaining to these categories is mentioned in the nextsections.

5.2.3.1. Tree e based MPC implementations in renewable energy.Tree e based MPC works with an assumption of time dependantevents can be synthesized from a rooted tree, where the mostrelevant possible disturbances can be captured. The concept of treee based MPC is quite similar to the multi e stage stochastic pro-gramming approach (refer Fig. 7). Each root to different nodes’paths represents a possible disturbance scenario, where thebranching of the paths symbolizes the different forecast possibil-ities and uncertainties along a given prediction horizon. Each nodeat a given point in time, t corresponds to a control action that can betaken at that time. One must note that the control action takenmust not be allowed to diverge before the bifurcation points. Thetree e based MPC utilizes the ensembles of forecasts and solves itby considering the sequences contained in the tree. Different paths/branches of the tree nodes are treated as individual deterministicproblems. The path with the least costs or the most efficient interms of given objective functions are implemented at current time,t as a control action. The process is repeated until the control op-timizations over the entire horizons have been obtained.

A hybrid robust and stochastic accelerated MPC have beenimplemented in the work of [128] with 24 h horizonwindow for EVintegrated microgrid energy management considering demandresponse. The authors have utilized the hybrid MPC with forecastscoupled with Benders decomposition (BD) method to

Table 10Tree e based stochastic MPC in renewable energy applications.

References Method Objective MPC Type Control horizon Samplingresolution

System's uncertainty Main Results Future Work

[128] Stochasticaccelerated MPC

Minimize total dailyoperational costs

Tree e based 16 h ½ hour EV charging demand, load,real e time electricity price,renewable energy output

The stochastic MPCoutperforms thedeterministic MPC by lowertotal daily operational costin all cases

Extending the proposedmethod with available EVcharging load predictionmodels

[129] Risk e aversestochastic MPC

Maximize profit andminimize risks (CVaR)

Tree e based 24 h 1 h Wind power forecasts, priceof energy

The proposed methodoutperforms all mentionedmethods and marginallyexpected profit comparedto perfect solution

Application of the proposedmethod to real e worldcases and other renewableapplications

[130] CVaR fault tolerantstochastic MPC

Optimize CVaR Tree e based 4 steps ahead 1 s Wind power forecasts The proposed method hasachieved a controlperformance of 40% higherthan the common Min e

Max performance index

Solving the proposedstochastic MPC in one stepto yield a higher practicalvalue

Table 11Chance - constrained stochastic MPC in renewable energy applications.

References Method Objective Type of MPC Controlhorizon

Samplingresolution

System's uncertainty Main Results Future Work

[131] Multi e time scalestochastic e heuristicMPC

Minimize weeklyoperational costs

Chance e constraint 12 h 5min/1 h PV power forecast, plug inEV, deferrable and non e

deferrable appliances insmart home

Shifting the hourescale and dayescaleappliances to the optimal hours andweek of the day can substantiallyreduce the weekly operational costs

Applying the proposed method inmulti e scale microgrids

[132] Stochastic two e stageMPC

Minimize cost of energyand emissions ofgreenhouse gases

Chance e constraint 6 h 1 h Renewable energyresources, demand

Experimental results have proven thefeasibility and implementation abilityof proposed stochastic MPC thatoutperforms the deterministic MPC

Analysing the scalability ofproposed framework andinvestigating distributed methods

[133] Stochastic warpingfunction MPC

Minimize wind powertracking error

Chance e constraint 1e12 h 5min Wind power forecasts The proposed stochastic MPCoutperforms the deterministic MPC inpower tracking errors

The proposed control system can beintegrated into currently existingsystem

[134] Stochastic e EMPC Minimize microgrids'operating costs

Chance e constraint 72 h 1 h Renewable supply, Loaddemand

The proposed method achieved a bettertrade e off between performance andcomputational efficacies in comparisonto centralized scheme

Incorporating the topology ofdistribution network, energyexchange between MG andfluctuating prices of energy

[135] Stochastic MPC Minimize operational costs Chance e constraint 24 h ½ hour RE generations, load,demands, EV, andelectricity prices

The stochastic MPC frameworkoutperforms the traditional day e

ahead programming strategy in termsof minimizing the operational costs

Applying the stochastic MPC in amulti scale microgrid systems

A.Zakaria

etal./

Renewable

Energy145

(2020)1543

e1571

1563


simultaneously reduce total operational cost in energy manage-ment as well as improve computational efficiency. Simulation re-sults showed that the proposed method outperforms the standarddeterministic MPC method with regard to total operational cost bya margin of around 10%. The algorithm of stochastic BD applied toMPC is shown below.

Algorithm: Stochastic BDAt each time step t doInitialize 1, Upper Bound (UB) ,Lower Bound (LB) , 0For 1 to NkSet 0,do Solve master problem in Equation (24) and determine a trail solution

, ,g t k tX

Update the value of LB Solve all sub-problems in Equation (25) with trail solution

, ,g t k tX

Update the value for UBif then /LB UB Add the new optimally cut associated with iteration to the master problem

Set , and repeat the solution procedures1else an optimal solution is obtained, and implement the control signals at time step tend for

Zm ¼ minXNk¼0

hlf Gg;tþkjtXg;tþkjt þ F

�Xg;tþk�1jt ;Xg;tþkjt

�þ qk

i

qm �XUp

ppp;tþkjt

XUw

wpw;tþkjt

hQm;p;w;tþkjt � εm;p;wGg;tþkjt

�Xm;g;tþkjt � Xm;g;tþkjt

�i(24)

Qm;p;w;tþkjt ¼ min�lpex;tþkjtE

p;rex;tþkjt þ lf G

p;rl;tþkjt

�(25)

Where lf is fuel price, Gg;t is the gas input of combined heat andpower (CHP) units, FðXg;tþk�1jt ;Xg;tþkjtÞ is penalty function used tocontrol the frequency changes during the on/off operating state,qmis Benders cut at iteration m, Qm;p;w;tþkjtdenote the sub-problemsvalue at iteration m under pth and wth scenario, Xm;g;tþkjt is the

Algorithm: SMPC algorithm of CVaR objective func1. Prepare the controller Ci

1.1 Generate m nereffidotgnidroccaseertoiranecs

1.2. Calculate corresponding Controllers Ci

2. Estimate VaR set 90% Solve SMPC problem in Equation (26) The VaR is given by Equations (27) and (28)3. Estimate SMPC of CVaR

For i=1:3

d1 2. . 1, * , ,..., ,%sCS f ones s nu

Calculate other parameters;end for

set T; %simulation timefor k=1:T

measure x k

solve CVaR SMPC problem in Equation (29) a

apply 1u k u

end for

trial solution at iteration m, εm;p;w represents the sensitivity for thecorresponding Qm;p;w;tþkjt , andEex;t indicates the energy purchasedfrom the external gird.

A risk e averse stochastic MPC based on real e time operationhas been developed by Ref. [129] for a wind energy generationsystem combined with a pumped hydro storage unit to maximize

profit and minimize risks in day e ahead bidding strategies. Au-thors have compared the results of stochastic MPC method withseveral other methods such as deterministic MPC, bid e followingheuristic and open e loop algorithms. The stochastic MPC methodoutperforms all other methods and reached an expected profitclose to the perfect information solution with a margin of around2%. Fault tolerant control problem of wind energy conversion sys-tems have been addressed by Ref. [130] using stochastic MPC basedon CVaR. Authors have implemented the Markov jump linearmodel to model randomness of the wind energy conversion sys-tems. A scenario e tree is created within the prediction horizon totransform the stochastic MPC problem to a deterministic MPC. Themethod produced a better fault tolerant control performance incomparison to the Min e Max performance index. The objectivefunction formulation of CVaR using SMPC algorithm is shownbelow.

tion

edontoort 1, 2,...,w k i i w k m

sthgiewsrotcevnoisice

nd obtain 1u

minu

Xi2T jT1∪S

pðxi � xrÞTQðxi � xrÞ þXi2T jS

piuTi Rui

s:t

8>><>>:

x1 ¼ xðkÞxi ¼ AðwðkÞÞxpreðiÞ þ BðwðkÞÞupreðiÞ þ DðwðkÞÞ þ D1ðwðkÞÞeðkÞ þ Iw*yrðkÞ; i2T jfT1g

GxxðkÞ þ GuuðkÞ � g; k ¼ 0; :::;N;cwðkÞ2W

(26)


Gxj;Gvj;Gdj;Gd1j;Gd3j (27)

abðuÞ ¼ minfa2ℝ : jðu;aÞ � bg (28)

minu;fvjgs

j¼1

½p1p2:::ps�m

s:t m � GxxðkÞ þ GxUðkÞ þ GdIw1þ Gd1ðIw1*eðkÞÞ þ Gd3*ðIw3*yrðkÞÞ � Lðr þ aÞm � GxxðkÞ � GvUðkÞ � GdIw1þ Gd1ðIw1*eðkÞÞ � Gd3*ðIw3*yrðkÞÞ þ Lðr þ aÞm � o; j ¼ 1; :::; sUðkÞ � uUðkÞ � u

(29)

Where the xk is the present state, pirepresents the realizationprobability of scenario i, Q and R denote weight matrixes, Gx2Rnxþnuand Gu2Rnxþnustand for coefficient matrixes used in stateand input constraints, f ðu;wÞis the estimation error, b is the prob-ability factor. By using the probability density of error,a can befound as a¼ b� VaR . The jumping information in Markov jumplinear model is denoted by Gxj;Gvj;Gdj;Gd1j;Gd3j .

5.2.3.2. Chance e constrained MPC implementations in renewableenergy. Chance e constrained MPC relies on the formulation ofoutput constraints with a given type ymin � y � ymax as chanceconstraints as shown below:

Prðymin � y � ymaxÞ � l (30a)

where PrðxÞ is the probability of an event X occurring, y is the

Table 12Multi e scenario stochastic MPC optimizations in renewable energy applications.

References Method Objective MPC Type Conthori

[136] Adaptive constrainedstochastic MPC

Minimize operationcosts

Multi - scenario 1e2

[137] Various stochastic MPC Compare multipletypes of MPC

All types 5 ste

output of a given process within the constraints of ymin & ymax, andl is the confidence level of such constraints that can be satisfied.

According to equation (2), the basic idea of a chance e con-strained MPC is to solve the optimization problem in each horizonwhile guarantying the satisfaction of the constraints with a certainprobability. It is to be noted that the chance e constrained MPCinvolves the careful selection of future output predictions and its

uncertainties. Since exact future output predictions can't possiblybe captured, uncertainties are represented in either of these twoways; which is either the uncertainty in future disturbances oruncertainty of the process outputs due to manipulated variables.Within this realm of solving probabilities and uncertainties inchance e constrained MPC, several recent publications have beenidentified and listed below.

A multi e time scale stochastic MPC combined with geneticalgorithm (GA) is proposed in Ref. [131] in order to performscheduling deferrable appliances and energy resources of a smarthome (SH) system. The stochastic parameters namely; solar irra-diances and its prediction uncertainties are forecasted using neuralnetwork toolbox in MATLAB. The uncertainties of the appliances’usage as well as the economic and technical constraints of otherenergy sources such as diesel generators, batteries, and PV panelsare also modelled by the author. The objective function is devel-oped for SH with a goal to minimize the value of the stochasticforward-looking objective function subject to various constraints. A

rolzon

Samplingresolution

System's uncertainty Main Results FutureWork

4 h 0.01e1 s Renewable energysources, electrical loads

The method produced aless conservativesolution compared tothe robust MPCapproach

N/A

ps ahead 30 s Renewable resources,Load, hydrogen e

based PEM electrolyserand fuel cells, lead acidbatteries' state ofcharge

Chance e constrainedMPC outperforms otherMPC types resulting in alower cost and lessenergy exchange in ahydrogen basedmicrogrid

N/A


total of 6 cost terms are taken into consideration to develop theobjective function including fuel costCF:DG

d;t , carbon esmsmions costCE:DGd;t , start up cost CSTU:DGshut down cost CSHD:DG of DG, switching

price of PEV battery CSW :PEVand cost or income due to the powerdistribution with the grid PGridd;t � p0DISCO

d;t

minFFLd;t ¼minXPV

Pd;t

FFLd;t � UPVd;t ; d2D; t2ft1; t2g;ct12T1;ct22T2

(30b)

Fd;t ¼

8>>>>>>>><>>>>>>>>:

hCF:DGd;t

iþhCE:DGd;t

iþh�

1� xDGd;t�1

�� xDGd;t�1 � CSTU:DG

iþhxDGd;t�1 �

�1� xDGd;t�1

�� CSHD:DG

iþhx0PEVd;t � CSW:PEV

iþhPGridd;t � p0DISCO

d;t

i

9>>>>>>>>=>>>>>>>>;d2D; t2

ft1; t2g;ct12T1;ct22T (31)

The multi e time scale MPC divided the control optimization inscale of minutes and hours in a weekly operation, where usage ofcertain appliances is dominant in their respective time scale. Theauthor has shown that the proposed MPC method has managed tonotably decrease the weekly operational cost of the smart homesystem.

An experimental case study was conducted by Ref. [132] in theoperation management of microgrids using stochastic MPC tooptimize the economic and environmental parameters. Un-certainties due to renewable energy resources and demand wereconsidered and the stochastic optimization was solved by usingmixed e integer linear programming toolbox via commercialsolvers. Experimental results have proven the feasibility andimplementation ability of stochastic MPC that outperforms thedeterministic MPC. Kou et al. [133] proposed a stochastic MPC forwind farm energy dispatch strategy with BESS using probabilisticwind power forecasts. The method considers the non e gaussianwind power forecast uncertainties using chance e constraintswarping function. The authors have shown that the proposedmethod outperforms the deterministic MPC method in terms ofpower tracking errors while maintaining the state of charge (SOC)of the battery within operational limits. The chance constraintoptimization problem is developed to enhance the wind powerdispatchability and lessen its oscillation, as shown in equation (32).In addition, SOC constraints and charge discharge power con-straints are assigned in order to protect the battery from beingovercharged and over discharged, as expressed in equation (33) andequation (34) respectively.

minuðkþhjkÞ;eðkþhjkÞ

J¼bXHk¼1

eðkþ hjkÞ2

þ ð1� bÞXH�1

h¼0

uðkþ hjkÞ2; Subject to (32)

Prhyðt þ kjtÞ � yref ðt þ kÞjt

� � eðt þ kjtÞi�a; h ¼ 1;2; :::;H

SOCmin � x2ðkþ hjkÞ � SOCmax;h ¼ 1;2; :::;H (33)

�PB;max � uðkþ hjkÞ � PB;max;h ¼ 1;2; :::;H (34)

Where eðkþ hjkÞis the set of auxiliary variables, yðt þ kjtÞ is thestochastic variables, yrefdenotes the reference, Pr is the probability,

aand b denote the confidence and trade-off parameter respectively.A new distributed chancee constraints stochastic EMPC scheme

has been presented in Ref. [134] for coordinated stochastic multiplemicrogrids energy management. The supply and demand side un-certainties were handled using the probabilistic forecasts ofrenewable generations and load of each microgrid that is in acooperation scheme with each other. The proposed method suc-cessfully reduced the system operating costs and achieved thesupply and demand balance in each microgrid within the controlhorizons through the development of distributed network operator(DNO) controller. DNO acts as an intermediary betweenmicrogrids,thus the energy selling between any two microgrids is performedindirectly through DNO. The mathematical expression of costfunction is presented in the following equation,

minEpur;DðkþhjkÞEsel;mðkþhjkÞEsel;DðkþhjkÞ

Epur;mðkþhjkÞ

JD¼XHD�1

h¼0

0BBBB@Epur;DðkþhjkÞhpur;Dþ

XMm¼1

Esel;mðkþhjkÞhsel;m

�Esel;DðkþhjkÞhsel;D�XMm¼1

Epur;mðkþhjkÞhpur;m

1CCCCA

(35)

Where,Epur;Dand Epur;m denote the energy purchased from the gridand DNO respectively, Esel;Dand Esel;m represent the energy soldback to the main grid and DNO respectively. hpur;Dand hpur;mstandfor energy price purchasing from the grid and DNO respectivelywhile hsel;Dand hsel;m signify the energy price selling to the maingrid and DNO respectively.

The optimal operation of a smart residential microgrid based onstochastic MPC has been conducted in the work of [135]. The res-idential microgrid comprised of renewable energy resources,distributed energy generators, energy storage, electrical vehicle,and smart loads. The uncertainties are modelled in a short e termforecasts of renewable energy generations, load demand, andelectricity prices. The proposed method aimed to reduce the totaldaily operational costs of the microgrid. The simulation results bythe authors have shown the economic advantages of the method incomparison to the traditional day e ahead programming approach.

5.2.3.3. Multi - scenario MPC implementations in renewable energy.Multi e scenario MPC utilizes multiple scenario generations withina given optimization horizon to implement a control action atpresent time, t. Similar to uncertainty modelling (Refer to Section3), the independent multiple scenarios generated are synthesizedfrom random input variables of PDFs to produce PDFs of outputvariables in representing the uncertainties. Ranges of solutionsexist, each with its own probability as represented in the outputPDFs. The most cost e effective scenario in terms of objectivefunctions are chosen to be the control action within the optimiza-tion horizon.

An adaptively constrained stochastic MPC has been proposed inRef. [136] for optimal dispatch of microgrid. The objective functionis formulated to minimize the total operation cost including cost ofrunning generator and cost of purchasing electricity form DG, asexpressed in the following equation.

minXTi¼1

fcconPconðt þ ijtÞg þ cGridðt þ ijtÞPGridðt þ ijtÞ (36)

Where T represents the length of time horizon, i denotes the timestep index, Pconðt þ ijtÞ stands for power discharge from thecontrollable generator in i-step ahead, cGridðt þ ijtÞ denotes the


electricity price for energy exchange in i-step ahead, PGridðt þ ijtÞ ispower exchange between MG and DG in i-step ahead.

The method adaptively/dynamically changed the rate ofconstraint violation in the microgrid operation to improve theperformance of the energy dispatch. In comparison to the robustMPCmethod, the authors have shown that themethod can improvethe dispatch performance (less conservative) in cases of uncertainrenewable generations and loads. Furthermore, with increment ofprediction horizon, computational efficacies were not significantlyaffected.

Stochastic MPC control strategies in a case of hydrogen e basedmicrogrid have been compared in the work of [137]. The threecategories of stochastic MPC mentioned in the previous paragraphswere compared in thework of the authors in an experimental setupincluding a PEM electrolyser, leade acid batteries, and a PEM fuelecell as the main equipment. For each category of the stochastic MPCeffectiveness, performances, advantages, and disadvantages werediscoursed. Authors have discussed extensively the valid criterianeeded when selecting the appropriate stochastic MPC method.

5.2.4. MPC's comparison and future trending5.2.4.1. MPC's comparison. It is apparent from the trending ofrecent stochastic MPC in renewable energy applications that thetree e based and chance e constrained MPC were the most usedmethods in recent studies. The multi e scenario MPC yielded arobust but overe conservative solutions. Therefore, this category ofMPC is not preferred due to the need of an accurate representationof the system, in which the tree e based and chance e constrainedMPC could provide better.

Furthermore, in cases of stochastic MPC applications, the pre-diction/control window is typically within 24 h. Despite the heavycomputational expenses of the tree e based compared to multi escenario MPC, the calculation time within the mentioned windowis still relatively inexpensive. The multi e scenario MPC is moresuitable in cases of huge numbers of scenarios needed to beconsidered (i.e. Optimization within 8760 h in a year, 1 e hour timestep, and multivariate properties). The multi e scenario MPC couldprovide a certain robustness of system's representation to the po-tential disturbances and provide a trade e off between the bestsolution and the computational expenses.

The chance e constrained MPC offers the lowest computationalexpenses compared to the other two. It formulates the optimizationproblem by considering the probabilities of the uncertaintieswithout adding the variables' size. In the work of [137], the chancee constrained MPC outperforms the other MPC methods by offer-ing a reduced computational time, lower operational costs, andminimal energy exchanges with the networks. These advantages ofchance e constrained MPC are one of the reasons of frequent usageof this MPCmethod as shown in recent literatures stated in Table 11.However, the chance e constrained MPC requires an explicit sta-tistical characterization of the systems’ disturbances. For the se-lection of the suitable MPC method, priority factors such asoperational costs and computational expenses must be taken intoconsiderations [137]. provided a general guideline in choosing thebest stochastic MPC for a given priority factors. Nonetheless, ingeneral categories of stochastic dynamic programming, an efficientmethod lies often on the specific problems at hand as stated byRef. [6].

The prediction/control window played an important role indetermining the accuracy of the solution as well as computationaltime. A long prediction/control window would mean a more ac-curate representation of unfolding events, thus yielding a greateraccuracy in finding the best solution. However, the computationalexpenses increase as the window increases. Trade e offs betweenprediction/control window and computational expenses must be

determined in order to produce the needed solution.

5.2.4.2. Future MPC's trending. The future trends in stochastic MPCare converging towards a multi e scale and multi e time basedoptimizations as stated in the works of [125,126,129]. In a renew-able energy system, where multiple sources of energy generationsare present, a realistic representation must consider these sourcesin order to provide an insight closer to real e world applications.Managing surges of dynamic demands and supplies from plug in EV(V2G), varying behaviours of energy consumers, smart appliances,demand response, and intermittent multiple renewable energyresources are the challenges that must be addressed together infuture smart grid e systems. In addition, these challenges are alltime e dependant variables in which, each of them possesses traitswith dominance in certain time e steps. Addressing the challengesin a multi e time scale approach could capture the undiscloseddynamic behaviour of the system.

In such systems where the dynamics are complex, multivariate,and time dependent, exact solutions are difficult to obtain. There-fore, approximate solutions to such cases are more feasible in theforms of ASDP. The works of [19,138] combines the stochasticmethod with a (meta)heuristic methods. The stochastic method ishybridized with genetic algorithm to produce ranges of relaxedsolutions. Trending in hybridization of stochastic and meta-heuristic methods are relatively new but promising in the field ofstochastic optimizations to improve the ASDP algorithms. Inter-ested readers are encouraged to read the works of [2,139e141] forrecent reviews of (meta)heuristic methods and intelligent searchesin the field of renewable energy applications.

6. Conclusions

Stochastic optimizations in renewable energy applications haveshown its successful implementations in recent surveys that arepresented in the paper. Almost always, based on the works of manyauthors, the stochastic optimization techniques exhibit enhancedperformances and can deliver accurate representations in capturingthe uncertainties of renewable systems. Despite its advantages, dueto numerous amounts of samplings and unfolding events, whichare discussed in the works of many authors to improve or developnovel algorithms in increasing the efficiency of stochastic optimi-zation techniques. Within these contexts, the relevant researchthemes going into the future based on stochastic optimization al-gorithms are concluded as follows:

i. Novel scenario generations and uncertainty modelling ap-proaches; These are necessary in renewable systems in-tegrations where trending in the future involves stochasticmulti e scale modelling. With rapid increment of data andsize of renewables' problem, perhaps model e driven ap-proaches alone could not fully address and cope with theunderlying complexity in vast multivariate and expandingrenewable systems. Data e driven scenario generationscould provide a pivotal role as highlighted in the works of[41,142].

ii. Unfolding dynamic uncertainties in multi e stage problems;Addressing dynamic probability issues as scenarios/newforecasts unfolds have been addressed by several authors[103,104] in the paper. Better weather and power forecastswhich provide information with dynamic uncertainties asevents unfold would incorporate a more robust real e timedecision e making strategy for generation companies inhandling stochastic renewable generations.

iii. Implementing new recent notable algorithms in the field ofrenewable energy optimizations; Recent work by Ref. [143]


in the form of proximal policy optimization (PPO) has showngreat promise in updating multiple epochs per data sample.The method boasts the ability of simple implementationsand great stability as well as better overall performance incomparison to its predecessor, trust region policy optimiza-tion (TRPO). The PPO algorithm has attracted many authorsespecially in the field of computational sciences. No works ofPPO have been published in renewable energy applications.

iv. Improvements of existing sampling and decompositionmethods; Parameters such as number of scenarios needed,scenario reduction techniques, quality of scenarios gener-ated, and relevant scenarios generated are still underextensive study as highlighted by the literatures in thesesections. Acceleration techniques and efficient cuts havebeen developed by several authors in the decompositionmethods approaches to speed up calculations (see Table 7).Where some sampling and decomposition methods provedto be advantageous, further testing of the methods to otherrenewable applications are still required.

v. Hybridizing existing methods with intelligent search (metae heuristic method); Especially in problems with higher di-mensions (Non e Linear), accurate representation ofrenewable systems is difficult. Intelligent searches find arelaxed approximation to a solution and can reducecomputational expenses while increase accuracy as high-lighted in the works of [19,138,144]. The works of metaheuristic method in renewable energy applications aremainly in the field of deterministic optimization problems[139].

While algorithms are important in solving the stochasticrenewable energy problems, future research areas in this field havealso been identified from the surveys conducted. The trendingthemes moving forward can be broken down in three maincategories:

i. Plug in EVs integration to microgrid; The surge of plug in EVsare expected in the nearest future as these vehicles are moreefficient and produces relatively less greenhouse gases [145].These EVs will lead to unique future challenges as well asopportunities in future MG systems. The plug in EVs aremostly stochastic problems as charging demand profiles varyfrom one user to another. Main themes regarding the inte-gration of Plug in EVs in microgrids are identified namely;Charging and scheduling of Plug in EVs [146e148], renew-able energy integration via vehicle to grid operation [149],and willingness of participation towards the usage of Plug inEVs [150]. Interested readers are encouraged to read therecent notable works mentioned pertaining to these high-lighted themes and its solution methodologies.

ii. Demand side management (DSM); Multiple authors havestarted to consider load demand as an active entity as sur-veyed. Results have shown reduction in peak demands, lowercosts, and reduction of generation capacities. However, thesuccess of DSM highly depends on the policies involved andactive participation of consumers. Recent review byRef. [151] identified the consumers as one of the three mainaspects of smart grid management. Authors have also high-lighted that the acceptance of DSM varies from one con-sumer to another. It is critical that the focus of futureresearches is consumer e centred to improve acceptance inDSM for a better management of the electrical grid. The mainfuture research directions identified regarding DSM and itsrelated recent works are; Consumer engagement methods

[152], accurate modelling of consumer's behaviour [142],security and privacy and scalability [155].

iii. Multie scale andmulti e timee scale distributed renewableenergy systems; vast amount of literatures have supportedthe claim that having hybrid or combination of renewablesystems would allow for a higher fraction of renewablegeneration in a distributed renewable energy system. How-ever, increasing the scale of distributed generation fromhousing, to district, and finally to national scale would meanaddressing new challenges such as ensuring the grid andmarket stability in a growing complex socio e techno e

economic system with underlying dynamic uncertaintiesand probabilities. Furthermore, each renewable component,consumer's appliances, and electricity market all havedifferent time e scales in which they are dominant.Addressing both multi e scale and multi e time e scaleproblems with high penetration of intermittent renewableresources in distributed generation are the future researchareas in the field.

The paper highlighted the recent and notable stochastic opti-mization approaches in renewable energy applications. The ad-vantages and challenges of stochastic optimization methods arecarefully evaluated, and its recent trending and future works aresummarized in this paper. An intuitive approach was presented toenlighten new researchers in venturing into the stochastic opti-mization methods within the domain of renewable energyapplications.

Acknowledgements

The authors would like to acknowledge the financial supportfrom BOLD grant provided by Universiti Tenaga Nasional (UNITEN),Malaysia and 20190101LRGS grant provided by Ministry of HigherEducation, Malaysia.

References

[1] IEA, Marke T Report Series Renewables 2O17 Marke T Report Series Re-newables 2O17, 2017.

[2] T. Tezer, R. Yaman, G. Yaman, Evaluation of approaches used for optimizationof stand-alone hybrid renewable energy systems, Renew. Sustain. EnergyRev. 73 (2017) 840e853. January.

[3] A. Ioannou, A. Angus, F. Brennan, Risk-based methods for sustainable energysystem planning : a review, Renew. Sustain. Energy Rev. 74 (2017) 602e615.December 2016.

[4] S. Spiecker, C. Weber, The future of the European electricity system and theimpact of fl uctuating renewable energy e a scenario analysis, Energy Policy65 (2014) 185e197.

[5] P.A. Gooding, E. Makram, R. Hadidi, Probability analysis of distributed gen-eration for island scenarios utilizing Carolinas data, Electr. Power Syst. Res.107 (2014) 125e132.

[6] Q.P. Zheng, J. Wang, S. Member, A.L. Liu, Stochastic optimization for unitcommitment d a review, IEEE Trans. Power Syst. 30 (4) (2015) 1913e1924.

[7] O. Altıntas, B. Okten, €O. Karsu, A.S. Kocaman, Bi-objective optimization of agrid-connected decentralized energy system, Int. J. Energy Res. 42 (2) (2018)447e465.

[8] Z. Abdmouleh, A. Gastli, L. Ben-Brahim, M. Haouari, N.A. Al-Emadi, Review ofoptimization techniques applied for the integration of distributed generationfrom renewable energy sources, Renew. Energy 113 (2017) 266e280.

[9] N. Sharma, Stochastic techniques used for optimization in solar systems : areview, Renew. Sustain. Energy Rev. 16 (2012) 1399e1411.

[10] M. Aien, M. Rashidinejad, M. Fotuhi-firuzabad, On possibilistic and proba-bilistic uncertainty assessment of power fl ow problem : a review and a newapproach, Renew. Sustain. Energy Rev. 37 (2014) 883e895.

[11] S.S. Reddy, V. Sandeep, C. Jung, Review of stochastic optimization methodsfor smart grid, Energy 11 (2) (2017) 197e209.

[12] A. Ali, N. Izzri, A. Wahab, I. Aris, J. Jasni, N. Abdalla, Computational techniquesfor assessing the reliability and sustainability of electrical power systems : areview, Renew. Sustain. Energy Rev. 80 (2017) 1175e1186. June 2016.

[13] M. Aien, A. Hajebrahimi, M. Fotuhi-Firuzabad, A comprehensive review onuncertainty modeling techniques in power system studies, Renew. Sustain.Energy Rev. 57 (2016) 1077e1089.

http://refhub.elsevier.com/S0960-1481(19)31101-2/sref1





















































[15] G.E. Constante-flores, S. Member, Data-Driven Probabilistic Power FlowAnalysis for a Distribution System with Renewable Energy Sources UsingMonte Carlo Simulation, 2017, pp. 1e8.

[16] D.D. Chen, J. Dean Chen, Monte - Carlo Simulation - Based StatisticalModeling, Springer, Singapore, 2016.

[17] G. Carpinelli, P. Caramia, P. Varilone, Multi-linear Monte Carlo simulationmethod for probabilistic load fl ow of distribution systems with wind andphotovoltaic generation systems, Renew. Energy 76 (2015) 283e295.

[18] C. Lucia, T. Borges, An overview of reliability models and methods for dis-tribution systems with renewable energy distributed generation, Renew.Sustain. Energy Rev. 16 (2012) 4008e4015.

[19] R. Dufo-L�opez, I.R. Crist�obal-Monreal, J.M. Yusta, Stochastic-heuristic meth-odology for the optimisation of components and control variables of PV-wind-diesel-battery stand-alone systems, Renew. Energy 99 (2016)919e935.

[20] S. Garshasbi, J. Kurnitski, Y. Mohammadi, A hybrid Genetic Algorithm andMonte Carlo simulation approach to predict hourly energy consumption andgeneration by a cluster of Net Zero Energy Buildings, Appl. Energy 179(2016) 626e637.

[21] J.M. Lujano-Rojas, G. Zubi, R. Dufo-L�opez, J.L. Bernal-Agustín, J.P.S. Catal~ao,Novel probabilistic optimization model for lead-acid and vanadium redoxflow batteries under real-time pricing programs, Electr. Power Energy Syst97 (2018) 72e84. June 2017.

[22] Y. Zheng, B.M. Jenkins, K. Kornbluth, C. Træholt, Optimization under uncer-tainty of a biomass-integrated renewable energy microgrid with energystorage, Renew. Energy 123 (2018) 204e217.

[23] A. Mohamed, S. Kazim, L. Alvarado, Optimal sizing for a hybrid power systemwith wind/energy storage based in stochastic environment, Renew. Sustain.Energy Rev. 59 (2016) 1149e1158.

[24] V.S. Lopes, C.L.T. Borges, S. Member, Impact of the combined integration ofwind generation and small hydropower plants on the system reliability, IEEETrans. Sustain. Energy 6 (3) (2015) 1169e1177.

[25] H. Hashemi-dezaki, M. Hamzeh, H. Askarian-abyaneh, H. Haeri-khiavi, Riskmanagement of smart grids based on managed charging of PHEVs andvehicle-to-grid strategy using Monte Carlo simulation, Energy Convers.Manag. 100 (2015) 262e276.

[26] P. Sarajcev, J. Vasilj, D. Jakus, Monte e Carlo analysis of wind farm lightning-surge transients aided by LINET lightning-detection network data, Renew.Energy 99 (2016) 501e513.

[27] R. Dufo-l�opez, E. P�erez-cebollada, J.L. Bernal-agustín, I. Martínez-ruiz, Opti-misation of energy supply at off-grid healthcare facilities using Monte Carlosimulation, Energy Convers. Manag. 113 (2016) 321e330.

[28] B.R. Prusty, D. Jena, An over-limit risk assessment of PV integrated powersystem using probabilistic load fl ow based on multi-time instant uncertaintymodeling, Renew. Energy 116 (2018) 367e383.

[29] K.-H. Chang, A quantile-based simulation optimization model for sizinghybrid renewable energy systems, Simul. Model. Pract. Theory 66 (2016)94e103.

[30] M. El Ha, O. Farges, J. J. B, Global optimization of solar power tower systemsusing a Monte Carlo algorithm : application to a redesign of the PS10 solarthermal power plant, Renew. Energy 119 (2018).

[31] E. Zio, M. Delfanti, L. Giorgi, V. Olivieri, G. Sansavini, Monte Carlo simulation-based probabilistic assessment of DG penetration in medium voltage dis-tribution networks, Int. J. Electr. Power Energy Syst. 64 (Jan. 2015) 852e860.

[32] G. Mokryani, A. Majumdar, B.C. Pal, Probabilistic method for the operation ofthree-phase unbalanced active distribution networks, IET Renew. PowerGener. 10 (2016) 944e954.

[33] M. Di Somma, G. Graditi, E. Heydarian-forushani, M. Sha, P. Siano, Stochasticoptimal scheduling of distributed energy resources with renewablesconsidering economic and environmental aspects, Renew. Energy 116 (2018)272e287.

[34] Y. Degeilh, G. Gross, Electrical Power and Energy Systems Stochastic simu-lation of power systems with integrated intermittent renewable resources,Electr. Power Energy Syst. 64 (2015) 542e550.

[35] U. Arnold, Economic risk analysis of decentralized renewable energy in-frastructures - a Monte Carlo Simulation approach, Renew. Energy 77 (2015)227e239.

[36] D. Pinheiro, E. Geraldo, A. Paulo, A. Traça, D. Almeida, Portfolio optimizationof renewable energy assets : hydro , wind , and photovoltaic energy in theregulated market in Brazil, Energy Econ. 64 (2017) 238e250.

[37] O. Hanbury, V.R. Vasquez, Life cycle analysis of geothermal energy for powerand transportation : a stochastic approach, Renew. Energy 115 (2018)371e381.

[38] S. Conti, S.A. Rizzo, Monte Carlo simulation by using a systematic approach toassess distribution system reliability considering intentional islanding, IEEETrans. Power Deliv. 30 (1) (2015) 64e73.

[39] M. Bugallo, V. Elvira, L. Martino, D. Luengo, J. Míguez, P.M. Djuric, AdaptiveImportance Sampling: the past, the present, the future, IEEE Signal Process.17 (2017) 60e78. July.

[41] Y. Chen, Y. Wang, D. Kirschen, B. Zhang, Model-Free renewable scenariogeneration using generative adversarial networks, IEEE Trans. Power Syst. 33(2018) 3265e3275.

[42] S. Talari, M. Sha, G.J. Os�orio, J. Aghaei, Stochastic modelling of renewableenergy sources from operators ’ point-of- view : a survey, Renew. Sustain.Energy Rev. 81 (2018) 1953e1965. June 2017.

[43] I. Goodfellow, Generative Adversarial Networks, Dep. d’informatique Rech.Oper. Univ. Montr., 2016.

[44] A. Creswell, T. White, V. Dumoulin, K. Arulkumaran, B. Sengupta,A.A. Bharath, Generative adversarial networks: an overview, IEEE SignalProcess. 18 (2018) 53e65. January.

[45] M. Arjovsky, S. Chintala, B. Leon, Wasserstein GAN, Courant Inst. Math. Sci.Faceb. AI Res. (2017) [stat. ML], 1701.07875.

[46] G. Qi, Loss-Sensitive Generative Adversarial Networks on Lipschitz, (MAPLE),Univ. Cent. Florida, 2018, pp. 1e34. Lab. Mach. Percept. Learn.

[47] A. Odena, Semi-Supervised Learning with Generative Adversarial Networks,Cornell Univ, 2016, pp. 1e3, arXiv 1606.01583v2 [stat.ML].

[48] J. Donahue, T. Darrell, P. Krahenbuhl, Adversarial Feature Learning, CornellUniv, 2017, pp. 1e18, arXiv 1605.09782v7 [cs.LG].

[49] X. Chen, Y. Duan, R. Houthooft, J. Schulman, I. Sutskever, P. Abbeel, InfoGAN :Interpretable Representation Learning by Information Maximizing Genera-tive Adversarial Nets, OpenAI, 2016 arXiv: 1606.03657v1 [cs.LG].

[50] A. Odena, C. Olah, J. Shlens, Conditional Image Synthesis with AuxiliaryClassifier GANs, Cornell Univ, 2017 arXiv 1610.09585v4 [stat.ML].

[51] L. Yu, W. Zhang, J. Wang, Y. Yu, SeqGAN: Sequence Generative AdversarialNets with Policy Gradient, Shanghai Jiao Tong Univ. Univ. Coll, London, 2016arXiv 1609.05473v6 [cs.LG].

[52] D. Berthelot, T. Schumm, L. Metz, “BEGAN : Boundary Equilibrium GenerativeAdversarial Networks, Google, 2017, pp. 1e10, arXiv:1703.10717v4 [cs.LG].

[53] M. Paridah, A. Moradbak, A. Mohamed, F. abdulwahab taiwo Owolabi,M. Asniza, S.H. Abdul Khalid, Improved Stochastic Modeling: an EssentialTool for Power System Scheduling in the Presence of Uncertain Renewables,in Intech, 2016.

[54] Q. Chen, L. Mili, Composite power system vulnerability evaluation tocascading failures using importance sampling and antithetic variates, IEEETrans. Power Syst. 28 (3) (2013) 2321e2330.

[55] R. Kardooni, S.B. Yusoff, F.B. Kari, L. Moeenizadeh, Public opinion onrenewable energy technologies and climate change in Peninsular Malaysia,Renew. Energy 116 (2018) 659e668.

[56] S. Ntanos, G. Kyriakopoulos, M. Chalikias, G. Arabatzis, M. Skordoulis, Publicperceptions and willingness to pay for renewable energy: a case study fromGreece, Sustain. Times 10 (3) (2018).

[57] Q. Li, X. Wang, S. Rong, Probabilistic load flow method based on modifiedLatin hypercube-important sampling, Energies 11 (11) (2018) 3171.

[58] Z. Shu, P. Jirutitijaroen, Latin hypercube sampling techniques for powersystems reliability analysis with renewable energy sources, IEEE Trans. Po-wer Syst. 26 (4) (2011) 2066e2073.

[59] O. Dahlblom, Control Variates for Monte Carlo-Pricing of Three-Asset SpreadOptions with Application in the Energy Markets Oscar Dahlblom Supervisedby, 2016.

[60] R. Sun, C. Singh, L. Cheng, Y. Sun, Short-term reliability evaluation usingcontrol variable based dagger sampling method, Electr. Power Syst. Res. 80(6) (2010) 682e689.

[61] Y. Wang, An adaptive importance sampling method for spinning reserve riskevaluation of generating systems incorporating virtual power plants, IEEETrans. Power Syst. 33 (5) (2018) 5082e5091.

[62] R. Rubinstein, The cross-entropy Method : a unified approach to combina-torial optimisation, motnte - Carlo simulation and machine learning, SmartEnviron. Technol. Protoc. Appl. 34 (5/6) (2005) 903.

[63] D.Y. Yang, J.G. Teng, D.M. Frangopol, Cross-entropy-based adaptive impor-tance sampling for time-dependent reliability analysis of deterioratingstructures, Struct. Saf. 66 (2017) 38e50.

[64] M. Leimeister, A. Kolios, A review of reliability-based methods for riskanalysis and their application in the off shore wind industry, Renew. Sustain.Energy Rev. 91 (April) (2018) 1065e1076.

[65] C. Hamon, M. Perninge, L. S€oder, An importance sampling technique forprobabilistic security assessment in power systems with large amounts ofwind power, Electr. Power Syst. Res. 131 (2016) 11e18.

[66] G. Wang, J. Zhao, F. Wen, Y. Xue, G. Ledwich, S. Member, Dispatch strategy ofPHEVs to mitigate selected patterns of seasonally varying outputs fromrenewable generation, IEEE Trans. Smart Grid 6 (2) (2015) 627e639.

[67] T. Egill, Generation adequacy analysis of multi-area power systems with ahigh share of wind power, IEEE Trans. Power Deliv. 33 (4) (2018)3854e3862.

[68] L. Wang, Q. Li, R. Ding, M. Sun, G. Wang, Integrated scheduling of energysupply and demand in microgrids under uncertainty : a robust multi-objective optimization approach, Energy 130 (2017) 1e14.

[69] A.M. Leite, C. Castro, Probabilistic assessment of spinning reserve via cross-entropy method considering renewable sources and transmission re-strictions, IEEE Trans. Power Syst. 33 (4) (2018) 4574e4582.

[70] P. Graf, K. Dykes, R. Damiani, J. Jonkman, P. Veers, Adaptive stratifiedimportance sampling : hybridization of extrapolation and importance sam-pling Monte Carlo methods for estimation of wind turbine extreme loads,Wind Energy Sci. 3 (1) (2018) 475e487.

[71] J. Huang, Y. Xue, Z.Y. Dong, K.P. Wong, An efficient probabilistic assessmentmethod for electricity market risk management, IEEE Trans. Power Syst. 27(3) (Aug. 2012) 1485e1493.

[72] D. Van Ravenzwaaij, P. Cassey, S.D. Brown, A Simple Introduction to MarkovChain Monte e Carlo Sampling, 2018.

[73] P.J. Green, K. Łatuszy, M. Pereyra, C.P. Robert, Bayesian computation : asummary of the current state , and samples backwards and forwards, Stat.




























































































































arxiv:1701.07875




































































































Comput. (2015) 835e862.[74] A. Gelman, D.B. Rubin, A. Gelman, D.B. Rubin, Inference from iterative

simulation using multiple sequences, Inst. Math. Stat. 7 (4) (2018) 457e472.[75] W.R. Gilks, S. Richardson, D.J. Spiegelhalter, Markov chain Monte Carlo in

practice, J. Appl. Stat. 25 (4) (1998) 575.[76] S.M. Kandil, H.E.Z. Farag, M.F. Shaaban, M.Z. El-sharafy, A combined resource

allocation framework for PEVs charging stations , renewable energy re-sources and distributed energy storage systems, Energy 143 (2018)961e972.

[77] H. Carstens, X. Xia, S. Yadavalli, Measurement uncertainty in energy moni-toring : present state of the art, Renew. Sustain. Energy Rev. 82 (2018)2791e2805. October 2017.

[78] S.P. Evans, P.D. Clausen, Modelling of turbulent wind flow using theembedded Markov chain method, Renew. Energy 81 (2015) 671e678.

[79] X. Zhang, B. Wang, Probability assessment for power system voltage stabilitymargin with renewable energy source using slice sampling, Int. Trans. Electr.Energy Syst. (2017) 1e13, no. March.

[80] A.F.M. Smith, G.O. Roberts, Bayesian computation via the Gibbs sampler andrelated Markov chain Monte Carlo methods, J. R. Stat. Soc. Ser. B 55 (1)(1993) 3e23.

[81] C.J.F. Ter Braak, A Markov chain Monte Carlo version of the genetic algorithmDifferential Evolution : easy Bayesian computing for real parameter spaces,Stat. Comput. 16 (2006) 239e249.

[82] M. Neal, Slice sampling, Inst. Math. Stat. 31 (3) (2003) 705e741.[83] S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Optimization by simulated annealing,

Science 220 (1983) 4598 (80-. ).[84] A. Street, F. Oliveira, J.M. Arroyo, Contingency-constrained unit commitment

with n K security Criterion : a robust optimization approach, S. Member, andA. Functions, IEEE Trans. Power Syst. 26 (3) (2011) 1581e1590.

[85] S. Choi, A. Hussain, H. Kim, Adaptive robust optimization-based optimaloperation of microgrids considering uncertainties in arrival and departuretimes of electric vehicles, Energies 11 (2646) (2018) 1e16.

[86] Y. Zhang, J. Le, F. Zheng, Y. Zhang, K. Liu, Two-stage distributionally robustcoordinated scheduling for gas- electricity integrated energy systemconsidering wind power uncertainty and reserve capacity con fi guration,Renew. Energy 135 (2019) 122e135.

[87] A. Hussain, V.-H. Bui, H. Kim, Multi-microgrids considering uncertainties,Energies 9 (2016) 278.

[88] M. Sharafi, T.Y. ElMekkawy, Stochastic optimization of hybrid renewableenergy systems using sampling average method, Renew. Sustain. Energy Rev.52 (2015) 1668e1679.

[89] Q.P. Zheng, J. Wang, P.M. Pardalos, A Decomposition Approach to the Two-Stage Stochastic Unit Commitment Problem, Springer , Sci. þ Bus. Media,2013, pp. 387e410.

[90] R.A. Jabr, Tight polyhedral approximation for mixed-integer linear pro-gramming unit commitment formulations, IET Gener., Transm. Distrib. 6(2012) 1104e1111. July.

[91] A. Papavasiliou, S.S. Oren, Multiarea stochastic unit commitment for highwind penetration in a transmission constrained network, Oper. Res. 61 (3)(2013) 578e592.

[92] M.P. Nowak, R. Udiger, M. Westphalen, A stochastic integer programmingmodel for incorporating day-ahead trading of electricity into hydro-thermalunit commitment, Optim. Eng. 6, 2005, pp. 163e176.

[93] H. Rashidizadeh-Kermani, M. Vahedipour-Dahraie, H. Najafi, A. Anvari-Moghaddam, J. Guerrero, A stochastic Bi-level scheduling approach for theparticipation of EV aggregators in competitive electricity markets, Appl. Sci. 7(10) (2017) 1100.

[94] M. Vahedipour-Dahraie, A. Anvari-Moghaddam, J.M. Guerrero, Evaluation ofreliability in risk-constrained scheduling of autonomous microgrids withdemand response and renewable resources, IET Renew. Power Gener. 12 (6)(2018) 657e667.

[95] M. Vahedipour-Dahraei, H.R. Najafi, A. Anvari-Moghaddam, J.M. Guerrero,Security-constrained unit commitment in AC microgrids considering sto-chastic price-based demand response and renewable generation, Int. Trans.Electr. Energy Syst. 28 (9) (Sep. 2018) e2596.

[96] M. Vahedipour-Dahraie, H. Rashidizadeh-Kermani, H.R. Najafi, A. Anvari-Moghaddam, J.M. Guerrero, Stochastic security and risk-constrained sched-uling for an autonomous microgrid with demand response and renewableenergy resources, IET Renew. Power Gener. 11 (14) (2017) 1812e1821.

[97] R. Hemmati, H. Saboori, S. Saboori, Assessing wind uncertainty impact onshort term operation scheduling of coordinated energy storage systems andthermal units, Renew. Energy 95 (2016) 74e84.

[98] V.S. Tabar, M.A. Jirdehi, R. Hemmati, Energy management in microgrid basedon the multi objective stochastic programming incorporating portablerenewable energy resource as demand response option, Energy 118 (2017)827e839.

[99] H. Alharbi, A. Indices, B. Parameters, Stochastic optimal planning of batteryenergy storage systems for isolated microgrids, IEEE Trans. Sustain. Energy 9(1) (2018) 211e227.

[100] S. Kwon, L. Ntaimo, N. Gautam, S. Member, Optimal day-ahead powerprocurement with renewable energy and demand response, IEEE Trans.Power Syst. 32 (5) (2017) 3924e3933.

[101] E. Mortaz, J. Valenzuela, Microgrid energy scheduling using storage fromelectric vehicles, Electr. Power Syst. Res. 143 (2017) 554e562.

[102] S. Hemmati, S.F. Ghaderi, M.S. Ghazizadeh, Sustainable energy hub design

under uncertainty using Benders decomposition method, Energy 143 (2018)1029e1047.

[103] E. Kianmehr, S. Nikkhah, A. Rabiee, Multi-objective stochastic model for jointoptimal allocation of DG units and network recon fi guration from DG owner’ s and DisCo ’ s perspectives, Renew. Energy 132 (2019) 471e485.

[104] A. Samimi, M. Nikzad, P. Siano, Scenario-based stochastic framework forcoupled active and reactive power market in smart distribution systemswith demand response programs, Renew. Energy 109 (2017) 22e40.

[105] P. Carpentier, J. Chancelier, G. Cohen, M. De Lara, Stochastic Multi - StageOptimization: at the Crossroads between Discrete Time Stochastic Controland Stochastic Programming, Springer, France, 2010.

[106] T. Shiina, J.R. Birge, Stochastic unit commitment problem, Int. Trans. Oper.Res. 11 (1) (Jan. 2004) 19e32.

[107] S. Takriti, J.R. Birge, E. Long, A stochastic model for the unit commitmentproblem, IEEE Trans. Power Syst. 1 (1996) 1497e1508. August.

[108] S. Sen, L. Yu, T. Genc, A stochastic programming approach to power portfoliooptimization, informs 54 (1) (2006) 55e72.

[109] M.P. Nowak, W. Romisch, Stochastic Lagrangian relaxation applied to powerscheduling in a hydro-thermal system under uncertainty, Ann. Oper. Res.100 (2001) 251e272.

[110] R. Jiang, S. Member, J. Wang, S. Member, M. Zhang, Two-stage minimaxregret robust unit commitment, IEEE Trans. Power Syst. 28 (3) (2013)2271e2282.

[111] R. Jiang, S. Member, J. Wang, Y. Guan, A. Sets, Power and pumped storagehydro, IEEE Trans. Power Syst. 27 (2) (2012) 800e810.

[112] M. Asensio, P.M. De Quevedo, G. Mu~noz-delgado, S. Member, J. Contreras,Joint distribution network and renewable energy expansion planningconsidering demand response and energy storage d Part I : stochastic pro-gramming model, IEEE Trans. Smart Grid 9 (2) (2018) 655e666.

[113] Y. Zhan, Q.P. Zheng, J. Wang, S. Member, Generation expansion planningwith large amountsof wind power via decision-dependent stochastic pro-gramming, IEEE Trans. Power Syst. 32 (4) (2017) 3015e3026.

[114] F. Santos, D.Z. Fitiwi, S. Member, A.W. Bizuayehu, M. Shafie-khah, M. Asensio,J. Contreras, C. Manuel, P. Cabrita, P.S. Catal, Novel multi-stage stochastic DGinvestment planning with recourse, IEEE Trans. Sustain. Energy 8 (1) (2017)164e178.

[115] G. Luiz, M. Fredo, E. Cristian, V. Luiz, D. Matos, Assessing solution quality andcomputational performance in the long-term generation scheduling problemconsidering different hydro production function approaches, Renew. Energy131 (2019) 45e54.

[116] H. Abdi, S.D. Beigvand, M. La Scala, A review of optimal power flow studiesapplied to smart grids and microgrids, Renew. Sustain. Energy Rev. 71 (2017)742e766. May 2015.

[117] J. G€onsch, M. Hassler, Sell or Store ? an ADP Approach to MarketingRenewable Energy, vol. 38, Springer, OR Spectr., 2016, pp. 633e660.

[118] J. Shin, J.H. Lee, M.J. Realff, Operational planning and optimal sizing ofmicrogrid considering multi-scale wind uncertainty, Appl. Energy 195(2017) 616e633.

[119] S. Elfwing, E. Uchibe, K. Doya, From free energy to expected energy :improving energy-based value function approximation in reinforcementlearning, Neural Netw. 84 (2016) 17e27.

[122] D.P. Bertsekas, Approximate policy iteration : a survey and some newmethods, J. Control Theor. Appl. 9 (2011) 310e335. January.

[123] G. Serale, M. Fiorentini, A. Capozzoli, D. Bernardini, A. Bemporad, Modelpredictive control ( MPC ) for enhancing building and HVAC system energyefficiency, Energies 11 (631) (2018) 1e35.

[124] W.R. Sultana, S. Kumar, S. Sukchai, S. Yamuna, D. Venkatesh, A review onstate of art development of model predictive control for renewable energyapplications, Renew. Sustain. Energy Rev. 76 (2017) 391e406. January 2016.

[125] N. Bazmohammadi, A. Tahsiri, A. Anvari-Moghaddam, J.M. Guerrero,A hierarchical energy management strategy for interconnected microgridsconsidering uncertainty, Int. J. Electr. Power Energy Syst. 109 (2019)597e608. October 2018.

[126] A. Afram, F. Janabi-sharifi, A.S. Fung, K. Raahemifar, Artificial neural network( ANN ) based model predictive control ( MPC ) and optimization of HVACsystems : a state of the art review and case study of a residential HVACsystem, Energy Build. 141 (2017) 96e113.

[127] A. Afram, F. Janabi-shari, Theory and applications of HVAC control systems eA review of model predictive control ( MPC ), Build. Environ. 72 (2014)343e355.

[128] Z. Ji, X. Huang, C. Xu, S. Houtao, Accelerated model predictive control forelectric vehicle integrated microgrid energy Management : a hybrid robustand stochastic approach, Energies 9 (973) (2016) 1e18.

[129] U. Yildiran, I. Kayahan, Risk-averse stochastic model predictive control-based real-time operation method for a wind energy generation systemsupported by a pumped hydro storage unit, Appl. Energy 226 (2018)631e643. October 2017.

[130] Y. Shi, X. Xiang, L. Wang, Y. Zhang, D. Sun, Stochastic model predictive faulttolerant control based on conditional value at risk for wind energy conver-sion system, Energies 11 (2018) 193.

[131] M. Rahmani-andebili, Scheduling deferrable appliances and energy re-sources of a smart home applying multi-time scale stochastic model pre-dictive control, Sustain. Cities Soc. 32 (2017) 338e347. January.

[132] A. Parisio, E. Rikos, L. Glielmo, Stochastic model predictive control for eco-nomic/environmental operation management of microgrids : an


































































































































































































































experimental case study, J. Process Control 43 (2016) 24e37.[133] P. Kou, F. Gao, X. Guan, Stochastic predictive control of battery energy

storage for wind farm dispatching : using probabilistic wind power forecasts,Renew. Energy 80 (2015) 286e300.

[134] P. Kou, D. Liang, L. Gao, Distributed EMPC of multiple microgrids for coor-dinated stochastic energy management, Appl. Energy 185 (2017) 939e952.

[135] Y. Zhang, T. Zhang, R. Wang, Y. Liu, B. Guo, ScienceDirect Optimal operationof a smart residential microgrid based on model predictive control byconsidering uncertainties and storage impacts, Sol. Energy 122 (2015)1052e1065.

[136] X. Guo, Z. Bao, Z. Li, W. Yan, Adaptively constrained stochastic model pre-dictive control for the optimal dispatch of microgrid, Energies 11 (243)(2018) 1e17.

[137] P. Velarde, L. Valverde, J.M. Maestre, C. Ocampo-martinez, C. Bordons, On thecomparison of stochastic model predictive control strategies applied to ahydrogen-based microgrid, J. Power Sources 343 (2017) 161e173.

[138] M. Rahmani-Andebili, Stochastic, adaptive, and dynamic control of energystorage systems integrated with renewable energy sources for power lossminimization, Renew. Energy 113 (2017) 1462e1471.

[139] S. Twaha, M.A.M. Ramli, A review of optimization approaches for hybriddistributed energy generation systems: off-grid and grid-connected systems,Sustain. Cities Soc. 41 (2018) 320e331. May.

[140] S.M. Zahraee, M. Khalaji Assadi, R. Saidur, Application of artificial intelligencemethods for hybrid energy system optimization, Renew. Sustain. Energy Rev.66 (2016) 617e630.

[141] B. Bhandari, K.-T. Lee, G.-Y. Lee, Y.-M. Cho, S.-H. Ahn, Optimization of hybridrenewable energy power systems: a review, Int. J. Precis. Eng. Manuf.Technol. 2 (1) (2015) 99e112.

[142] Y. Chen, An unsupervised deep learning approach for scenario forecasts, Inst.Electr. Electron. Eng. Inc. (IEEE), Piscataw. (2018) 1e7.

[143] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, O. Klimov, “Proximal Policy

Optimization Algorithms,” Prox. Policy Optim. Algorithms, Cornell Univ. Libr.arXiv.org, Ithaca, 2017, pp. 1e12.

[144] M. Sharafi, T.Y. ElMekkawy, Stochastic optimization of hybrid renewableenergy systems using sampling average method, Renew. Sustain. EnergyRev. 52 (2015) 1668e1679.

[145] S. Howell, Y. Rezgui, J. Hippolyte, B. Jayan, H. Li, Towards the next generationof smart grids : semantic and holonic multi- agent management of distrib-uted energy resources, Renew. Sustain. Energy Rev. 77 (2017) 193e214.March.

[146] A. Ovallle, A. Hably, S. Bacha, Grid Optimal Integration of Electric Vehicles :Examples with Matlab Implementation, Springer, Poland, 2018.

[147] D. Wang, J. Coignard, T. Zeng, C. Zhang, S. Saxena, Quantifying electric vehiclebattery degradation from driving vs . vehicle-to-grid services, J. PowerSources 332 (2016) 193e203.

[148] A.W. Thompson, Economic implications of lithium ion battery degradationfor Vehicle-to- Grid ( V2X ) services, J. Power Sources 396 (2018) 691e709.June.

[149] S. Rajakurna, F. Shahnia, A. Ghosh, Plug in Electric Vehicles in Smart Grids,Springer, Singapore, 2015.

[150] J. Geske, D. Schumann, Willing to participate in vehicle-to-grid ( V2G )? Whynot !, Energy Policy 120 (2018) 392e401. May.

[151] O. Ellabban, H. Abu-rub, “Smart grid customers ’ acceptance and engage-ment : an overview, Renew. Sustain. Energy Rev. 65 (2016) 1285e1298.

[152] A. Mengolini, F. Gangale, J. Vasiljevska, Exploring community-oriented ap-proaches in demand side management projects in Europe, Sustainability 8(2016) 1266.

[155] B. Amarasekara, C. Ranaweera, R. Evans, A. Nirmalathas, Computers andOperations Research Hierarchical aggregation method for a scalable imple-mentation of demand side management, Comput. Oper. Res. 96 (2018)188e199.

















































































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