3964 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4...

3964 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013

A Robust Optimization Approach for theInterdependency Analysis of Integrated EnergySystems Considering Wind Power Uncertainty

Alberto Martinez-Mares and Claudio R. Fuerte-Esquivel, Senior Member, IEEE

Abstract—As power generation plants which use wind energyare increasingly integrated into existing electric power systems, itbecomes important to evaluate how the wind power uncertaintiesaffect the power system’s operation as well as its interdependencywith those infrastructures utilized to transport the various forms ofprimary energy that is converted into electric energy. This paperproposes a robust optimization model for analyzing the interde-pendency between natural gas, coal and electricity infrastructuresconsidering their operation constraints and wind power uncertain-ties. The optimizationmodel obtains an uncertainty-immunized so-lution in a unified framework based on the balance of nodal energyflows, which remains feasible and nearly optimal for all values ofuncertain data. Case studies are presented to verify the effective-ness of the proposed solution for a multi-energy system composedby the IEEE-118 test system coupled to a 15-nodes natural gas net-work and a 4-nodes coal distribution system as well as for the reallife Belgian natural gas and electricity infrastructures.

Index Terms—Coal, multi-energy systems, natural gas, powersystems, robust optimization, uncertainties, wind power.

NOMENCLATURE

A. Sets

Number of coal sources.

Number of compressors in the natural gassystem.

Number of buses in the coal system.

Number of buses in the electrical system.

Number of buses with controlled magnitudevoltage.

Number of buses in the natural gas system.

Number of railways in the coal system.

Number of natural gas sources.

Manuscript received September 09, 2012; revised January 15, 2013 and April03, 2013; acceptedMay 12, 2013. Date of publicationMay 27, 2013; date of cur-rent version October 17, 2013. This work was supported by CONACyT,Méxicounder Grant 94349 and Research Project 106198. Paper no. TPWRS-01036-2012.The authors are with the Electrical Engineering Faculty, Universidad Mi-

choacana de San Nicolás de Hidalgo (UMSNH), Morelia, Michoacán, 58000,México (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2013.2263256

B. Constants

Coal cost per Ton from the source .

Diesel cost per liter.

32.2178, gravitational constant .

1015, natural gas gross heating value(BTU/SCF).

33.4, diesel gross heating value (MJ/liter).

27.5, coal gross heating value(MBTU/Ton).

Natural gas cost per SCFH from the source.

C. Parameters

Wind-rotor swept area .

Susceptance elements of the nodaladmittance matrix (pu).

Gas specific heat ratio (dimensionless).

Wind-turbine power coefficient(dimensionless).

Railroad parameters.

Inner diameter of pipe from to(inches).

Compressor’s parasitic efficiency(dimensionless).

Pipes’ efficiency (dimensionless).

Conductance elements of the nodaladmittance matrix (pu).

Water gross head for basin .

Nominal water gross head for basin .

Pipelines’ inlet and outlet elevation (feet).

Length of pipe from to (miles).

Railroad length (miles).

Lower coal limit injection in mine-mouthat bus .

0885-8950 © 2013 IEEE

MARTINEZ-MARES AND FUERTE-ESQUIVEL: ROBUST OPTIMIZATION APPROACH FOR THE INTERDEPENDENCY ANALYSIS 3965

Upper coal limit injection in mine-mouthat bus .

Lower natural gas limit injection at bus .

Upper natural gas limit injection at bus .

Active electric power load at bus .

Reactive electric power load at bus .

Compression ratio (dimensionless).

Gas average temperature for pipe fromto .

Base temperature for natural gas system.

Gas temperature at node .

Average speed and efficiency of train inrailroad from to .

Lower limit of available resources at basin.

Upper limit of available resources at basin.

Wind speed (m/seg).

Supercompressibility factor(dimensionless).

Air density .

Base pressure for natural gas system(PSIA).

Gas specific gravity (dimensionless).

Compressor’s process efficiency(dimensionless).

D. Variables

Coal mass flow in train from to (tons).

Coal supplied by a mine-mouth at bus(tons).

Coal extracted by load at bus (tons).

Gas flowing in compressor from to(SCFH).

Gas supplied by a source at bus (SCF).

Gas extracted by load at bus (SCF).

Nominal active power of wind generator(pu).

Nodal voltage magnitude at buses and(pu).

Nodal pressures at nodes and (PSIA).

Nodal voltage angle at buses and (rad).

I. INTRODUCTION

R ENEWABLE energy resources are receiving consider-able attention in the continued growth and development

of electric power systems, being the wind power production thefastest growing type of renewable energy [1]. However, unlikefossil- and hydro-based power plants where the rate of gener-ation is controllable, the ability to control the output of windturbines is limited, and the capacity of a wind farm changes ac-cording to wind speeds [2] such that the wind energy convertedinto electric power has to be consumed immediately and is notdispatchable. In this context, as wind power becomes an impor-tant portion of generation portfolios, evaluating how the windpower uncertainty impacts the economic generation dispatch ofconventional power plants becomes necessary [3], [4] as wellas the existing interdependency between the electricity networkand those infrastructures used to transport the various forms ofprimary energy that is converted into electric energy [5].A relevant number of recent works propose the combined

modeling of natural gas, coal and electricity infrastructures fora unified analysis of energy flows and the coordinated opti-mization of the coupled energy systems [6]–[13]; see [14] fora description of these proposals. The only proposal that an-alyzes interdependencies between coal, natural gas and elec-tricity sub-systems considers the networks’ modeling based onnodes and arcs where the technical operating parameters areomitted [7]. To avoid this drawback, we propose to representall these energy infrastructures in a unified single frame of ref-erence based on the balance of nodal energy flows, but whichconsiders their corresponding network and operating parame-ters. On the other hand, only two publications report the impactof wind generation on the existing interdependency between en-ergy infrastructures [15], [16]. The impact of wind generationon a multi-energy system composed of electricity, coal and nat-ural gas infrastructures is reported in [15] for a multi-time pe-riod of study, while a direct current model of the electrical net-work is assumed in [16] to assess the impact of wind generationon the British gas network for a multi-time period of analysis.Despite the contributions of all these proposals in the under-standing of how the operation of each infrastructure influencesor is correlated to the state of the other, their mathematical for-mulations are based on the assumption that all the data, whichinclude wind power generation, are precisely known at the timewhen the solution must be determined: the uncertainties are ig-nored. The data uncertainty is invariably present in the analysisof multi-energy systems so that a small perturbation in the datavalues assumed for the analysis may lead to non-optimality oreven infeasibility of the current solution. The latter may occurwhen critical constraints are violated in such a way that the cur-rent solution is completely meaningless from a practical view-point. Therefore, the next natural step in the study of multi-en-ergy infrastructures consists of developing a mathematical op-timization model considering the effect of data uncertainties onthe optimality and feasibility of the solution. In this context,the contribution of this paper is a mathematical formulation todeal with the problem of considering wind power uncertainty inthe study of interdependencies between energy infrastructures,which has not already been tackled.


The problem of considering uncertainties can be addressedby sensitivity analysis or stochastic programming [17], [18].The objective of the former approach consists of understandingthe effect of small data perturbations on a solution obtained byan optimization approach that ignores the data uncertainty andfinds the interval of data values in which the current solution re-mains optimal [17]. However, when a large number of data un-certainties or large perturbations are present in the model, thisanalysis may be rendered impractical. On the other hand, thegoal of the stochastic optimization is to find the feasibility ofthe solution, with at least some specified probability, in terms ofprobabilistic constraints assuming that the distributions of un-certain data are known; however, a fundamental problem as-sociated with this approach is the difficulty in accurately esti-mating these distributions, and that can destroy the convexityproperties of the model [18]. In view of the difficulties of theaforementioned approaches, a new framework to explore datauncertainty in optimization, referred to as the robust optimiza-tion (RO) approach, has been proposed by several authors [17],[19]–[21]. In proposals [20] and [21], a set-based robust opti-mization is formulated assuming that uncertain parameters be-long to a bounded uncertainty set, and a robust solution is onethat is feasible for the worst-case value of the parameters withinthat uncertainty set. Robust optimization considering ellipsoidaluncertainty sets has been investigated in [20], while Bertsimasand Sim have investigated the case where the uncertainty set isa polyhedron [21]. On the other hand, the key idea of [17] and[19] is to define the problem data, both deterministic and un-certain data, by a finite set of scenarios, each of which is a de-terministic set containing some of the possible values that maybe realized for the uncertain data. In this case, the approach as-sumes that the values of uncertainty parameters are known for alimited number of scenarios and that the total sum of the prob-abilities of occurrence of each scenario is equal to one. There-fore, the discrete probability distribution for the limited numberof scenarios is known. An ROmodel is then formulated in orderto find an uncertainty-immunized solution that remains feasibleand nearly optimal for all scenarios that the uncertain data coulddefine: the RO is a “methodology capable of detecting caseswhen data uncertainty can heavily affect the quality of the nom-inal solution, and in these cases to generate a robust solution,one that is immunized against the effect of data uncertainty”[20]. A critical point of this approach is that a significant gainin the model’s robustness, over the given set of possible uncer-tainties, is achieved at the expense of losing optimality with re-spect to the objective value; however, this shortcoming has beenovercome by introducing a function that embodies a trade-offbetween the objective function value and its variability over thegiven set of scenarios. In addition, a feasibility penalty functionis used to relax some constraints under some of the scenarios[17], [19].The need for robustness in several problems associated with

the planning and operation of electric power systems has beenrecognized. For example, the RO approach has been applied tothe power capacity expansion problem [19], and more recentlyto the optimization of the penetration levels of plug-in hybridelectric vehicles into the transport sector [22] and to the optimalscheduling problem of an energy hub [23]. As a follow-up to this

analysis, this paper proposes a robust optimization model to as-sess how uncertainties associated with wind speed forecasts af-fect the economic and secure operation of a multi-energy systemcomposed of natural gas, coal and electricity infrastructures,which are coupled at multiple nodes of the electricity networkthrough thermal power plants. The state of each infrastructureis formulated for a time horizon corresponding to a single timeperiod (snapshot) based on the balance of nodal energy flowsconsidering security constraints. An analytical RO frameworkis then formulated as an integrated, generalized, single periodnetwork flow model, which is capable of simulating the eco-nomic and secure operation of the overall energy system takingdirectly into account the uncertainty of wind power generation.In this case, the interdependency between the electricity andprimary energy infrastructures is considered more realisticallybecause the power dispatch of generation units is adjusted ac-cording to the variability and uncertainty of the wind power out-puts in order to achieve the total power balance in the electricpower system.To the best of the authors’ knowledge, the proposed idea has

not been explored, and it is structured in the rest of the paperas follows: Section II presents the static models associated withthe natural gas, coal and electricity sub-systems composing theintegrated multi-energy system. Section III formulates the ROmodel of the overall energy system considering the uncertaintyof wind speed forecast. The application of the proposed model ispresented in Sections IV and V, where the impact of uncertain-ties on the economic and secure operation of the overall systemis discussed. Finally, the main conclusions and contributions ofthis proposal are summarized in Section VI.

II. MULTI-ENERGY SYSTEM FORMULATION

The gas, coal and electricity sub-systems that comprise anintegrated multi-energy system are represented as network flowmodels, based on an energy balance formulation. The mathe-matical models associated with each energy infrastructure aredescribed below.

A. Coal System

Coal is transported from the mine-mouth to the electric powerplants over long distances and in large quantities, with diesellocomotives as the dominant transportation mode [24]. For thepurposes of this paper, the coal transportation system is modeledby a linear network composed of nodes and railways. Nodes areused to represent the system’s facilities such as coal mines, coalpiles and/or coal-based thermal power plants which are con-nected through railways.The steady-state operating point of the coal infrastructure is

formulated by the coal flow balance equation that must be sat-isfied at each node of the system: the total amount of coal trans-ported by the rails connected to a nodal point must be equal tothe total coal demanded at that node. The balance equation isthen given by

(1)


Note that the coal supplied from each mine-mouth is limitedby the available resource:

(2)

Based on the coal flow computed by (1), calculating the loco-motives’ energy consumption is possible. There are several pro-posals for calculating this energy consumption based on the dis-tance, weight and average train speed in which both geographicand driving aspects are assumed constant [25]. On the otherhand, a methodology described in [26] recommends estimatingthe energy consumption on each railway based on statistic data,considering that the driving conditions, type of locomotive andaverage speed, among others parameters for a given railway arepractically the same. The computation of energy consumptionproposed in [25] is also used in this work and is given by (3).This consumed energy is expressed in joules in such a way thatthe diesel required by locomotives is computed by (4):

(3)

(4)

B. Natural Gas System

For the purpose of this paper, the natural gas system is mod-eled as a nonlinear network composed of pipelines, compres-sors, sources and loads. The equations associated with thesecomponents are given below, and their details are explicitly re-ported in [14]. The gas flow balance is also formulated to assessthe equilibrium point condition at each network’s node.1) Pipeline Equation: The gas flow in a pipeline connected

between gas nodes and is computed by

(5)

where

(6)

(7)

(8)

(9)

(10)

(11)

2) Compressor Station Equations: The compression sta-tion is formulated by those equations computing its energy

consumption and its compression ratio. Hence, a compressorconnected between nodes and is represented by

(12)

(13)

When the energy required by the compressor to provide thepressure needed to transport the gas is supplied from the thnode of the electric power system, the power extracted fromthis infrastructure is computed by (14); otherwise, this energyis assumed to be extracted from the natural gas network and iscalculated by the quadratic polynomial (15), whose coefficients, and define the efficiency in the energy conversion

process from the thermal energy contained in the gas to the HPrequired by the compressor

(14)

(15)

3) Gas Nodal Balance Equations: The gas flow balancemustbe satisfied at each network’s node such that the gas injected by asource, the gas extracted by industrial loads and gas-fired powerplants, as well as gas flowing through pipelines and compressorsconnected to the node must add up to zero. This equilibrium isexpressed by

(16)

Note that the natural gas supplied at nodal sources should bedelimited by the available resource by

(17)

C. Wind Energy System

The increasing level of wind penetration has augmented theconcern about its impact on the power system’s performanceand the effect on the economic dispatch of generators due to thevariability of wind [27]. The power output of a wind turbine isgiven by (18), which indicates that large fluctuations of windpower may occur because of wind speed variations [28]:

(18)

Fig. 1 shows a typical wind power curve as a function ofthe wind speed, which can be used to express the output powerobtained from the wind generator by a polynomial function.

D. Hydraulic Energy System

Pumped-storage hydropower plants consist of two waterreservoirs located at different elevations and connected by a


Fig. 1. Typical power curve for a wind-turbine.

penstock system; when the water is released from the upperreservoir, the potential energy in the water is converted toelectricity in the hydroplant. For purposes of this paper, theavailable hydrologic resource is considered known, and the ele-vation of the surface of the upper reservoir is assumed constant.This last assumption is regarded as correct because the studyis performed for a single time period (snapshot); however, thisbecomes an important issue in multi-time analysis [29]. Thecoupling between the water reservoir and the electric powersystem is given by the water rate (19) that relates the amount ofwater needed to generate a specified active power and whosecoefficients , and define the efficiency in the energyconversion process from the potential energy contained in thewater volume to the electrical energy generated by the groupturbine-generator:

(19)

The water resources available in each basin are delimited by

(20)

E. Electric Power System

A completeACmodel for the electricity system is representedby the well-documented power flowmismatch equations, whichare formulated for both active and reactive powers and take thefollowing form at node [29]

(21)

(22)

where and correspond to the active and reac-tive power generated from thermal, wind or hydropowerplants: ,

, isgiven by (14) and

(23)

(24)

F. Coupling at Thermal Units

The relationship between the electricity system and the fossilfuel primary energy networks is mathematically formulated bythe heat rate (25), which relates the thermal energy contained innatural gas (or coal) that is transformed into electric power fromthe th node of the primary energy infrastructure to the th nodeof the electricity network. The coefficients , and define theefficiency in this energy conversion process

(25)

Additionally, the fossil fuel quantity required for the energydemanded by the heat rate curve is computed by

(26)

Note that if data associated with the energy primary infra-structure is not available, the same approach, based on the heatrate equation as well as the cost and limits of the primary energyresource, can be employed to represent the coupling betweenthe primary resource used by the power plant and the electricitysystem as with nuclear or petroleum driven generators.

III. ROBUST OPTIMIZATION MODEL

Mathematically, the single-period, deterministic multi-en-ergy OPF is stated by the following constrained nonlinearoptimization problem:

(27)

where the mapping is the objective func-tion to be optimized, ; the set ofequality constraints corresponds to thenodal energy balance in the primary energy and electricity in-frastructures, ; theset of inequality constraints corresponds todecision variables with lower and upper limits given by and, respectively; and the parameter vector is only composed ofdeterministic data. The point in the search space thatsatisfies all constraints is defined as the feasible point .When data take values different than the nominal or expected

values, the computed optimal solution considering the nominaldata may no longer be optimal or even feasible. Hence, the goalof robust optimization is to provide a solution that remains fea-sible and near optimal when data changes within a prescribeduncertainty set.The robust optimization counterpart of (27) is formulated

by introducing a set of scenarios , each ofwhich is a deterministic set with the probability of occurrence, , and composed of the values that uncertain

data take in the analyzed scenario as well as by the fixed values


of deterministic data: . Note that the as-sumptions and imply that a dis-crete probability distribution for the selected limited number ofscenarios is known.The optimal solution of (27) is robust with respect to opti-

mality if it remains close to the optimal solution of any scenarioof the input data, and it is robust in terms of feasibility if allconstraints are satisfied for all possible values that may be re-alized for the uncertain parameters. However, it is unlikely thatthe solution remains near optimal and feasible for all possiblescenarios, such that a trade-off between solution and model ro-bustness could be achieved by introducing a function to controlthe variability in the optimal solution for all realizations of ,and/or by relaxing the constraints in a controlled way to permit acertain degree of infeasibility [19]. This latter option is not con-sidered in this paper: all equality constraints are regarded as ac-tive at any solution since they must be satisfied unconditionallyat any operating point. Furthermore, the active set of inequalityconstraints consists of those variables to be explicitly enforcedto specified values in a particular feasible solution. Therefore,the robust optimization model is formulated as

(28)

The objective function is composed of the expected optimalsolution of over all possible scenarios plusa constant times the variance of . In addition, the sets ofequality and inequality constraints of the multi-energy system,which must be simultaneously satisfied for all scenarios ,are shown in (29)–(40) at the bottom of the page.

The set of decision variables for the ROproblem is given as follows: for the coal system,

; for the gas system,;

and finally for the electric power system.

IV. STUDY CASE

The robust optimization model described in Section III is ap-plied to the analysis of a multi-energy system in order to as-sess the impact of wind power uncertainties on the cost associ-ated with the production of coal and natural gas, which directlyaffects the existing interdependency between primary energyinfrastructures and the electricity system. Furthermore, an ROmodel is formulated and solved to achieve an “almost” constantactive power generation from gas-fired plants, over all windpower scenarios, in order to maintain invariant the operatingconditions in the natural gas infrastructure.The energy system is composed of gas, coal and electricity

networks, the first two networks are depicted in Figs. 2 and3, while their data are reported in [14] and [15], respectively.The four compressors in the natural gas system are driven bygas turbines, and the gas is tapped from the inlet node of thecompressor station. The electrical network corresponds to theIEEE 118-bus system [30], which is coupled to the other pri-mary energy infrastructures through thermal, hydro and windpower generators at those nodes reported in Table I. The param-eters for heat rate curves in thermal units and water curves forhydraulic units have been chosen to reflect the typical efficiencybehavior in each unit. The generation limits for the thermal andhydro generators are and ,

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)


Fig. 2. Natural gas infrastructure.

Fig. 3. Coal infrastructure diagram.

TABLE IIEEE-118 TEST SYSTEM, GENERATORS

respectively, with a base of 100 MVA. Finally, the water avail-able for each hydroelectric generator is computed by (19) con-sidering the coefficients and rate powers given in Table I, and aconstant gross head for all hydro plants is considered.Based on the idea reported in [31], we only consider five dif-

ferent wind speed scenarios of 5, 8, 10, 12 and 15 m/s to rep-resent the wind speed forecast uncertainty. For purposes of thispaper, the probability of each chosen scenario is hypotheticallyselected as 10, 10, 20, 50 and 10%, respectively. Note that ourassumption of five wind speed scenarios with hypothetical prob-abilities of occurrences does not affect the proposed RO for-mulation because these data are considered input parameters in

the proposed formulation: our proposal does not lose its gener-ality. The same wind power curve characteristic is assumed forall wind generators, which is approximated by the eighth-de-gree polynomial function (41), whose coefficients are reportedin Table I. Therefore, each wind generator produces an activepower defined by (42)

(41)

(42)

The RO model (28) associated with this multi-energy systemis composed of 303 variables, 306 equality constraints and 298inequality constraints per wind speed scenario; furthermore, it isnumerically solved using the software GAMS/MINOS [32] forthe following three study cases detailed below: ) A base casefor the cost of fossil fuel production considering no penalizationin the dispersion between scenarios, i.e., ; ) Case )but considering to penalize the dispersion of thetotal cost of fossil fuel production between scenarios; and ) Acase that minimizes the fossil fuel consumed by thermal powerplants considering a penalty factor in the range ofto penalize the cost variance of natural gas used to generate

electricity. The following reference prices are considered forall study cases: Diesel is 1 USD/liter, natural gas is 4 USD/MSCF and coal is 80.15 USD/Ton [33]. A coal heating valueof 12 500 Btu/lb has also been considered for all studies, whichcorresponds to the category of sub-bituminous coal [33].

A. Case A

The objective function for each scenario is defined as the totalcost of primary resources of fossil fuels injected by all naturalgas and coal sources:

(43)

The results of the cost of primary resources and the activepower generation dispatch are summarized in Fig. 4 for eachwind speed scenario. The CPU time for this simulation was 1.6s. The cost and generation dispatch are indicated by bars andsolid lines, respectively. Note that the increment of active powerproduced by wind generators has a major impact on the totalgeneration associated with coal-fired power plants, which is re-duced by approximately 10 pu; meanwhile the generation dueto natural gas plants is only reduced by less than 2 pu. Theseresults are caused by the efficiency energy conversion process,fossil fuel prices and gross heating value of each generationtechnology. As expected, the optimization solution provides noimmunization to the wind speed forecast uncertainty, ;


Fig. 4. Generation dispatch and operation cost (Case A).

Fig. 5. Standard deviation and operating costs by scenario (Case B).

there then exists a dispersion of the operation cost between sce-narios, which is about 30 KUSD between the first and last sce-narios.

B. Case B

Study case is repeated but considers different penalty fac-tors, , on the operation cost variance. This penal-ization allows an uncertainty-immunized solution for all windspeed scenarios but at a suboptimal operation cost as demon-strated in Fig. 5. The total operating cost of each scenario tendsto the same value, which is themost expensive one as the penaltyfactor is increased. In this case, the same robust optimal opera-tion cost is obtained for scenarios 2 to 5 from a penalty factorof ; at this value of the operation cost is 112.4KUSD. On the other hand, all scenarios have the cost of 118.8KUSD for a penalty factor of , which corresponds toworst-case RO solution. The execution time for each simula-tion was 25 s of CPU time, except for the case where . Thestandard deviation of the robust operating costs is also shown inFig. 5, which is calculated from the existing dispersion betweenthe costs of a set of scenarios related to the same value of .The standard deviation of active power dispatches over all

possible scenarios is shown in Fig. 6, as a function of the penaltyvalue . A close relation between the dispersion of the total op-eration cost shown in Fig. 5 and the total power dispatched bycoal-fired plants shown in Fig. 6 is observed: the higher penalty

Fig. 6. Standard deviation for power dispatch between scenarios.

Fig. 7. Active power and standard deviation of coal-fired units.

factor, the smaller dispersion in the operation cost and totalcoal-based power generation associated with the set of scenariosfor wind speed. This smaller dispersion is because the coal-firedpower generation rises with increments in the values of the pe-nalization factor, as shown in Fig. 7. This increment in the par-ticipation of coal-fired units provokes a reduction in the amountof active power supplied by the hydro and natural gas genera-tion units in order to meet the electric demand and transmissionlosses, as shown in Figs. 8 and 9, respectively.The optimal costs obtained by the proposed approach and the

expected cost value are numerically reported in Tables II and III,respectively. The latter was computed by performing an optimalpower flow study for each wind speed scenario considering asan objective function the minimization of the total cost of pri-mary resources of fossil fuels given by (43), and the expectedcost value was then calculated considering the result of each sce-nario and its corresponding probability of occurrence. Since theobjective function (28) of the RO approach includes a weightedpenalization term of the cost variance, if this term is set to zero,the total RO cost is similar to the expected cost.On the other hand, the robust operation cost of each scenarioand the total RO cost tend toward the same value as the pe-

nalization factor increases its value, such that the optimal solu-tions are not affected by the volatility of wind power: an uncer-


Fig. 8. Active power and standard deviation of gas-fired units.

Fig. 9. Active power and standard deviation of hydroelectric units.

TABLE IICOSTS OBTAINED BY THE RO APPROACH

TABLE IIICOSTS OBTAINED BY THE EXPECTED VALUE APPROACH

TABLE IVELECTRIC POWER GENERATED BY GAS-FIRED PLANTS (PU),

Fig. 10. Active power and standard deviation of coal-fired units.

tainty-immunized solution is obtained for all scenarios of windspeeds.Lastly, since the optimization is performed over the cost of

primary energy sources, large industrial loads embedded at thenatural gas and coal networks can have an undesirable effecton the amount of fossil fuel that can be supplied to the thermalpower plants, constraining their power dispatch. This statementis illustrated in Table IV which reports the active power dis-patched by gas-fired generators for two different values of nat-ural gas industrial demands. As expected, a decrement in the ac-tive power dispatched by these plants occurs when the demandof natural gas increases.When hydroelectric resources are reduced, or even elimi-

nated, the deficit in active power will be compensated by thenatural gas-fired and coal-fired generators. In order to show thisinterdependency between the primary energy infrastructures,the case study described in this section has been repeatedconsidering a reduction of 20% in the hydraulic resources.The active power generated from each type of primary en-

ergy is shown in Fig. 10–12 together with the profile of the cor-responding standard deviations; an increase in the active powerdispatched from thermal units is observed in Figs. 10 and 11, asa result of the decrement in the participation of the hydroelectricplants shown in Fig. 12. Furthermore, the active power gener-ated by gas-fired plants differs between wind speed scenariosand values of the penalization factor, such that the standard de-viation of these active powers increases in value. These inter-dependencies also cause the nodal gas pressures along the gasinfrastructure to not remain constant for the different conditionsof generation dispatches.


Fig. 11. Active power and standard deviation of gas-fired units.

Fig. 12. Active power and standard deviation of hydro-electric units.

C. Case C

The cost minimization of fossil-fuel consumption by thermalpower plants is considered as the objective function in this studycase as given by

(44)

In addition, the cost variance of natural gas consumed byeach gas-fired plant is penalized in order to set a similar op-eration condition in the natural gas network for all wind speedscenarios. Therefore, the objective function of the robust modelis now given by (45) to achieve both goals

(45)

The results obtained are reported in Table V for the nodalpressures in the natural gas network and for the active powerdispatched by gas-fired plants considering all wind speed sce-narios and three different values of the penalty factor. Theseresults illustrate how the robust optimization solution steer themulti-energy system to an equilibrium point where the opera-tion in the natural gas network is immune to the uncertaintiesin wind speed forecast, as the penalty factor is increased. Thisimmunity is achieved because each gas-fired plant has a similarpower dispatch; in such a way that their gas consumption re-mains constant over all wind speed scenarios, as also indicatedin that table. The CPU execution time for each simulation cor-responds to those reported for cases A and B.

V. RO APPLICATION IN THE BELGIAN NATURALGAS AND ELECTRICITY INFRASTRUCTURES

In this case study the integrated Belgian natural gas andelectricity infrastructures are considered to show the appli-cability of the proposed approach. The main components ofthe 20-nodes natural gas network are 24 pipelines, eight gasnonelectric loads, seven sources and two compressors. Data ofthese components and the gas network topology are reportedin [13] and [14]. The electrical network is represented by anequivalent model composed of 32 buses, 16 loads, 25 transmis-sion lines, 15 transformers and seven generation units, threeof which are non-voltage regulating generators [34]. For thepurpose of this paper, these three generators have been consid-ered as wind power plants, while the rest of the power plantshave been considered gas-fired thermal generators that connectboth energy infrastructures at those natural gas and electricnodes reported in Table VI. The parameters of the heat rateequations are also reported in Table VI and have been selectedconsidering that an efficiency close to 65% can be attained forthe rated capacity of each generator. Lastly, the wind speedscenarios described in §V are also considered for this analysis.The objetive function to be minimized is the cost of natural

gas consumed for thermal units, as given by

(46)

The formulated RO counterpart consists of 83 variables, 102equality constraints and 86 inequality constraints per wind sce-nario and penalizes the variance of this cost in order to set asimilar operation cost for all wind speed scenarios. Therefore,the objective function of the robust model is

(47)

Since the active power required to attain the load-generationbalance for each wind speed scenario is supplied only by thegas-fired generators, the total active power delivered by theseunits for a given scenario remains almost constant for differentvalues of , as shown in Fig. 13. The consequence of this patternof generation is that the standard deviation for the total activepower is almost constant for every value of , as also shown


TABLE VNATURAL GAS NODAL PRESSURE AND ACTIVE POWER GENERATED BY GAS-FIRED PLANTS, (CASE C)

TABLE VICIGRE-32 BELGIUM TEST SYSTEM, GENERATORS

Fig. 13. Active power and standard deviation of gas-electric units.

in Fig. 13. Finally, the existing difference of this power for dif-ferent values of is due to the transmission losses re-location,as a result of the penalization of the cost variance.

On the other hand, since the gas used for electric generationis the only variable affecting the penalized cost, the reductionin the cost variance is achieved by reducing the variance of thetotal amount of natural gas consumed by all thermal generatorsat a given set of wind speed scenarios, as shown in Fig. 14. Notethat even though gas units have a constant production for allvalues of , the corresponding gas consumption follows a com-pletely different profile because of the way in which these gen-erators are dispatched for different values of : we can generatethe same amount of active power considering different gener-ation dispatches, which means a different consumption of nat-ural gas. When we select a low value for the penalization factor,thereby putting more weight on the total expected cost whilewe are less concerned about the cost variance, the balance inload-generation is attained through the most economic powergenerating schedule, which implies lower natural gas consump-tion. On the other hand, a higher value of results in an increasein the active power dispatch of the most expensive generators,which means higher gas consumption. Lastly, the optimal re-sults obtained for all wind speed scenarios and three differentvalues of are shown in Table VII for the natural gas nodalpressures and for the active powers generated by thermal units.In order to show the prowess of the proposed approach to ob-tain an uncertainty-immunized solution, Table VIII reports therobust operation cost for each scenario and the total operationcost for each set of scenarios considering different values of thepenalization factor .Lastly, the correlation between the standard deviations of the

natural gas consumed by thermal units and the total robust op-eration cost of the multi-energy system is shown in Fig. 15.As a final remark, the concepts of RO are related to the prac-

tical operation of the multi-energy system as follows: the RO


TABLE VIIBELGIAN NATURAL GAS NODAL PRESSURE AND ACTIVE POWER GENERATED BY GAS-FIRED PLANTS

Fig. 14. Natural gas consumption and standard deviation by generators.

TABLE VIIIOPERATION COSTS OBTAINED BY THE RO APPROACH

problem has been formulated in such a way that its RO solu-tion is robust in terms of feasibility over all possible scenarios,

Fig. 15. Natural gas consumption and standard deviation by generators.

independent of the value of the penalization factor , whichmeans that the system operation robustness is ensured becauseall constraints imposed for each energy infrastructure are sat-isfied over all possible scenarios. On the other hand, a seriesof robust solutions have been reported with different values ofthe penalization factor . The solution to be selected in termsof solution robustness is based on the operators’ preference orsome preferable criteria. For this case, a criterion will be basedon the desired trade-off between the total expected cost and thecost variance. If the operators want the total expected cost tobe stable over all possible scenarios, the solution to be selectedcorresponds to the one where the cost variance is minimized,e.g., the solution corresponding to and an RO cost of101 220. Furthermore, for this selected RO solution, the power


system operator has the information regarding how the genera-tors must be dispatched for each wind speed scenario, as shownin Table VII, to satisfy the network operation conditions and toobtain the corresponding expected cost very close to the RO so-lution.

VI. CONCLUSION

The problem of considering wind power uncertainties withinthe context of the integrated optimization analysis of a multi-en-ergy system composed by natural gas, coal and electricity infra-structures has been addressed in this paper. An RO model hasbeen derived from the first principles and has been implementedto obtain an optimal operating point of the overall multi-en-ergy system, taking into account the influence of uncertaintiesin the optimality and feasibility of the solution without makingany distributional assumptions; therefore, an uncertainty-immu-nized solution is achieved for any realization of the wind powerscenarios. Numerical examples have been reported to illustratehow the economic and secure operation of the multi-energysystem is optimized over all wind power scenarios. The secureoperation of the natural gas network is achieved by keeping thenodal gas pressures unchanged along the network for all pos-sible variations of electric power flows.In all these analyses, the model robustness has been priori-

tized in order to satisfy all physical and operation constraintsassociated with each infrastructure at the cost of losing opti-mality with respect to the best objective value.

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Alberto Martínez-Mares received the B.Eng. degree (Hons) from InstitutoTecnológico de Morelia, México, in 1999, and the M.Sc. degree from Univer-sidad Autonoma de Nuevo Léon, México, in 2002. He is currently pursuing thePh.D. degree at Universidad Michoacana, Morelia, México.

Claudio R. Fuerte-Esquivel (M’91–SM’07) received the Ph.D. degree fromthe University of Glasgow, Glasgow, U.K., in 1997.Currently, he is a full-time Professor at Universidad Michoacana, Morelia,

México.

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