GA PPT

Medium Term Stochastic Hydrothermal Coordination

Model

Engineering Systems Division

Andres Ramos

http://www.iit.upcomillas.es/aramos/

[email protected]

[email protected]

MIT, Cambridge, MA, Jan 23-27, 2012

ModelESD.S30 Electric Power System Modeling for a Low Carbon

Economy

Topic objectives

• To understand

– What is a medium term hydrothermal coordination model

• Purpose

• How to use

– How stochasticity is modeled

• Scenario tree

Medium Term Stochastic Hydrothermal Coordination Model 1

• Scenario tree

– What techniques are used for solving the stochastic problem

• Stochastic optimization

References• State-of-the-art in hydro scheduling

– J.W. Labadie Optimal Operation of Multireservoir Systems: State-of-the-Art Review JOURNAL OF WATER

RESOURCES PLANNING AND MANAGEMENT MARCH/APRIL 2004 pp. 93-111

• Hierarchy of planning models

– A. Ramos, S. Cerisola, J.M. Latorre A Decision Support System for Generation Planning and Operation in

Electricity Markets in the book P.M. Pardalos, S. Rebennack, M.V.F. Pereira and N.A. Iliadis (eds.) Handbook of

Power Systems pp. 797-817 Springer December 2009 ISBN 9783642024924


– S. Cerisola, J.M. Latorre, A. Ramos Stochastic Dual Dynamic Programming Applied to Nonconvex Hydrothermal

Models European Journal of Operational Research 218 (2012) 687–697 10.1016/j.ejor.2011.11.040

– A. Ramos, S. Cerisola, J.M. Latorre, R. Bellido, A. Perea and E. Lopez A decision support model for weekly


– A. Ramos, S. Cerisola, J.M. Latorre, R. Bellido, A. Perea and E. Lopez A decision support model for weekly

operation of hydrothermal systems by stochastic nonlinear optimization in the book G. Consiglio, M. Bertocchi

(eds.) Stochastic optimization methods in finance and energy Springer

• Scenario tree generation

– J.M. Latorre, S. Cerisola, A. Ramos Clustering Algorithms for Scenario Tree Generation. Application to Natural

Hydro Inflows European Journal of Operational Research 181 (3): 1339-1353 Sep 2007

• Object-oriented simulation

– J.M. Latorre, S. Cerisola, A. Ramos, R. Bellido, A. Perea Creation of Hydroelectric System Scheduling by

Simulation in the book H. Qudrat-Ullah, J.M. Spector and P. Davidsen (eds.) Complex Decision Making: Theory

and Practice pp. 83-96 Springer October 2007 ISBN 9783540736646

Contents

Medium term stochastic hydrothermal coordination model


• Mathematical formulation



• Case study

Generation planning functions

Functions

New liberalized

market functions

• Strategic bidding:- Energy- Power reserve- Other ancillary services

• Objectives:- Market share

- Price

• Budget planning • Future derivatives market bids

• Capacity investments

• Risk management • Long term contracts:

- Fuel acquisition- Electricity selling


Scope

Traditional regulated operation functions

Short termMedium termLong term

• Fuel management• Annual reservoir and seasonal pumped storage hydro management- Water value assessment

• Capacity investments- Installation- Repowering

• Maintenance• Energy management- Nuclear cycle- Multiannual reservoirs

• Startup and shut-down of thermal units

• Pumped storage hydro operation

• Economic dispatch

Hydro scheduling models

• Nowadays, under a deregulated framework electric companies manage their own generation resources and need detailed operation planning tools

• In the next future, high penetration of intermittent generation is going to stress the electric system operation


electric system operation

• Storage hydro and pumped storage hydro plantsare going to play a much more important role due to their flexibility and complementary use with intermittent generation

Medium term optimization model. Characteristics

• Hydroelectric vs. hydrothermal models

– Hydroelectric model deals only with hydro plants

– Hydrothermal model manages simultaneously both hydro and thermal plants

• Thermal units considered individually. So rich marginal cost information for guiding hydro scheduling


scheduling

• No aggregation or disaggregation process for hydro input and output is needed

• It is very difficult to obtain meaningful results for each hydro plant because:

– It requires a huge amount of data and

– The complexity of hydro subsystems

Optimization-simulation combination

• Use the model in an open-loop control mechanism with rolling horizon

1. First, planning by stochastic optimization

2. Second, simulation of the random parameters

1. Stochastic optimization

– Determines optimal scheduling policies taking into account the uncertainty


2. Simulation

– Evaluates possible future outcomes of random parameters given the optimal policies obtained previously

• We are going to focus on

– STOCHASTIC OPTIMIZATION MODELS

Medium term optimization model. Overview

• Determines:– The optimal yearly operation

of all the thermal and hydro power plants

– Taking into account multiple basins and multiple cascaded reservoirsconnected among them

START

Stochastic MarketEquilibrium ModelStochastic MarketEquilibrium Model

HydrothermalCoordination Model


Monthly hydro basin and thermal plant production

Adj

ustm

ent

START





Adj

ustm

ent

START





Adj

ustm

ent

START





Adj

ustm

ent


– Satisfying the demand and other technical constraints

• Cost minimization model because the main goal is medium term hydro operation. Suitable for profit maximization for a market agent (known prices) END

Stochastic Simulation Model


Weekly hydro plant production

Daily hydro unit production

Coincidence?

Adj

ustm

ent

Adj

ustm

ent

yes

no

END





Coincidence?

Adj

ustm

ent

Adj

ustm

ent

yes

no

END





Coincidence?

Adj

ustm

ent

Adj

ustm

ent

yes

no

END





Coincidence?

Adj

ustm

ent

Adj

ustm

ent

yes

no

Medium term optimization model. Results

• Operation planning

– Fuel consumption, unit (thermal, storage hydro and pumped storage hydro) and/or technology operation

– CO2 Emissions

– Reservoir management

– Targets for short term models (water balance)


• Economic planning

– Annual budget

– Operational costs

– System marginal costs

– Targets for short term models (water value)

Medium term optimization model. Main modeling assumptions

• System characteristics and data that are known with certainty (deterministic)

– Technical characteristics of existing power plants

– Multiple cascaded reservoirs

– Net load demand (includes intermittent generation and imported/exported power)

– Availability of generation units as a reduced rated power


– Availability of generation units as a reduced rated power

– Fuel costs

• Uncertain or stochastic data

– Unregulated hydro inflows

• Hydro plants are limited in both energy and power output

• Transmission network doesn’t affect the optimal operation of the units (it is not represented)

Solution methods

• Deterministic approaches:

– Network Flows

– LP

– NLP

– MILP

• commitment of thermal or hydro units

• piecewise linear approximation of water head effects


• piecewise linear approximation of water head effects

• Stochastic approaches:

– Stochastic Dynamic Programming (SDP)

– Stochastic Linear Programming. Decomposition approaches (Benders, Lagrangian Relaxation, Stochastic Dual Dynamic Programming)

– Stochastic Nonlinear Programming

Contents

• Medium term stochastic hydrothermal coordination model

Hydroelectric system modeling




• Case study

Hydroelectric Dam


Source: Environment Canada

Hydro unit modeling difficulties

• Stochasticity in natural water inflows

• Topological complexities in waterways

• Nonlinearities in production function. Head dependency: energy production depends on the water reserve at the reservoir and originally on the water inflows


on the water inflows

– Important when changes in reservoir levels are significant for the time scope of the model

• Complex operation constraints by other uses of water (irrigation, minimum and maximum river flow, minimum and maximum reservoir levels, sporting activities)

Contents


Hydroelectric system modeling Topology

Water head effect

Inflow stochasticity




• Case study

Lower Tagus basin


Source: Iberdrola

Lower Tagus basin


Source: Iberdrola

Spanish Duero basin


Source: Iberdrola

Sil basin


Source: Iberdrola

Multiple basins

• Hydro subsystem is divided in a set of independent hydro basins:


InflowBasin 1

Basin 2

Basin 3

Storage hydro plant

Run of the river plant

Spanish hydro subsystem

• Very diverse system:

– Hydro reservoir volumes from 0.15 to 2433 hm3

– Hydro plant capacity from 1.5 to 934 MW


Almendra reservoir (2433 hm3)

Contents



Water head effect





• Case study

Nonlinearity

• The power output of each hydro generatordepends on the flow (water discharge)through the turbine and on the water head(difference between the reservoir and drainlevels).


efficiency theoretical value

p g q hη ρ= ⋅ ⋅ ⋅ ⋅

Waterdensity

Gravity acceleration

Flow

Net water head

Characteristic curve

Maximumoutput

elec

tric

al p

ow

erp

[MW

]

elec

tric

al p

ow

erp

[MW

]


water dischargeq [m3/s]


Some times, head dependency canbe neglected.But in some plants this dependencycan be significant even in a week.

Characteristic surface

Maximumoutput


elec

tric

al p

ow

erp

[MW

]

ele

ctric

al p

ower

[M

W]


q [m /s]

120 140 160 180


net hydraulic headh [m]

ele

ctric

al p

ower

p [M

W]

( , )i i i ip q h= Φ

outputWaterflow

Net water head

Contents



Water head effect

Inflow stochasticity: Definition


Inflow stochasticity: Definition


• Case study

Stochasticity or uncertainty

• Origin

– Future information (e.g., prices or future demand)

– Lack of reliable data

– Measurement errors

• In electric energy systems planning

– Demand (yearly, seasonal or daily variation, load growth)


growth)

– Hydro inflows

– Availability of generation or network elements

– Electricity or fuel prices

• Uncertainty relevance for each time scale

Stochasticity sources

• Natural hydro inflows (clearly the most important factor in the Spanish electric system) Year Hydro energy Index % of being

TWh exceeded

2000 26.2 0.90 62%

2001 33.0 1.14 27%

2002 21.0 0.73 88%

2003 33.2 1.15 26%

2004 24.6 0.85 68%

2005 12.9 0.45 100%

2006 23.3 0.82 74%


• Changes in reservoir volumes are significant because of:

– stochasticity in hydro inflows

– chronological pattern of inflows and

– capacity of the reservoir with respect to the inflows

2006 23.3 0.82 74%

2007 18.3 0.64 93%

2008 18.8 0.67 91%

2009 22.1 0.78 77%

Hydro inflows and hydro output


Source: REE

Output: stochastic reservoir levels

Wet year

Dry yearMean year


Source: REE

Natural hydro inflows: (monthly) historical series

25000

30000

35000

40000


0

5000

10000

15000

20000

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

134

141

148

155

162

169

176

183

190

197

204

211

218

225

232

239

246

253

260

267

274

281

288

295

302

309

316

323

330

Cau

dal [

m3/

s] 1970-1997

media

mínima

máxima

Natural hydro inflows: (monthly) historical series

20000

25000

30000

19701971197219731974197519761977197819791980198119821983


0

5000

10000

15000

1 2 3 4 5 6 7 8 9 10 11 12

Cau

dal [

m3/

s]

198319841985198619871988198919901991199219931994199519961997mediamedia+2 desv tipicamedia-desv tipicamaximominimo

Probability density function f(x)

0.100

0.120

0.140

0.160

0.180f(x)


0.000

0.020

0.040

0.060

0.080

0.100

010

0020

0030

0040

0050

0060

0070

0080

0090

0010

000

1100

012

000

1300

014

000

1500

016

000

1700

018

000

1900

020

000

2100

022

000

2300

024

000

2500

026

000

2700

028

000

2900

030

000

3100

032

000

3300

034

000

3500

036

000

3700

038

000

3900

0

Caudal [m3/s]

Cumulative distribution function F(x)

0.600

0.700

0.800

0.900

1.000F(x)


0.000

0.100

0.200

0.300

0.400

0.500

010

0020

0030

0040

0050

0060

0070

0080

0090

0010

000

1100

012

000

1300

014

000

1500

016

000

1700

018

000

1900

020

000

2100

022

000

2300

024

000

2500

026

000

2700

028

000

2900

030

000

3100

032

000

3300

034

000

3500

036

000

3700

038

000

3900

0

Caudal [m3/s]

Water inflows

• Several measurement points in main different river basins

• Partial spatial correlation among them

• Temporal correlation in each one

• No establish method for obtaining a unique multivariate probability tree


Contents



Water head effect

Inflow stochasticity: Modeling alternatives


Inflow stochasticity: Modeling alternatives


• Case study

Deterministic vs. Stochastic Optimization

• Deterministic

– Parameters known with certainty (it can be the mean value)

• Stochastic

– Parameters modeled as stochastic variables with known distributions


known distributions

• Historical

• Discrete

• Continuous ⇒ simulation

Alternatives for modeling the uncertainty (i)

• Wait and see o scenario analysis o what-if analysis

– Decisions are taken once solved the uncertainty

– The problem is solved independently for each scenario

– The scenario with mean value of the parameters is just an special case

– A priori, decisions will be different for each scenario


– A priori, decisions will be different for each scenario (anticipative, clairvoyant, non implementable)

– Solution of an scenario can be infeasible in the others

Alternatives for modeling the uncertainty (ii)

• Heuristic criteria

– Robust decisions will be those appearing in multiple deterministic optimal plans (for many scenarios)

– Flexible decisions will be those than can be changed along time once the uncertainty is being solved


Alternatives for modeling the uncertainty (iii)

• Here and now

– Decisions have to be taken before solving uncertainty

– Non anticipative decisions (only the available information so far can be used, no future information)

– The only relevant decisions are those of the first stage, given that are the only to be taken immediately


given that are the only to be taken immediately

– Stochastic solution takes into account the stochasticitydistribution

– It allows to include risk averse attitudes, penalizing worst cases

– STOCHASTIC OPTIMIZATION

Example: hydrothermal coordination problem

• Scenario analysis

– Run the model supposing that the natural inflows will be the same as any of the previous historical inflows(i.e., year 1989 or 2004, etc.) for the time scope

– Run the model supposing that the natural inflows for each period will be exactly the mean of the historical values (i.e., average year) for the time scope



– Run the model taking into account that the distribution of future natural inflows will be the same as it has been in the past

Multistage stochastic optimization

• Taking optimal decisions in different stages in presence of random parameters with known distributions

• General formulation of the problem:

∫Ω∈∈= )()·,(min),(min ωωω dPxfxfE

XxP

Xx


• Uncertainty is represented by a scenario tree

Solution of a stochastic model

Stochastic parameters

Scenario tree generation


Stochastic optimization

Stability of thestochastic solution?

YES

NO

Stability of the stochastic solution

• The main stochastic solutions (i.e., the first-stage ones) must be robust against the uncertainty modeling (structure and number of scenarios of the tree)

• A tree must be generated such as the solution of the stochastic model ought to be independent


of the stochastic model ought to be independent of it

• Analyze the stochastic solutions for different scenario trees

Contents



Water head effect

Inflow stochasticity: Scenario tree definition


Inflow stochasticity: Scenario tree definition


• Case study

Alternatives for modeling the stochastic parameters

• Discrete probability function (i.e., scenario tree)

• Continuous or historical probability function that generates the tree by sampling (simulating) in each time period


Probability tree or scenario tree

• Tree: represents how the stochasticity is revealed over time, i.e., the different states of the random parameters and simultaneously the non anticipative decisions over time. Correlation among parameters should be taken into account

• Scenario: any path going from the root to the


• Scenario: any path going from the root to the leaves

• The scenarios that share the information until a certain period do the same into the tree (non anticipative decisions)

Scenario tree

• Nodes: where decisions are taken.

• Scenarios: instances of the random process.


Stage 1 Stage 2 Stage 3 Stage 4 scenario

Scenario tree example

wet

wet

dry

Period 1 Period 2

Inflow: 25 m3/s Prob: 0.55



In each node a decision is made and afterwards stochastic


wet

wet

dry

dry




stochastic parameters are revealed

Scenario recombining tree example

wet

dry

Period 1 Period 2


Inflow: 30 m3/sProb: 0.60



wet


dry

dry





wet

Scenario recombining tree example

wet

dry

Period 1 Period 2





wet


dry

dry




stochastic parameters are revealed wet

Recombining tree

• The inflows depend on the scenarios in each period.

– In the tree in period 2 there are four outcomes, 30, 25, 20 y 10 m3/s.

– In the recombining tree, in period 2 there are only two outcomes, 30 y 15 m3/s.


Issues on uncertainty representation

• Tree based in

– Historical series (usually in a reduced number)

– Synthetic series

• Tree

– Recombining

– Not recombining


• Comparison

– Statistical properties (moments, distances) of the original series and the scenario tree

– Results of the stochastic optimization in the first stage

Scenario tree trade-off

• Big scenario tree and

simplified electric system operation problem

– Where do we branch the tree?

• Small scenario tree and


• Small scenario tree and

realistic electric system operation problem

Where is it important to branch the tree?

• Where there are huge variety of stochastic values

– Winter and spring in hydro inflows

• Short-term future will affect more that long-term future

– If the scope of the model is from January to December branching in winter and spring will more relevant than branching in autumn


branching in autumn

Contents



Water head effect

Inflow stochasticity: Scenario tree generation


Inflow stochasticity: Scenario tree generation


• Case study

Scenario tree generation (i)

• Univariate series (one inflow)

– Distance from the cluster centroid to each series from a period to the last one

• Multivariate series (several inflows)

– Distance from the multidimensional cluster centroid to each series of each variable from a period to the


to each series of each variable from a period to the last one

Scenario tree generation (ii)

• There is no established method to obtain a unique scenario tree

• A multivariate scenario tree is obtained by neural gas clustering technique that simultaneously takes into account the main stochastic series and their spatial and temporal dependencies.


• Very extreme scenarios can be artificially introduced with a very low probability

• Number of scenarios generated enough for yearly operation planning

Common approach for tree generation

• Process divided into two phases:

– Generation of a scenario tree.

Neural gas method.

– Reduction of a scenario tree.

Using probabilistic distances.


Clustering in two dimensions

Inflow 2 [m3/s]

45

Centroid

Historical inflows

Historical density function


Centroids have the minimum distance to their corresponding points

Their probability is proportional to the number of points included in the centroid

Inflow 1 [m3/s]5025

25functionDiscrete density function

Scenario tree generation

• Idea

– Minimize the distance of the scenario tree to the original series

– Predefined maximum tree structure(2x2x2x2x1x1x1x1x1x1x1x1, for example)

– Extension of clustering technique to consider many inflows and many periods


inflows and many periods

• J.M. Latorre, S. Cerisola, A. Ramos Clustering Algorithms for Scenario Tree Generation. Application to Natural Hydro InflowsEuropean Journal of Operational Research 181 (3): 1339-1353 Sep 2007

Neural gas algorithm (i)

• Soft competitive learning method

– All the scenarios are adapted for each new series introduced

– Decreasing adapting rate

• Iterative adaptation of the centroid as a function of the closeness to a new series randomly chosen

• Modifications to this method:


• Modifications to this method:

– Initialization: considers the tree structure of the centroids

– Adaptation: the modification of each node is the average of the corresponding one for belonging to each scenario

Natural inflows (V)

• Data series for one hydro inflow:

3500

4000

4500

5000

Aportaciones [m3/s]


1 5 9 13 17 21 25 29 33 37 41 45 490

500

1000

1500

2000

2500

3000

Etapa

Natural inflows (VI)

• Initial scenario tree for one hydro inflow:

3500

4000

4500

5000

Aportaciones [m3/s]


1 5 9 13 17 21 25 29 33 37 41 45 490

500

1000

1500

2000

2500

3000

Etapa0 4 8 12 16 20 24 28 32 36 40 44 48

Etapa

Natural inflows (VII)

• Reduced scenario tree for one hydro inflow:

3500

4000

4500

5000

Aportaciones [m3/s]


0 4 8 12 16 20 24 28 32 36 40 44 48Etapa

1 5 9 13 17 21 25 29 33 37 41 45 490

500

1000

1500

2000

2500

3000

Etapa

Natural inflows: scenario tree

800

1000

1200

1400N

atu

ral

In

flo

ws [

m3/

s]

Serie1

Serie2

Serie3

Serie4

Serie5

Serie6

Serie7

Serie8

800

1000

1200

1400N

atu

ral

In

flo

ws [

m3/

s]

Serie1

Serie2

Serie3

Serie4

Serie5

Serie6

Serie7

Serie8


-200

0

200

400

600

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

Week

Natu

ral

In

flo

ws [

m

Serie8

Serie9

Serie10

Serie11

Serie12

Serie13

Serie14

Serie15

Serie16

-200

0

200

400

600

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

Week

Natu

ral

In

flo

ws [

m

Serie8

Serie9

Serie10

Serie11

Serie12

Serie13

Serie14

Serie15

Serie16

Contents



Mathematical formulation



• Case study


• Objective function

– Minimize the total expected variable costs plus penalties for energy and power not served

• Variables– BINARY: Commitment, startup and shutdown of thermal units

– Thermal, storage hydro and pumped storage hydro output

– Reservoir levels

• Operation constraints


• Operation constraints

– Inter-period

• Storage hydro and pumped storage hydro scheduling

Water balance with stochastic inflows

– Intra-period

• Load and reserve balance

• Detailed hydro basin modeling

• Thermal, storage hydro and pumped-storage hydro operation constraints

• Mixed integer linear programming (MIP)

Indices

• Time scope

– 1 year

• Period

– 1 month

• Subperiod

Period

Subperiod

p

s


• Subperiod

– weekdays and weekends

• Load level

– peak, shoulder and off-peak

Subperiod

Load level

s

n

Demand (5 weekdays)

Chronological Load Curve (5 Working Days)

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

Dem

and

[MW

]

Load Duration Curve

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

Dem

and

[MW

]

Load Duration CurveChronological Load Curve


01 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117

Hours

0

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117

HoursLoad Duration Curve in 3 Load Levels

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117

Hours

Dem

and

[MW

]

Load Duration Curve in 3 Load Levels

Demand

• Monthly demand with several load levels

– Peak, shoulder and off-peak for weekdays and weekends

• All the weekdays of the same month are similar (same for weekends)

4500

5000

Demand [ ]

Duration [ ]psn

psn

MW D

h d


0

500

1000

1500

2000

2500

3000

3500

4000

4500

p01 p02 p03 p04 p05 p06 p07 p08 p09 p10 p11 p12

Dem

and

[MW

]

Month

WeekDay.n01

WeekDay.n02

WeekDay.n03

WeekEnd.n01

WeekEnd.n02

WeekEnd.n03

Technical characteristics of thermal units (t)

• Maximum and minimum output

• Fuel cost

• Slope and intercept of the heat rate straight line

• Operation and maintenance (O&M) variable cost

– No load cost = fuel cost x heat rate intercept

– Variable cost = fuel cost x heat rate slope + O&M cost

• Cold startup cost


• Cold startup cost

• Equivalent forced outage rate (EFOR)

Max and min output [ ] ,

No load cost [€ / ]

Variable cost [€ / ]

Startup cost [€]

[ . .]

t t

t

t

t

t

MW p p

h f

MWh v

sr

EFOR p u q

Technical characteristics of hydro plants (h)

• Maximum and minimum output

• Production function (efficiency for conversion of water release in m3/s to electric power MW)

• Efficiency of pumped storage hydro plants

– Only this ratio of the energy consumed to pump the water is recovered by turbining this water


3

Max and min output [ ] ,

Production function [ / ]

Efficiency [ . .]

h h

h

h

MW p p

kWh m c

p u η

Technical characteristics of hydro reservoirs (r)

• Maximum and minimum reserve

• Initial reserve

– Final reserve = initial reserve

• Stochastic inflows

3Max and min reserve [ ] ,r r

hm r r


3

3

Max and min reserve [ ] ,

Initial and final

Stochasti

rese

c inf

r

lows

ve [ ]

[ / ]

r r

r

pr

hm r r

hm r

m s iω

′

Scenario tree example

wet

wet

dry

Period 1 Period 2






wet

wet

dry

dry





Scenario tree. Ancestor and descendant

( ) ( 02, 03) ( 03, 03)a p sc a p scω ω ′ ∈ ∈

Scenario 1Scenario 1

Scenario 2( 02, 01)p sc

( 03, 01)p sc

Tree relations

Scenario

Period

Scenario tree ( , )

p

p

ω

ω

Tree structure


Scenario 3

Scenario 4

Scenario 3

Scenario 2( 02, 01)p sc

( 02, 03)p sc

( 03, 04)p sc

( 03, 03)p sc( 01, 01)p sc

( 03, 02)p sc

3

Scenario probability [ . .]

[ /Stochastic i s ]nflowp

pr

p u p

m s i

ω

ω

Tree data

Hydro topology

r1 r2

r3

h1 h2


r3

h3 Hydro plant upstream of reservoir ( ) ( , ) ( 1, 3)

Pumped hydro plant upstream of reservoir ( ) ( , ) ( 3, 2)

Reservoir upstream of hydro plant ( ) ( , ) ( 2, 2)

Reserv

h up r hur h r h r

h up r hpr h r h r

h dw r ruh r h r h

∈

∈

∈

oir upstream of pumped hydro plant ( ) ( , ) ( 3, 3)

Reservoir upstream of reservoir ( ) ( , ) ( 1, 3)

h dw r rph r h r h

r up r rur r r r r

∈

′ ∈

Other system parameters

• Energy not served cost

• Operating power reserve not served cost

• Operating power reserve

Energy not served cost [€ / ]

Operating power reserve not served cost [€ / ]

Operating reserve [ ]

MWh v

MW v

MW O

′

′′


1Operating reserve [ ]

psMW O

Variables

• Commitment, startup and shutdown of thermal units (BINARY)

• Production of thermal units and hydro plants

• Consumption of pumped storage hydro plants

Production of a thermal or hydro unit [ ] ,psnt psnh

MW P Pω ω

Commitment, startup and shut 0,own ,1d ,pst pst pstA SR SDω ω ω


• Consumption of pumped storage hydro plants

• Reservoir levels

• Energy and power not served

Consumption of a hydro plant [ ]psnh

MW C ω

Energy and power not served [ ] ,psn ps

MW ENS PNSω ω

3Reservoir level [ ]pr

hm Rω

Constraints: Operating power reserve

Committed output of thermal units

+ Maximum output of hydro plants

+ Power not served

>= Demand

+ Operating reserve for peak load level, subperiod, period and scenario

Committed output of thermal units

+ Maximum output of hydro plants

+ Power not served

>= Demand

+ Operating reserve for peak load level, subperiod, period and scenario


+ Operating reserve for peak load level, subperiod, period and scenario+ Operating reserve for peak load level, subperiod, period and scenario

1 1t pst h ps ps pst h

p A p PNS D O psω ωω+ + ≥ + ∀∑ ∑

Constraints: Generation and load balance

Generation of thermal units

+ Generation of storage hydro plants

– Consumption of pumped storage hydro plants

+ Energy not served

= Demand for each load level, subperiod, period and scenario

Generation of thermal units

+ Generation of storage hydro plants

– Consumption of pumped storage hydro plants

+ Energy not served

= Demand for each load level, subperiod, period and scenario


= Demand for each load level, subperiod, period and scenario= Demand for each load level, subperiod, period and scenario

/psnt psnh psnh h psn psn

t h h

P P C ENS D psnω ω ω ωη ω+ − + = ∀∑ ∑ ∑

Constraints: Production in consecutive load levels

Output of a unit in shoulder ≤ Output of a unit in peak

Output of a unit in off-peak ≤ Output of a unit in shoulder

Output of a unit in shoulder ≤ Output of a unit in peak

Output of a unit in off-peak ≤ Output of a unit in shoulder

1

1

psn t psnt

psn h psnh

P P psnt

P P psnh

ω ω

ω ω

ω

ω

+

+

≤ ∀

≤ ∀


Demand

Hours

Peak

Shoulder

Off-peak

• All the weekdays of the same month are similar (same for weekends)

• Commitment decision of a thermal unit

Constraints: Commitment, startup and shutdown

Period p-1 Period p Period p+1

1


Weekdays Weekdays WeekdaysWeekend Weekends s+1 s s+1 s

1

0

• Startup of thermal units can only be made in the transition between consecutive weekend and weekdays

Constraints: Commitment, startup and shutdown

1 1( )

pst p s t pst pstA A SR SD p astω ω ωω

ω ωω− +

′ ′ ∈− = − ∀

Commitment of a thermal unit in a weekday

– Commitment of a thermal unit in the weekend of previous period

≥ Startup of a thermal unit in this weekday

Commitment of a thermal unit in a weekday

– Commitment of a thermal unit in the weekend of previous period

≥ Startup of a thermal unit in this weekday


• Shutdown only in the opposite transition

Commitment of a thermal unit in a weekend

– Commitment of a thermal unit in the previous weekday

≥ Startup of a thermal unit in this weekend

- Shutdown of a thermal unit in this weekend

Commitment of a thermal unit in a weekend

– Commitment of a thermal unit in the previous weekday

≥ Startup of a thermal unit in this weekend

- Shutdown of a thermal unit in this weekend

1 1pst p s t pst pst− +

1 1 1ps t pst ps t ps tA A SR SD pstω ω ω ω

ω+ + +− = − ∀

Constraints: Commitment and production

Production of a thermal unit

≤ Commitment of a thermal unit x the maximum output reduced by availability rate


≤ Commitment of a thermal unit x the maximum output reduced by availability rate


≥ Commitment of a thermal unit x the minimum output


≥ Commitment of a thermal unit x the minimum output


(1 ) (1 )pst t t psnt pst t tA p q P A p q psntω ω ω

ω− ≤ ≤ − ∀

≥ Commitment of a thermal unit x the minimum output reduced by availability rate

≥ Commitment of a thermal unit x the minimum output reduced by availability rate

• If the thermal unit is committed (Aϖpst = 1) it can produce

between its minimum and maximum output• If the thermal unit is not committed (Aϖ

pst = 0) it can’t produce

Constraints: Water balance for each reservoirReservoir volume at the beginning of the period

– Reservoir volume at the end of the period

+ Natural inflows

– Spills from this reservoir

+ Spills from upstream reservoirs

+ Turbined water from upstream storage hydro plants

– Turbined (and pumped) water from this reservoir

+ Pumped water from upstream pumped hydro plants = 0 for each reservoir,

Reservoir volume at the beginning of the period

– Reservoir volume at the end of the period

+ Natural inflows

– Spills from this reservoir

+ Spills from upstream reservoirs

+ Turbined water from upstream storage hydro plants

– Turbined (and pumped) water from this reservoir




period and scenario


period and scenario

1( )

( ) ( )

( )

(

/ /

0 )/

p r pr pr pr prr up r

psn psnh h psn psnh hsn sn

h up r h dw r

psn psnh hsn

h up r

R R i S S

d P c d P c

C c r ad p

ω ω ω ωω

ω ω

ωω ω ω

′−′∈

∈ ∈

∈

′− + − +

+ −

′ ∈+ = ∀

∑

∑ ∑

∑

Constraints: Operation limits

Reservoir volumes between limits for each hydro reservoirReservoir volumes between limits for each hydro reservoir

Power output between limits for each unitPower output between limits for each unit

0

r pr r

r Pr r

r R r pr

R R r r

ω

ω

ω

ω

≤ ≤ ∀

′= = ∀


0 (1 )

0 ,psnt t t

psnh psnh h

P p q psnt

P C p psnh

ω

ω ω

ω

ω

≤ ≤ − ∀

≤ ≤ ∀

Commitment, startup and shutdown for each unitCommitment, startup and shutdown for each unit

, , 0,1pst pst pstA SR SD pstω ω ω

ω∈ ∀

Multiobjective function

• Minimize

– Thermal unit expected variable costs

– Expected penalties introduced in the objective function for energy and power not served

p t pst p psn t pst p psn t psntpst psnt psnt

p srSR p d f A p d v Pω ω ω ω ω ω

ω ω ω

+ +∑ ∑ ∑


for energy and power not served

p psn psn p pspsn ps

p d v ENS p v PNSω ω ω ω

ω ω

′ ′′+∑ ∑

Short Run Marginal Cost (SRMC)

• Dual variable of generation and load balance [€/MW]

– Change in the objective function due to a marginal increment in the demand

• Short Run Marginal Cost = dual variable / load level

:psnt psnh psnh psn psn

t h hpsn

P P C ENS D psnω ω ωω ωσ ω+ − + = ∀∑ ∑ ∑


• Short Run Marginal Cost = dual variable / load level duration. Expressed in [€/MWh]

/psnpsn psn

SRMC d psnω ωωσ= ∀

Water value

• Dual variable of water balance for each reservoir [€/hm3]

– Change in the objective function due to a marginal increment in the reservoir inflow

1( )

/ /

p r pr pr pr prr up r

psn psnh h psn psnh h

R R i S S

d P c d P c

ω ω ω ω

ω ω

ω

′−′∈

′− + − +

+ −

∑

∑ ∑


• Turbining water has no variable cost. However, an additional hm3 turbined allows to substitute energy produced by thermal units with the corresponding variable cost (this is called water value)

( ) ( )

( ) ( )

/ /

/ / 0 : ( )

psn psnh h psn psnh hsn sn

h up r h dw r

psn h psnh h psn h psnh hsn sn

h dw r h up

p

r

r

d P c d P c

d P c d P c p arω ω ωη η ωπ ω ω

∈ ∈

∈ ∈

+ −

− + = ∀ ′ ∈

∑ ∑

∑ ∑

Contents






Case study

StarGen Lite Medium Term Stochastic Hydrothermal Coordination Model

(http://www.iit.upcomillas.es/aramos/StarGenLite_SHTCM.zip)

• Files

– Microsoft Excel interface for input and output data StarGenLite_SHTCM.xlsm

– GAMS file StarGenLite_SHTCM.gms

• How to use it

– Select the GAMS directory

– Run the model Run


– Run the model

– The model creates

• tmp.xlsx with the output data and

• StarGenLite_SHTCM.lst as the listing file of the GAMS execution

– Load the results into the Excel interface

Run

Load results

StarGenLite_SHTCM (i)

$Title StarGen Lite Medium Term Stochastic Hydrothermal Coordination Model (SHTCM)

$ontext

Developed by

Andrés RamosInstituto de Investigación TecnológicaEscuela Técnica Superior de Ingeniería - ICAIUNIVERSIDAD PONTIFICIA COMILLASAlberto Aguilera 2328015 Madrid, Spain

Model name

Authorship and version


28015 Madrid, [email protected]

January 6, 2012

$offtext

$onempty onmulti offlisting

* solve the optimization problems until optimalityoption OptcR = 0 ;

Allow declaration of

empty sets and multiple

declaration. Suppress

listing

Obtain the optimal solution

StarGenLite_SHTCM (ii)

* definitions

setsp periodp1(p) first periodpn(p) last periods subperiods1(s) first subperiodn load leveln1(n) first load levelsc scenariosca (sc ) scenarioscp (sc,p ) tree defined as scenario and periodscscp(sc,p,sc) ancestor sc2 of node (sc1 p)

Set definition


scscp(sc,p,sc) ancestor sc2 of node (sc1 p)scsch(sc,sc,p) descendant (sc2 p) of node sc1scscr(sc,p,sc) representative sc2 of node (sc1 p)spsn(sc,p,s,n) active load levels for each scenariopsn ( p,s,n) active load levels

g generating unitt (g) thermal unith (g) hydro plantr reservoirrs(r) storage reservoirruh(r,g) reservoir upstream of hydro plantrph(r,g) reservoir upstream of pumped hydro planthur(g,r) hydro plant upstream of reservoirhpr(g,r) pumped hydro plant upstream of reservoirrur(r,r) reservoir 1 upstream of reservoir 2

alias (sc,scc,sccc), (r,rr)

StarGenLite_SHTCM (iii)

parameterspDemand ( p,s,n) hourly load [GW]pOperReserve( p,s,n) hourly operating reserve [GW]pDuration ( p,s,n) duration [h]pCommitt (sc,g,p,s ) commitment of the unit [0-1]pProduct (sc,g,p,s,n) production of the unit [GW]pEnergy (sc,g,p,s,n) energy of the unit [GWh]pReserve (sc,r,p ) reservoir level [hm3]pSRMC (sc, p,s,n) short run marginal cost [M€ per GWh]pWV (sc,r,p ) water value [M€ per hm3]

pEFOR (g) EFOR [p.u.]pMaxProd (g) maximum output [GW]pMinProd (g) minimum output [GW]


pMinProd (g) minimum output [GW]pMaxCons (g) maximum consumption [GW]pSlopeVarCost(g) slope variable cost [M€ per GWh]pInterVarCost(g) intercept variable cost [M€ per h]pStartupCost (g) startup cost [M€]pMaxReserve (r) maximum reserve [km3]pMinReserve (r) minimum reserve [km3]pIniReserve (r) initial reserve [km3]pProdFunct (g) production function [GWh per km3]pEffic (g) pumping efficiency [p.u.]pInflows (r,sc,p) inflows [km3]pInflOrg (r,sc,p) inflows original [km3]pENSCost energy non-served cost [M€ per GWh]pPNSCost power non-served cost [M€ per GW ]

pProbsc (sc,p) probability of a given node

lag(p) backward counting of periodscaux scenario number

Parameter

definition

StarGenLite_SHTCM (iv)

variablesvTotalVCost total system variable cost [M€]

binary variablesvCommitt (sc,p,s, g) commitment of the unit [0-1]vStartup (sc,p,s, g) startup of the unit [0-1]vShutdown (sc,p,s, g) shutdown of the unit [0-1]

positive variablesvProduct (sc,p,s,n,g) production of the unit [GW]vConsump (sc,p,s,n,g) consumption of the unit [GW]vENS (sc,p,s,n ) energy non served [GW]

Variables


vENS (sc,p,s,n ) energy non served [GW]vPNS (sc,p,s ) power non served [GW]vReserve (sc,p, r) reserve at the end of period [km3]vSpillage (sc,p, r) spillage [km3]

equationseTotalVCost total system variable cost [M€]eOpReserve(sc,p,s,n ) operating reserve [GW]eBalance (sc,p,s,n ) load generation balance [GW]eMaxOutput(sc,p,s,n,g) max output of a committed unit [GW]eMinOutput(sc,p,s,n,g) min output of a committed unit [GW]eProdctPer(sc,p,s,n,g) unit production in same period [GW]eStrtUpPer(sc,p,s, g) unit startup in same periodeStrtUpNxt(sc,p,s, g) unit startup in next periodeWtReserve(sc,p, r) water reserve [km3] ;

Equation

definition

StarGenLite_SHTCM (v)* mathematical formulation

eTotalVCost .. vTotalVCost =e= sum[(spsn(sc,p ,s,n) ), pProbSc(sc,p)*pDuration(p,s,n)*pENSCost *vENS (sc,p,s,n )] +sum[(scp (sc,p),s ), pProbSc(sc,p) *pPNSCost *vPNS (sc,p,s )] +sum[(scp (sc,p),s, t), pProbSc(sc,p) *pStartupCost (t)*vStartup(sc,p,s, t)] +sum[(spsn(sc,p ,s,n),t), pProbSc(sc,p)*pDuration(p,s,n)*pInterVarCost(t)*vCommitt(sc,p,s, t)] +sum[(spsn(sc,p ,s,n),t), pProbSc(sc,p)*pDuration(p,s,n)*pSlopeVarCost(t)*vProduct(sc,p,s,n,t)] ;

eOpReserve(spsn(sc,p,s,n1(n))) .. sum[t, pMaxProd(t)*vCommitt(sc,p,s, t)] + sum[h, pMaxProd( h) ] + vPNS(sc,p,s ) =g= pDemand(p,s,n) + pOperReserve(p,s,n) ;eBalance (spsn(sc,p,s, n )) .. sum[g, vProduct(sc,p,s,n,g)] - sum[h, vConsump(sc,p,s,n,h)/pEffic(h)] + vENS(sc,p,s,n) =e= pDemand(p,s,n) ;

eMaxOutput(spsn(sc,p,s,n),t) $pMaxProd(t) .. vProduct(sc,p,s,n,t) / pMaxProd(t) =l= vCommitt(sc,p,s,t) ;eMinOutput(spsn(sc,p,s,n),t) $pMinProd(t) .. vProduct(sc,p,s,n,t) / pMinProd(t) =g= vCommitt(sc,p,s,t) ;

eProdctPer(spsn(sc,p,s1(s),n),g) .. vProduct(sc,p,s+1,n,g) =l= vProduct(sc,p,s,n,g) ;

eStrtUpPer(scp(sc,p),s1(s),t) .. vCommitt(sc,p,s+1,t) =e= vCommitt(sc ,p ,s ,t) + vStartup(sc,p,s+1,t) -vShutdown(sc,p,s+1,t) ;eStrtUpNxt(scp(sc,p),s1(s),t) $[not p1(p)] .. vCommitt(sc,p,s ,t) =e= sum[scscp(sc,p,scc), vCommitt(scc,p-1,s+1,t)] + vStartup(sc,p,s ,t) -


eStrtUpNxt(scp(sc,p),s1(s),t) $[not p1(p)] .. vCommitt(sc,p,s ,t) =e= sum[scscp(sc,p,scc), vCommitt(scc,p-1,s+1,t)] + vStartup(sc,p,s ,t) -vShutdown(sc,p,s ,t) ;

eWtReserve(scp(sc,p), r) .. sum[scscp(sc,p,scc), vReserve(scc,p-1,r)] + pIniReserve(r) $p1(p) - vReserve(sc,p,r) +pInflows(r,sc,p) - vSpillage(sc,p,r) + sum[rur(rr,r), vSpillage(sc,p,rr)] +sum(s,n), pDuration(p,s,n)*sum[hur(h,r), vProduct(sc,p,s,n,h)/pProdFunct(h)] -sum(s,n), pDuration(p,s,n)*sum[ruh(r,h), vProduct(sc,p,s,n,h)/pProdFunct(h)] +sum(s,n), pDuration(p,s,n)*sum[hpr(h,r), vConsump(sc,p,s,n,h)/pProdFunct(h)] -sum(s,n), pDuration(p,s,n)*sum[rph(r,h), vProduct(sc,p,s,n,h)/pProdFunct(h)] =e= 0 ;

model SHTCM / all / ;SHTCM.solprint = 0 ; SHTCM.holdfixed = 1 ;

Mathematical

formulation of

equationsModel includes

all the equationsReduced solution output Eliminate fixed variables

StarGenLite_SHTCM (vi)* read input data from Excel and include into the model

file TMP / tmp.txt /$onecho > tmp.txt

i="%gams.user1%.xlsm"r1=indiceso1=indicesr2=paramo2=paramr3=demando3=demandr4=oprreso4=oprresr5=durationo5=durationr6=thermalgeno6=thermalgenr7=hydrogeno7=hydrogenr8=reservoiro8=reservoirr9=inflowso9=inflowsr10=tree

Read input from Excel

named ranges and

write into text files


r10=treeo10=tree

$offecho$call xls2gms m @"tmp.txt"

sets$include indices;$include paramtable pDemand(p,s,n)$include demandtable pOperReserve(p,s,n)$include oprrestable pDuration(p,s,n)$include durationtable pThermalGen(g,*)$include thermalgentable pHydroGen(g,*)$include hydrogentable pReservoir(r,*)$include reservoirtable pInflows(r,sc,p)$include inflowstable pScnTree(sc,*)$include tree

execute 'del tmp.txt indices param demand oprres duration thermalgen hydrogen reservoir inflows tree' ;

Input from text files

into GAMS

Delete read text files

StarGenLite_SHTCM (vii)

* determine the first and last period and the first subperiod

p1(p) $[ord(p) = 1] = yes ;s1(s) $[ord(s) = 1] = yes ;n1(n) $[ord(n) = 1] = yes ;pn(p) $[ord(p) = card(p)] = yes ;psn(p,s,n) $pDuration(p,s,n) = yes ;lag(p) = card(p) - 2*ord(p) + 1 ;

* assignment of thermal units, storage hydro and pumped storage hydro plants

t (g) $ pThermalGen(g,'FuelCost' ) = yes ;

First period, first subperiod

first load level, …

Defining thermal and hydro


t (g) $ pThermalGen(g,'FuelCost' ) = yes ;h (g) $[not pThermalGen(g,'FuelCost' ) ] = yes ;rs(r) $[ pReservoir (r,'MaxReserve') > 0] = yes ;

Defining thermal and hydro

units and reservoirs

StarGenLite_SHTCM (viii)

* scaling of parameters

pDemand (p,s,n) = pDemand (p,s,n) * 1e-3 ;pOperReserve(p,s,n) = pOperReserve(p,s,n) * 1e-3 ;pENSCost = pENSCost * 1e-3 ;pPNSCost = pPNSCost * 1e-3 ;

pEFOR (t) = pThermalGen(t,'EFOR' ) ;pMaxProd (t) = pThermalGen(t,'MaxProd' ) * 1e-3 * [1-pEFOR(t)] ;pMinProd (t) = pThermalGen(t,'MinProd' ) * 1e-3 * [1-pEFOR(t)] ;pSlopeVarCost(t) = pThermalGen(t,'OMVarCost' ) * 1e-3 +

pThermalGen(t,'SlopeVarCost') * 1e-3 * pThermalGen(t,'FuelCost') ;pInterVarCost(t) = pThermalGen(t,'InterVarCost') * 1e-6 * pThermalGen(t,'FuelCost') ;pStartupCost (t) = pThermalGen(t,'StartupCost' ) * 1e-6 * pThermalGen(t,'FuelCost') ;

Scaling of parameters


pMaxProd (h) = pHydroGen (h,'MaxProd' ) * 1e-3 ;pMinProd (h) = pHydroGen (h,'MinProd' ) * 1e-3 ;pMaxCons (h) = pHydroGen (h,'MaxCons' ) * 1e-3 ;pProdFunct (h) = pHydroGen (h,'ProdFunct' ) * 1e+3 ;pEffic (h) = pHydroGen (h,'Efficiency' ) ;pMaxReserve (r) = pReservoir (r,'MaxReserve' ) * 1e-3 ;pMinReserve (r) = pReservoir (r,'MinReserve' ) * 1e-3 ;pIniReserve (r) = pReservoir (r,'IniReserve' ) * 1e-3 ;

pInflows(r,sc,p) = pInflows (r,sc,p ) * 1e-6 * 3.6*sum[(s,n), pDuration(p,s,n)] ;pInflOrg(r,sc,p) = pInflows (r,sc,p ) ;

* if the production function of a hydro plant is 0, it is changed to 1 and scaled to 1000* if the efficiency of a hydro plant is 0, it is changed to 1

pProdFunct(h) $[pProdFunct(h) = 0] = 1e3 ;pEffic (h) $[pEffic (h) = 0] = 1 ;

StarGenLite_SHTCM (ix)

* bounds on variables

vProduct.up (sc,p,s,n,g) = pMaxProd(g) ;vConsump.up (sc,p,s,n,g) = pMaxCons(g) ;

vENS.up (sc,p,s,n ) = pDemand(p,s,n) ;

vReserve.up(sc,p,r) = pMaxReserve(r) ;vReserve.lo(sc,p,r) = pMinReserve(r) ;vReserve.fx(sc,p,r) $pn(p) = pIniReserve(r) ;

Bounds on variables


StarGenLite_SHTCM (x)

* define the nodes of the scenario tree and determine ancestor sc2 of node (sc1 p) and descendant (sc2 p) of node sc1

scp ( sc,p ) $[ord(p) >= pScnTree(sc,'FirstPeriod') ] = yes ;scscp(scp(sc,p),scc) $[ord(p) > pScnTree(sc,'FirstPeriod') and ord(scc) = ord(sc) ] = yes ;scscp(scp(sc,p),scc) $[ord(p) = pScnTree(sc,'FirstPeriod') and ord(scc) = pScnTree(sc,'Ancestor')] = yes ;scsch(sc,scp(scc,p)) $scscp(scc,p,sc) = yes ;

pProbSc(sc,pn(p)) = pScnTree(sc,'Prob') ;loop (p $[not p1(p)],

pProbSc(scp(sc,p+lag(p))) = sum[scsch(sc,scc,p+(lag(p)+1)), pProbSc(scc,p+(lag(p)+1))] ;) ;

* delete branches with probability 0 and define the active load levels

scp ( sc,p ) $[pProbSc(sc,p) = 0 ] = no ;scscp( sc,p ,scc) $[pProbSc(sc,p) = 0 or pProbSc(scc,p-1) = 0] = no ;


scscp( sc,p ,scc) $[pProbSc(sc,p) = 0 or pProbSc(scc,p-1) = 0] = no ;scsch(sc,scc,p ) = yes $scscp(scc,p,sc) ;spsn (scp(sc,p),s,n) $psn (p,s,n) = yes ;

* determine the representative sc2 of node (sc1 p) for non existing scenarios in the tree

loop (sc,scaux = ord(sc) ;loop (p,

scscr(sc,p+lag(p),scc) $[ord(scc) = scaux] = yes ;SCA(scc) $[ord(scc) = scaux] = yes ;scaux = sum[scscp(sca,p+lag(p),scc), ord(scc)] ;SCA(scc) = no ;

) ;) ;SCA(sc) $sum[p, pProbSc(sc,p)] = yes ;

Building the scenario tree

StarGenLite_SHTCM (xi)* solve stochastic hydrothermal coordination model

solve SHTCM using MIP minimizing vTotalVCost ;

* scaling of results

pCommitt(sca,t, p,s ) = sum[scscr(sca,p,scc), vCommitt.l (scc,p,s, t) ] + eps ;pProduct(sca,g,psn(p,s,n)) = sum[scscr(sca,p,scc), vProduct.l (scc,p,s,n,g) ]*1e3 + eps ;pEnergy (sca,g,psn(p,s,n)) = sum[scscr(sca,p,scc), vProduct.l (scc,p,s,n,g)*pDuration(p,s,n) ]*1e3 + eps ;pReserve(sca,rs(r),p ) = sum[scscr(sca,p,scc), vReserve.l (scc,p, r) ]*1e3 + eps ;pSRMC (sca, psn(p,s,n)) = sum[scscr(sca,p,scc), eBalance.m (scc,p,s,n )/pDuration(p,s,n)/pProbSc(scc,p)]*1e3 + eps ;pWV (sca,rs(r),p ) = sum[scscr(sca,p,scc), eWtReserve.m(scc,p, r) /pProbSc(scc,p)] + eps ;

* data output to xls file

put TMP putclose 'par=pProduct rdim=2 rng=Output!a1' / 'par=pEnergy rdim=2 rng=Energy!a1' / 'par=pReserve rdim=2 rng=WtrReserve!a1' / 'par=pWV rdim=2 rng=WtrValue!a1' / 'par=pSRMC rdim=1 rng=SRMC!a1' / 'par=pCommitt rdim=2 rng=UC!a1'execute_unload 'tmp.gdx' pProduct pEnergy pReserve pWV pSRMC pCommittexecute 'gdxxrw.exe tmp.gdx SQ=n EpsOut=0 O="tmp.xlsx" @tmp.txt'

Solve the optimization

problem


execute 'gdxxrw.exe tmp.gdx SQ=n EpsOut=0 O="tmp.xlsx" @tmp.txt'execute 'del tmp.gdx tmp.txt'

$onlisting Scaling the results

Write output to Excel

Menu


Input Data. Indices


Input Data. Cost of energy or power not served


Input Data. Demand, operating reserve and duration


Input Data. Thermal and hydro parameters


Scenario tree

( 01, 01)p sc

Scenario 1

( 02, 01)p sc

( 03, 01)p sc

( 04, 01)p sc

( 05, 01)p sc

( 06, 01)p sc

Scenario 2

( 03, 02)p sc ( 05, 02)p sc


( 03, 03)p sc

( 01, 01)p sc

Scenario 3

( 02, 03)p sc

( 04, 03)p sc

( 05, 03)p sc

( 06, 03)p sc

( 02, 02)p sc ( 04, 02)p sc ( 06, 02)p sc

Input Data. Inflows and scenario tree


Medium term optimization model. Results

• Operation planning

– Fuel consumption, unit (thermal, storage hydro and pumped storage hydro) and/or technology operation

– CO2 Emissions

– Reservoir management

– Targets for short term models (water balance)


• Economic planning

– Annual budget

– Operational costs

– System marginal costs

– Targets for short term models (water value)

Output Data. Thermal unit commitment

5

6

7

8

9C

omm

itted

The

rmal

Uni

ts

FuelOilGas

OCGT_3

OCGT_1

CCGT_4

CCGT_3

CCGT_2


0

1

2

3

4

p01 p01 p02 p02 p03 p03 p04 p04 p05 p05 p06 p06 p07 p07 p08 p08 p09 p09 p10 p10 p11 p11 p12 p12

Com

mitt

ed T

herm

al U

nits

Period

CCGT_1

ImportedCoal_Bituminous

ImportedCoal_SubBituminous

BrownLignite

DomesticCoal_Anthracite

Nuclear

Output Data. Production

50

60

70

80

90

100

2500

3000

3500

4000

4500

Mar

gina

l Cos

t [€/

MW

h]

Out

put [

MW

]

StorageHydro3_Basin1



FuelOilGas

OCGT_3

OCGT_2

OCGT_1

CCGT_4

CCGT_3


0

10

20

30

40

50

0

500

1000

1500

2000

p01

p01

p01

p01

p01

p01

p02

p02

p02

p02

p02

p02

p03

p03

p03

p03

p03

p03

p04

p04

p04

p04

p04

p04

p05

p05

p05

p05

p05

p05

p06

p06

p06

p06

p06

p06

p07

p07

p07

p07

p07

p07

p08

p08

p08

p08

p08

p08

p09

p09

p09

p09

p09

p09

p10

p10

p10

p10

p10

p10

Mar

gina

l Cos

t [

Out

put [

MW

]

Load Level

CCGT_2

CCGT_1



BrownLignite


Nuclear

RunOfRiver

SRMC

Output Data. Energy

800000

1000000

1200000

1400000E

nerg

y [M

Wh]




RunOfRiver

FuelOilGas

OCGT_3

OCGT_2

OCGT_1

CCGT_4


0

200000

400000

600000

p01

p01

p01

p01

p01

p01

p02

p02

p02

p02

p02

p02

p03

p03

p03

p03

p03

p03

p04

p04

p04

p04

p04

p04

p05

p05

p05

p05

p05

p05

p06

p06

p06

p06

p06

p06

p07

p07

p07

p07

p07

p07

p08

p08

p08

p08

p08

p08

p09

p09

p09

p09

p09

p09

p10

p10

p10

p10

p10

p10

Ene

rgy

[MW

h]

Load Level

CCGT_4

CCGT_3

CCGT_2

CCGT_1



BrownLignite


Nuclear

Output Data. Reservoir level and water value

20

25

30

35

500

600

700

800

Wat

er v

alue

[€/

hm3]

Res

ervo

ir le

vel [

hm3]

Reservoir1_Basin1 Reservoir2_Basin1 Reservoir3_Basin1 Reservoir1_Basin1 Reservoir2_Basin1 Reservoir3_Basin1


0

5

10

15

0

100

200

300

400

p01 p02 p03 p04 p05 p06 p07 p08 p09 p10 p11 p12

Wat

er v

alue

[

Res

ervo

ir le

vel [

hm3]

Period

Output Data. Short Run Marginal Cost (SRMC)

40

50

60

70€/

MW

h]


0

10

20

30

p01

p01

p01

p01

p01

p01

p02

p02

p02

p02

p02

p02

p03

p03

p03

p03

p03

p03

p04

p04

p04

p04

p04

p04

p05

p05

p05

p05

p05

p05

p06

p06

p06

p06

p06

p06

p07

p07

p07

p07

p07

p07

p08

p08

p08

p08

p08

p08

p09

p09

p09

p09

p09

p09

p10

p10

p10

p10

p10

p10

SR

MC

[€

Load Level

Stochastic measures

• Expected value with perfect information (EVWPI) o Wait and See(WS)– Weighted mean of the objective function of each scenario knowing that is

going to happen (for minimization problems always lower or equal than the objective function for the stochastic problem)

• Value of the stochastic solution (VSS)– Difference between the objective function of the expected value for the

mean value solution of the stochastic parameters EEV and that of the stochastic problem RP


stochastic problem RP

• Expected value of perfect information (EVPI) o mean regret– Weighted average of the difference between the stochastic solution for

each scenario and the perfect information solution in this scenario (always positive for minimization)

• EVPI = RP - WS

• VSS = EEV - RP

• WS <= RP <= EEV EVPI >= 0 VSS >= 0

Stochastic measures

sc01 sc02 sc03 Expected StochasticGeneration RunOfRiver in p01 MWh 107136 107136 107136 107136 107136Generation StorageHydro_Basin1 in p01 MWh 79200 67356 82629 79200 78741Generation StorageHydro_Basin2 in p01 MWh 37466 17600 44903 37466 12602Generation StorageHydro_Basin3 in p01 MWh 124281 86400 148800 118110 92787Reserve StorageHydro_Basin1 end p01 hm3 328 368 317 328 330Reserve StorageHydro_Basin2 end p01 hm3 452 518 427 452 535Reserve StorageHydro_Basin3 end p01 hm3

779 800 734 800 800Total Hydro Generation in p01 MWh 348083 278492 383467 341912 291265Total Reserve end p01 hm3 1560 1686 1478 1581 1665Total System Variable Cost M€ 1123.997 1144.447 1103.624 1129.624 1130.284


EWPI or WS EEV VSS EVPI1130.140 1130.360 0.077 0.144

• Stochasticity in hydro inflows is not relevant from the point of view of total variable cost

• But it is important for defining the operation of the first period

How to use of a medium term stochastic hydrothermal coordination model

• Run in a rolling mode (i.e., the model is run each week with a time scope of several months up to one year)

• Only decisions for the closest period are of interest (i.e., the next week). The remaining


interest (i.e., the next week). The remaining decisions are ignored

Summary

• Purpose of a medium term stochastic hydrothermal coordination model

– Characteristics

– Overview

– Results for operation planning and economic planning

– Main modeling assumptions




– General structure

– Parameters, variables, equations, objective function

– Short run marginal cost, water value

• Case study with StarGenLite_SHTCM

– Input data

– Output data

Task assignment

• Compute numerically the water value for a particular period and reservoir by running twice the hydrothermal model and compare this value with the water value determined by the model as the dual variable of the water balance constraint. Apply it to one reservoir in period 1 and another reservoir in period 7.


period 7.

• Introduce intermittent generation into the model

– Play with the number to observe the complementaritybetween hydro and intermittent generation

• Evaluate all the stochastic measures of considering stochasticity of hydro inflows


Prof. Andres Ramos

http://www.iit.upcomillas.es/aramos/

[email protected]

[email protected]

Date post:	30-Oct-2014
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