Stochastic Unit Commitment - Federal Energy … Watson, IBM.pdfIBM Research Stochastic Unit...

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© 2009 IBM Corporation

Stochastic Unit Commitment

Sanjeeb Dash, Joao Goncalves, Jay(ant) Kalagnanam, Ali Koc, Ming Zhao, BAMS, IBM ResearchContact: [email protected]

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© 2009 IBM Corporation2

Unit Commitment Problem (Distributed Generation)

MINIMUM DOWN TIME

2448 72

DEMAND

CO2 EMISSION

BASEUNITS

PEAKING UNITS

Integer programming problem with uncertain demand & supply-> Stochastic optimization

The heat rate of a unit is a (nonlinear) function of load -> nonlinear optimization- maintenance improves heat rate and hence CO2 emissions

SOLAR

WIND

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Summary

• Formulations• Primal heuristics for stochastic unit commitment• Branch-and-cut for (stochastic) unit commitment

• Cuts for linear-cost unit commitment• Cuts for nonlinear-cost unit commitment• Computational results

• Scenario generation using DeepThunder forecasts• Stochastic unit commitment vs. spinning reserves• Further research

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Formulation:

Variables:

Data:

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Minimum up/down Constraints

Formulation:

where

A better formulation (D. Rajan and S. Takriti (2005) )

This formulation improves computation time dramatically

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Comparison of Improved FormulationJ. Goez et al (2008)

590

300

3290

46

1.98

0.04

General

Convex hull

4

510

400

2240

62

11.71

0.11

General

Convex hull

3

10

0

314

1

14.7

0.12

General

Convex hull

2

150

0

436

1

15.3

0.13

General

Convex hull

1

B&B nodesTotal TimeLP Time FormulationInstance

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Some additional constraint classes

Ramping constraints– gi,t + ki ≥ gi,t+1 gi,t - mi ≤ gi,t+1; ki & mi are ramp up/down

rates

Spinning reserves

Modeling storage

Power flow constraints– DC : Linear

– AC: Nonlinear

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Stochastic Unit Commitment with Linear Cost FunctionsFormulation:

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t0

t1

t2

s1

s2

s3

s4

s1 s2

s3 s4

If two scenarios are indistinguishable up to time t, then the decisions for both scenarios by time t should be the same.

If , then we add equalities

. We call the collection of this equalities as nonanticipativity constraints

(s1 s2 s3 s4 )

Bundle (nonanticipativity) Constraints

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Unit Commitment with Nonlinear Cost Functions

Formulation:

where

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Some heuristics to generate initial (feasible) solutions

Lagrangian relaxations– Relax the bundle constraints and add a penalty

term for violations– Solve the s subproblems independently as a

starting point LP rolling horizon heuristic

– Keep only one bundle as binary and relax the remaining

– Start with t0 and roll forward fixing previous periods

– Provides good initial feasible solutions

0.13%15526

0.12%14023

0.10%14920

0.10%13117

0.05%13114

0.06%12211

0.03%1378

0.00%1375

0.00%882

gapTime (s)Scenarios

LP rolling horizon heuristic

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Semi-continuous Knapsack

We study PS = conv(S), whereS = { x Rn : aj xj ≥ b, xj {0}[lj , uj] }

Assumption: aj = 1

Proposition:PS is full dimensional

( More general semi-continues cuts see I. R. de Farias “ semi-continues cuts for Mixed-Integer Programming” )

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Cover Inequality

Definition:Let C N. We say that C is a cover if

jC uj < b

Proposition: Let C be a cover. Then the inequality

is valid for PS.Zhao and Kalagnanam (09) strengthen this cover inequality, develops

cover inequalities for semi-SOS2 knapsack, and proposes heuristic separation algorithms.

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Computational Results (Linear cost function)

• CPLEX 11.0 is used as an LP solver,• The instance: 32 units and 72 periods,• The instance are terminated after at most 7,200 CPU

seconds.• Cover and flow cover cuts are turned off for testing

instances with user cuts.

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Computational Results – basic comparison

4.2

With user + mir cuts

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Computational Results – optimal solution

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Computational Results – root node

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Conclusions• With user cuts, all the instance are solved to optimality and the

time reduction is more than 80% in average.

• The difficulty lies on closing optimality gap.

• By adding cutting planes in the initial formulation, one can take advantages of dynamic search.

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Computational Results –Nonlinear cost functions

• CPLEX 11.0 is used as an LP solver,• The instance: 72 periods, and 4 segments in piecewise

linear functions,• The instance are terminated after at most 3,600 CPU

seconds.• SOS2 concept is enforced by introducing binary

variables to take advantages of CPLEX, since the model already has lots of binary variables.

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Computational Results – basic comparison

10.2

With user + mir cuts

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Computational Results – optimal solution

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Modeling Wind Intermittency

Demand is uncertain, i.e. dt is a random variable

Wind energy is forecast using weather models– Wind speed and direction can be forecast but with uncertainty

– For each farm, generation gi,t is a random variable

Assume that wind energy (subject to technical cut-in constraints) has to be used (regulatory)– A must-take constraint

Therefore the total demand can be written as

– Dt = dt – i gi,t (a new random variable)

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Modeling Wind Intermittency

t0

t1

t2

s1

s2

s3

s4

s1 s2

s3 s4

(s1 s2 s3 s4 )

The forecast for weather to be generated from Deep Thunder – 24-72 hr horizon

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25© 2009 IBM Corporation25

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Modeling Wind Intermittency Scenario Reduction using Kantorovich Distance

• Consider two sets of sample paths – P = (Pi),= i = 1,…,n – Q = (Qj), j = 1,…,m– where Pi=( ,…, ), Qj=( ,…, ),– and Pi has probability , and Qj has probability .

• Kantorovich distance between the two is defined as D(P,Q) = min

s.t.

where .

1ip T

ip 1jq T

jq

ji

ijijcx,0ijx

ij

ij px

ji

ij qx

ip jq

T

t

tj

tiij qpc

1||


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Stochastic Unit Commitment vs. Spinning Reserves

• Base demand is taken 2500 MWH for each 85 periods.

• Stochasticity is in the wind power. All of the wind power is used to meet the demand. Thus, the net demand to be met by the other units is stochastic.

• A wind-farm instance with 200 wind mills is considered.

• Using a scenario reduction technique, wind-power scenario tree is generated with 5, 10, 20 scenarios.


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Wind speed forecasts –Instance 1Wind Speed Forecast for 67 Locations (Instance 1)

-10

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60 70 80 90

Hours

Spee

d (m

ph)

C


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Wind speed forecasts –Instance 2Wind Speed Forecast for 64 Locations (Instance 2)

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90

Hours

Spee

d (m

ph)


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Implicit Reserves –Instance 1Implicit Reserves (Instance 1)

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

0 10 20 30 40 50 60 70 80 90

Periods

Impl

icit

rese

rves

StochasticExp: 0%Exp: 10%Exp: 20%

Costs

Stochastic: $ 1,105,955

Exp-20%: $1,131,847

Impr: 2.29%


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Unmet Demand –Instance 1Unmet Demand (Instance 1)

-20.00

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

0 10 20 30 40 50 60 70 80 90

Periods

Expe

cted

unm

et d

eman

d

StochasticExp: 0%Exp: 10%Exp: 20%


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Implicit Reserves –Instance 2Implicit Reserves (Instance 2)

-100.00%

0.00%

100.00%

200.00%

300.00%

400.00%

500.00%

600.00%

700.00%

0 10 20 30 40 50 60 70 80 90

Periods

Impl

icit

Res

erve

s StochasticExp: 0%Exp: 10%Exp: 20%Exp: 30%Exp: 40%Exp: 50%

Costs

Stochastic: $ 1,131,510

Exp-50%: $1,237,099

Impr: 8.54%


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Unmet Demand –Instance 2Unmet Demand (Instance 2)

-20.00

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

0 10 20 30 40 50 60 70 80 90

Periods

Expe

cted

unm

et d

eman

d

StochasticExp: 0%Exp: 10%Exp: 20%Exp: 30%Exp: 40%Exp: 50%

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Further Research

Real-life problems– Handling power flows equations (linear vs non-linear)

– Hundreds of units

– Storage constraints

Scenario reduction

Scaling the problem size– Decomposition methods: branch-and-price

– Parallel computing

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