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IBM Research
© 2009 IBM Corporation
Stochastic Unit Commitment
Sanjeeb Dash, Joao Goncalves, Jay(ant) Kalagnanam, Ali Koc, Ming Zhao, BAMS, IBM ResearchContact: [email protected]
IBM Research
© 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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© 2009 IBM Corporation3
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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© 2009 IBM Corporation24
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||
26© 2009 IBM Corporation26
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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.
27© 2009 IBM Corporation27
IBM Research
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
28© 2009 IBM Corporation28
IBM Research
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)
29© 2009 IBM Corporation29
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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%
30© 2009 IBM Corporation30
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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%
31© 2009 IBM Corporation31
IBM Research
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%
32© 2009 IBM Corporation32
IBM Research
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%
IBM Research
© 2009 IBM Corporation33
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