Matheuristics for the discrete unit commitment problem with min-stop

Problem statement and related state-of-the-art MIP formulation Constructive matheuristics VNS Local Search with MIP neighbourhoods Implementation results Conclusion and perspectives

Matheuristics for the discrete unit commitment problem with min-stop ramping constraints Nicolas DUPIN, El-Ghazali TALBI University Lille 1

Matheuristics 2016, Brussel, 7 September 2016

Nicolas DUPIN, El-Ghazali TALBI

Matheuristics for the discrete UCP


Plan 1

Problem statement and related state-of-the-art

2

MIP formulation

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Constructive matheuristics

4

VNS Local Search with MIP neighbourhoods

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Implementation results

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Conclusion and perspectives




Plan 1


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MIP formulation

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Energy Management, electricity production Electricity is not storable on a large scale. The global production has to match exactly the demand at any time. Demands are very fluctuating, with seasonality effects. Production means are technically constrained to modulate their production. Different scales for Energy management optimization from long term strategies to short term production optimization.




Unit Commitment Problem, UCP UCP: Planning the production of electric power generating units, meeting the demands, in order to minimize the operations costs. Cost function: Proportional cost to the power generated, fixed costs, start up costs. Demand constraints on every time step: Demands in power. (forecasted) Demands in reserves, primary and secundary. (imposed) Decision variables: Ptu : power generated by unit u at time step t. xtu : set-up binary variables. ytu : start-up binary variables. Nicolas DUPIN, El-Ghazali TALBI



MILP formulation

min

XX u∈U t∈T

u Cprop Ptu +

XX

Cfixu xtu +

u∈U t∈T

XX

u Cstart ytu

u∈U t∈T

s.t: u ∀u ∈ U, ∀t ∈ [[1, T ]], xtu − xt−1 6 ytu X ∀t ∈ [[1, T ]], Ptu = Dt

(1) (2)

u∈U

∀u ∈ U, ∀t ∈ [[0, T ]], xtu .Pmintu 6 Ptu ∀u ∈ U, ∀t ∈ [[0, T ]],

Ptu

6

xtu .Pmaxtu

u ∀u ∈ U, Pu0 = Pinit



(3) (4) (5)


Minimum stops constraints for fossil power plants




Our discrete UCP: UCPd

We model only discrete operating points: Discrete power generated. (only integer variables) Demands are knapsack constraints. Dynamic constraints: Min up- min down constraints. (start up) Minimum stops on discrete operating points. (in operation)




Minimum stops constraints for UCPd

$ (i)+

∆u

=

(i)

δu +

P(u, i + 1) − P(u, i)

%

|gradmontee (u)|


$ (i)−

, ∆u

=

(i)

δu +

P(u, i) − P(u, i − 1) |gradbaisse (u)|


% .


Plan 1


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MIP formulation

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Modeling issues

2 definitions of variables can be considered: state vs level variables. Isomorphic structures, isomorphic constraints but different ILP resolution characteristics. Exact formulation and MIP analyses published in EURO J Comput Optim: N Dupin, Tighter MIP formulations for the discretised unit commitment problem with min-stop ramping constraints Nicolas DUPIN, El-Ghazali TALBI



Variable definition

(i)

(i)

Level variables lu,t : lu,t = 1 iff unit u generates at time step t on a point j > i. (1)

Set up variable: xu,t = lu,t . Precedence constraints: (i−1)

∀u, t, i, lu,t

(i)

6 lu,t




Start up variables for each operating point Start up variables for each operating point: (i)+

yu,t = 1 if unit u is ramped up from point i − 1 at time step t − 1 to point i at time step t. (i)− yu,t = 1 if unit u is ramped down from point i + 1 at time step t − 1 to point i at time step t. Linking constraints: ∀u, t, i,

(i)

(i+1) lu,t−1

∀u, t, i,

(i−1)−

∀u, t, i, yu,t

(i)

(i)+

(6)

(i)− yu,t

(7)

lu,t − lu,t−1 6 yu,t −

(i+1) lu,t

(i)+

6 (i)

(i)

= yu,t + lu,t−1 − lu,t



(8)


X

min x,y

(1)

(1)+

CuF xu,t + CuS yu,t

u,t

(i)

CuP

(Pu,i − Pu,i−1 )xu,t (9)

(i)

xu,0

∀u, t, i,

(i) xu,t

∀u, t, i,

(i) xu,t (i) xu,t

∀u, t, i, ∀u, t,

− Pt

∀u, t,

Pt

(i) xu,t−1

+

= 11i 0 >i u

(i−1)− yu,t

(1)+ y 0 t 0 =t−∆on u +1 u,t

P

t+∆+ u,i t 0 =t+1

(i+1)+

∀t, ∀t,

P

∀t,

P

+

P

t+∆− u,i t 0 =t+1

(i−1)−

yu,t 0

u,i (Pu,i

− Pu,i−1 )

(i) xu,t

1 u,i (Ru,i

−

1 Ru,i−1 )

(i) xu,t

−

2 Ru,i−1 )

(i) xu,t

2 u,i (Ru,i

6

(11)

>

(i+1) xu,t+1

(12)

=

(i)+ yu,t

(13)

(1)

(14)

61− (i+1)

+ xu,t

(10)

(i−1) xu,t+1

6 xu,t

(1)+ y 0 t 0 =t−∆off u +1 u,t

yu,t 0 P

∀u, t, i,

X u,t,i

∀u, i,

∀u, t, i,

+

(1) xu,t−lu

(15)

6 xu,t

(i)

(16)

> DtP

(17)

>

DtR1

(18)

>

DtR2

(19)

(i)+ (i)− (i) yu,t ,TALBI yu,t , xu,tMatheuristics for the discrete ∈ {0,UCP 1} Nicolas DUPIN, El-Ghazali

(20)


Plan 1


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MIP formulation

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Variable fixing heuristics

(i)

Following fixing strategies using the LP relaxed solutions x˜u,t were implemented: (i)

(i)

(i)

hFix01: For all variable such that x˜u,t ∈ {0, 1} , we fix xu,t = x˜u,t . (i)

(i)

hFix0 : For all variable such that x˜u,t = 0, we fix xu,t = 0. hFct01 : Same strategy as hFix1, applying only for i = 1. hFct0 : Same strategy as hFix0, applying only for i = 1.




hSelect : we remove the power plants u in which production is systematically null in the continuous relaxation, ie such as for all t, (i) x˜u,t = 0. hTube : power plants are first removed using hSelect. For the remaining (i+1) (i−1) (i) units, if x˜u,t = 1 (resp x˜u,t = 0), we fix xu,t = 1 (resp 0). hTube3 : we consider three scenario for the demands :the real one, an increased demand and an underestimated demand. For these three scenario we calculate the continuous relaxation, and we fix the common variables of the hTube strategy fixation. Noting a > b that fixations of a contains fixations of b, hFix01 > hFix0 > hTube > hTube3 > hSelect.




Illustration




Relax-and-Fix heuristics Decompositions truncate the problem and/or relax continuously a subset of variables to have easier resolutions as follows: The dcpDay strategy fixes the first day production, relaxing continuously the variables relative to the second day. Then we resolve the MIP on the second day with the remaining variables. The dcpLev strategy builds a solution level by level following index i. (i) Once variables with index i 0 < i are fixed, variables xu,t are calculated (i 0 )

using the MIP with variables xu,t integer for i 0 6 i and fixed to their (i 0 )

previous value for i 0 < i. Variables xu,t for i 0 > i are continuously relaxed. dcpMid strategy also proceeds level by level, following index i. First iteration consider only the variables of the middle operating points Nfct (u) Mid(u) = b c, with other variables continuously relaxed. A middle 2 variable fixed to 0 (resp 1) involves indeed that the variables of the upper (i) (i+1) levels (lower respectively) are also fixed. with constraints xu,t > xu,t . Nicolas DUPIN, El-Ghazali TALBI



Adding more flexibility with slight modifications of the previously fixed decisions improves the solution quality and avoids infeasibilities. With dcpMid or dcpLev strategies, only 1 -stable variables are fixed, as defined below. With dcpDay, the first step considers binary variables for t ∈ [1, 54] and fixes the variables for t ∈ [1, 42] (assuming T = 96). (i)

Definition: k-stable variables xu,t have incumbent values (i) (i) (i) (i) (i) xu,t−k = · · · = xu,t−1 = xu,t = xu,t+1 = · · · = xu,t+k .




Algorithm : Parallelisable Relax-and-Fix decomposition First step: Variable elimination hSelect unit selection using the LP relaxation. Second step : Two independent MIP are computed: Mid(u) MIP 2.1: only variables xu,t are integer for t ∈ [[1, 48]] . Mid(u) are integer for t ∈ [[49, 96]]. MIP 2.2: only variables xu,t Third step : Three independent MIP are computed: (i) MIP 3.1: only variables xu,t are integer for t ∈ [[1, 32]]. Integer 1-stable variables of MIP 2.1 are fixed to their optimal value in MIP 2.1. (i) MIP 3.2: only variables xu,t are integer for t ∈ [[65, 96]]. Integer 1-stable variables of MIP 2.2 are fixed to their optimal value in MIP 2.2. Mid(u) MIP 3.3: only variables xu,t are integer. The optimal values in MIP 2.1 (resp 2.2) are fixed for t ∈ [[1, 36]] (resp t ∈ [[65, 96]]). Final step: the partial solutions are merged in one computation fixing: (i) • xu,t for t ∈ [[1, 24]] (resp t ∈ [[79, 96]]) are fixed to its value in MIP 3.1 (resp 3.2). (Mid(u)) • xu,t for t ∈ [[25, 32]] (resp t ∈ [[65, 78]]) are fixed if they are 1-stable and equal in the optimal values in MIP 3.3 and MIP 3.1 (resp MIP 3.2). (Mid(u)) • xu,t for t ∈ [[33, 64]] are fixed only if their optimal value in MIP 3.3 are 1-stable Nicolas DUPIN, El-Ghazali TALBI



Plan 1


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MIP formulation

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Variable Neighbourhood Descent with MIP neighbourhoods Input: an initial solution, a set and order of neighbourhoods to explore Initialisation: currentSol = initSol, N =initial neighbourhood. while the stopping criterion is not met define the MIP with incumbent currentSol and the neighbourhood N define currentSol as warmstart currentSol = solveMIP(MIP,timeLimit( N )) N = nextNeighborhood(N ) end while return currentSol =⇒ A steepest descent VND algorithm.




ILP neighbourhoods

Usually, VNS choose a solution in a given neighbourhoods. Our approach Fix a subset of the integer variables of the current solution. ILP Resolution limited in the Branch & Bound enumeration (number of nodes, time limit, stopping criteria with the gap lower - upper bound . . . ). Specialized parametrization of the ILP resolution.

=⇒ The cost of the solution is given with the cost given by the ILP solver. The solution has at least the same cost than the original one (solution of this MIP).




How to parametrize an MIP solver?

General mode mipemphasis to feasibility, to activate at most ILP heuristics. Node limit / time limit. Limit cutting plane passes. Increase the node frequency to try ILP heuristics. Aggregator: detect and suppress the implied fixed variables or useless constraints, leading to a smaller MIP. Warmstart: ILP Resolution easier with an initial solution given to the solver.




Generic neighbourhoods

Generic MIP neighbourhoods: RINS, Local Branching . . . (i)

Fixation of variables along the multi index structure of variables xu,t : Remove unused thermal units in the current solution. (selection in u) Fix the start up decisions. (selection in i first level) Fix the decisions outside the time windows peaks (selection in t).




Time window fixing and propagated fixing




Variable fixing when fixing mid power level




Neighbourhoods using special dynamic structures Fixing based on coupling constraints (dynamic constraints for index t, and precedence constraint for index i): 1 translation: fix variable in t iff the correspondent value is the same at t − 1 and t + 1. Tube fixing: fix a variable in i iff the correspondent value is the same at i − 1 and i + 1. (Corridor ?) More flexible decisions that could be crossed with other neighbourhoods.




Variable fixing with 1-translation




Tube fixing




Mid level variable fixing with 1-translation




Plan 1


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MIP formulation

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Variable fixing and primal heuristics for real size instances

=⇒ Truncating to 15 min, it is more efficient to fix variables smartly. Nicolas DUPIN, El-Ghazali TALBI



Numerical results in 5 minutes on a dataset with only thermal units Data Frontal hSelect hTube3 D67 2,24% 1,81% 1,59% D57 3,18% 1,76% 1,73% D30 0,42% 0,31% 0,06% D32a 1,02% 1,31% 0,65% D32b 0,56% 0,55% 0,30% D80a 1,20% 0,57% 0,25% D80b 0,70% 0,66% 0,69% D82a 1,39% 0,76% 0,33% D82b 0,89% 0,87% 0,16% Total 1,36% 0,98% 0,58% Bold values emphasize the best results


hTube 1,60% 1,64% 0,03% 0,36% 0,16% 0,28% 0,52% 0,27% 0,89% 0,69%

dcpMid 1,99% 1,84% 0,66% 1,24% 0,46% 0,42% 0,93% 0,64% 0,38% 0,86%

dcpLev 1,87% 1,60% 0,09% 0,74% 0,34% 0,16% 0,78% 3,05% 0,20% 1,26%


dcpMix 1,97% 1,86% 0,62% 1,15% 0,50% 0,42% 0,97% 0,21% 0,16% 0,70%

VND 1,57% 1,75% 0,03% 0,21% 0,00% 0,24% 0,76% 0,29% 0,19% 0,52%


Numerical results in 5 minutes on real world instances

Formulations Instances data 10-01 data 12-01 data 17-03 data 08-05 data 20-06 data 30-07 data 08-09 data 31-10

UCPd Proj 5,13 % 2,38% 2,39% 0,49% 6,17% 0,21% 1,82% 1,03%

Frontal gap dual 1,79 % 1,35% 1,07% 0,23% 10,17% 2,35% 5,41% 1,38 %

Frontal gap Primal 1,35% 0,65% 0,37% 0,08% 7,57% 1,63% 0,75% 1,38%


hSelect gap dual 1,53% 1,21% 0,92% 0,16% 5,24% 0,98% 5,02 % 0,00%

hSelect gap Primal 1,09% 0,51% 0,22% 0,00% 2,50% 0,24% 0,34% 0,00%



Numerical results in 5 minutes considering tertiary reserve

Frontal gap Primal 6,38% 2,76% 1,69% 8

Frontal gap dual 7,04% 3,44% 1,84% 8

Formulations Instances data 12-01 data 17-03 data 08-05 data 20-06 data 30-07 data 08-09 data 31-10

2,58% 12,21% 3,28%

1,85% 7,89% 3,28%


hSelect gap dual 2,87% 1,06% 0,43% 8,95% 0,98% 5,56 1,42%

hSelect gap Primal 2,18% 0,36% 0,28% 6,32% 0,24% 0,90% 1,42%



Plan 1


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MIP formulation

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Conclusion In the short truncated time of operational processes, matheuristics outclass significantly truncated exact approach. Efficient hard-fixing strategies based on problem structure. VND with ILP neighbourhoods combine useful advantages: uses the MIP modelling facilities easy definition and resolution with diverse and large neighbourhoods efficient for a very constrained problem converge to very good local minimums RINS neighbourhoods different from physical intuition

=⇒ Home-made matheuristics knowing problem structures are better than blind generic primal MIP heuristics, advantage to know and exploit some problem knowledge




Perspectives

Tackle more difficult instances: Extension of the results to a production fleet with hydraulic units Tuned manually in the order of neighbourhoods (key point). Automatic tuning with irace would be particularly interesting. Parallelization is a key point to improve the approach: Providing quickly good solutions in parallel decomposition schemes. Implementing ILP neighbourhoods within a parallel Variable Neighbourhood Search (VNS) scheme.




Questions?

Thank you for your attention

