REFERAT: MPC. Computational Results. John Bagterp Jørgensen

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1 REFERAT: Følgende punkter blev diskuteret: a. sagsbehandlingstid i forbindelse med tilsagn The Dantzig-Wolfe Method forb. Large-Scale ansættelse af og indhentning af visum Economic til PhD studerende RW MPC Computational Results Laura Standardi, Leo Emil Sokoler, Niels af de 4 PhD Kjølstad studerende på AAU Poulsen og DTU bliver uændret and John Bagterp Jørgensen DTU Compute Department of Applied Mathematics and Computer Science Technical University of Denmark June 10, Forlængelse af projektet Opstarten af projektet har været udsat for en forsinkelse pga. En anmodning om forlængelse frem til skal skrives til EBST - ansvarlige e 2. Der er behov for en korrektion af budgettet. Forhold som skal tages i betragtning er a. En operationalisering og en detaljering af budgettet for at synliggøre støtteb omkostninger b. Den ønskede forlængelsen af projektet input fra alle sendes til MLL (igennem google.docs budgetfil, adgang rekvirer MLL) c. Danfoss reducerer engagementet i projektet MLL modificerer budgetfordelingen til at den svarer til den nye situation; fin 3. Næste afrapportering er 1. marts For de fleste partnere er dette FØRSTE afrap Derfor skal der allerede nu opbygges de nødvendige procedurer og indsamles de for ta. Startende med dokumentation for tidsforbrug. Alle vil blive kontaktet af Ole Sloth Energy som indleder denne indsamling af data og kontrol af procedurer. 4. Der skal fremmes samarbejde mellem de PhD studerende: a. Mulighed for korte besøg (evt en uge) på universiteter og virksomheder, b. Workshops for PhD studerende blev diskuteret, c. KVCA medlemsseminar med fokus på emner fra projektet evt. i september 2 5. Næste møde Smart & Cool: d. 11 juni i Nordborg (Danfoss) WP4 1

2 Outline 1 Status 2 Large-scale energy systems 3 Economic MPC 4 Dantzig-Wolfe decomposition 5 Results 6 Conclusion & Future Works WP4 2

3 Status Status WP4 Paper submitted to: CDC 2013 NPC 2013 (abstract) ISGT 2013 Spring 2013: TA activity at Stochastic Adaptive Control From now until August 2014: research. WP4 3

4 Large-scale energy systems Large-scale energy systems x k+1 = Ax k + Bu k +Gw k + Ed k y k = Cx k +v k z k = C z x k (1a) (1b) (1c) WP4 4

5 Large-scale energy systems Power units Bounds on manipulated variables u k u min u k u max u min u k u max Bounds on output z k : [r min,k, r max,k ] R k = {ˆr min,k+j k, ˆr max,k+j k } N j=1 r min,k s k z k r max,k + s k, s k 0 every time s k is non-zero a penalty cost must be paid. Cost of operating the system: 1 φ = lim N N N c k u k + ρ k s k k=0 F k = { c k+j k, ρ k+j+1 k } N 1 j=0 WP4 5

6 Large-scale energy systems Dynamically decoupled systems P = {1,..., P } of dynamically decoupled systems x i,k+1 = A i x i,k + B i u i,k + G i w k + E i d k y i,k = C i x i,k + v i,k z i,k = C z,i x i,k i P i P i P that jointly create the following measurement, ỹ k, and output, z k : ỹ k = z k = P i=1 P i=1 C i x i,k +ṽ k C z,i x i,k i P and j N we obtain the following linear program WP4 6

7 Economic MPC Economic MPC min φ k = P i=1 N 1 φ i,k + j=0 ˆ ρ k+j+1 kˆ s k+j+1 k s.t. ˆx i,k+j+1 k = A iˆx i,k+j k + B i û i,k+j k + E i ˆdk+j k ẑ i,k+j+1 k = C z,iˆx i,k+j+1 k u min,i û i,k+j k u max,i u min,i û i,k+j k u max,i ẑ i,k+j+1 k + ŝ i,k+j+1 k ˆr min,i,k+j+1 k ẑ i,k+j+1 k ŝ i,k+j+1 k ˆr max,i,k+j+1 k ŝ i,k+j+1 k 0, ˆ s k+j+1 k 0 P ˆ z k+j+1 k = C iˆx i,k+j+1 k i=1 ˆ z k+j+1 k + ˆ s k+j+1 k ˆ r min,k+j+1 k, ˆ z k+j+1 k ˆ s k+j+1 k ˆ r max,k+j+1 k WP4 7

8 Economic MPC D k,f k,r k Regulator u k Power units z k x k k Sensors y k Kalman filter Economic MPC WP4 8

9 Dantzig-Wolfe decomposition Dantzig-Wolfe decomposition The control problem obtained has the following block-angular matrix: min φ = {q i } P i=1,s s.t. P c iq i + d s i=1 F 1 F 2... F P E g G 1 q 1 G 2 q 2 h 1 h GP q P. h s P I 0 In order to describe the Dantzig-Wolfe algorithm we need: Define Master problem Define subproblems WP4 9

10 Dantzig-Wolfe decomposition Theorem of convex combinations The Dantzig-Wolfe algortihm is built on the theorem of convex combinations: Let Q = {q Gq h} be nonempty, closed and bounded, i.e. a polytope. The extreme points of Q are denoted v j with j {1, 2,..., M}. Q i = {q i G i q i h i } M i q i = λ ij v j i M i j=1 j=1 λ ij 0 λ ij = 1 j = 1, 2,..., M i WP4 10

11 Dantzig-Wolfe decomposition Master problem The Master problem (MP) equivalent to the block-structural control problem, is: min λ φ = s.t. P +1 P +1 i=1 j=1 M i j=1 M i i=1 j=1 M i λ ij 0, λ ij = 1, f ij and p ij are defined as f ij λ ij p ij λ ij g π i = 1, 2,..., P + 1 ρ i = 1, 2,..., P + 1; j = 1, 2,..., M i f ij = c iv j i, p ij = F i v j i WP4 11

12 Dantzig-Wolfe decomposition Reduced master problem Reduced Master Problem is introduced as a MP but with l number of vertices, where l M i : P +1 min φ = λ s.t. P +1 i=1 j=1 i=1 j=1 l f ij λ ij l p ij λ ij g l λ ij = 1, i = 1, 2,..., P + 1 j=1 λ ij 0, in which l M i for all i {1, 2,..., P + 1}. i = 1, 2,..., P + 1; j = 1, 2,..., l WP4 12

13 Dantzig-Wolfe decomposition Subproblems KKT conditions: P +1 M i i=1 j=1 M i j=1 p ij λ ij g 0 π 0 λ ij 1 = 0 λ ij 0 κ ij = [ c i F i π ] v j i ρ i 0 They are already satisfied for i = 1, 2,..., P + 1 and j = 1, 2,..., l. For all i = 1, 2,..., P + 1 and j = l + 1, l + 2,..., M i = min i ψ i ρ i 0 where ψ i = min v j i [ ci F i π ] v j i i = 1, 2,..., P + 1 v j i is an extreme point of the polytope Q i = {q i : G i q i h i }. WP4 13

14 Dantzig-Wolfe decomposition Stopping criteria Therefore the subproblems are ψ i = min q i φ = [ c i F i π ] qi s.t. G i q i h i ψ i ρ i 0 for all i = 1, 2,..., P + 1 = l qi = v j i λ ij i = 1, 2,..., P + 1 (2) j=1 ψ i ρ i < 0 for some i {1, 2,..., P + 1} = we augment the Reduced Master Problem with the new extreme points, vi l+1, obtained by solution of the subproblems. WP4 14

15 Dantzig-Wolfe decomposition Algorithm Compute the initial feasible vertex for the Master Problem. If any points is found then STOP. l=1. Converged=false While Converged == f alse Solve the l th RMP Solve all the subproblems (i = 1, 2,..., P + 1) considering the π and ρ i from the MP. Ifφ i ρ i 0 i {1, 2,..., P + 1} Coverged = true The optimal solution is given by (2) Else Compute the coefficients for the new columns in the RMP l = l + 1 EndIf EndWhile f i,l+1 = c ivi l+1, p i,l+1 = F i vi l+1 WP4 15

16 Dantzig-Wolfe decomposition Initial feasible vertex Usually it is found computing Simplex Phase I (finding a feasible vertex in this way, may be just as expensive as computing the optimal solution of the control problem). The Dantzig-Wolfe algorithm is a part of a MPC controller. A feasible initial vertex for our MP may be defined as { q 0 i = q i,min } P i=1 s = max { g } P F i qi 0, 0 i=1 WP4 16

17 Dantzig-Wolfe decomposition Suboptimal MPC - Early termination Obj. Func Number of extreme points l The DW algorithm can be stopped before to a suboptimal solution. Such early termination reduces computational time. WP4 17

18 Results Total power portfolio 30 Power Demand interval Power production Time WP4 18

19 Results Suboptimal MPC - Early termination In average, the algorithm needs 25 extreme points. 100 [%] Number of extreme points [l] WP4 19

20 Results Comparison with state of the art of LP solvers cpu time [sec] MOSEK [IP] Gurobi [AS] Gurobi [IP] CPLEX [AS] CPLEX [IP] DW DEC Nr. of units WP4 20

21 Conclusion & Future Works Conclusions The algorithm developed in this work: In closed-loop simulation the controller keeps the total power produced inside the customers demand. The computational time highly decreases implementing the early termination technique. The controller performes faster than any other LP solvers. WP4 21

22 Conclusion & Future Works Future works Future works are going to be deal with: Subotpimal MPC: how to reduce extra costs. Include other energy sources, as EVs, refrigeration systems and heat pumps. Include DW algorithm in a MATLAB toolbox to solve large-scale energy systems. WP4 22