Economic MPC for large and distributed energy systems

Transkript

1 Economic MPC for large and distributed energy systems WP4 March 24, / 27

2 Outline Problem definition Economic MPC Decomposition techniques Dantzig-Wolfe Reduced Dantzig-Wolfe algorithm Computational results Performances: Dantzig-Wolfe Vs. reduced Dantzig-Wolfe Future works 2 / 27

3 Current section Problem definition Economic MPC Decomposition techniques Dantzig-Wolfe Reduced Dantzig-Wolfe algorithm Computational results Performances: Dantzig-Wolfe Vs. reduced Dantzig-Wolfe Future works 3 / 27

4 Large and distributed energy systems Electric cars Solar panels Operations Wind turbines Houses Industries Thermal power plant x k+1 = Ax k + Bu k + Gw k y k = Cx k + v k z k = C z x k w k N iid (0, R ww,k ) v k N iid (0, R vv,k ) 4 / 27

5 Power Units Bounds on manipulated variables u k Bounds on output z k u min u k u max u min u k u max 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 Cost of operating the system 1 φ = lim N N N c k u k + ρ k s k k=0 F k = { ĉ k+j k, ˆρ k+j+1 k } N 1 j=0 5 / 27

6 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 i,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 6 / 27

7 Kalman filter and certainity equivalence Kalman filter computes: ˆx k k, ˆx k+1+j k and ẑ k+1+j k The certainty equivalence assumption = mean value predictions for all variables. To compute the optimal control trajectory we need: Mean value of the forecasts R k and F k, filtered state ˆx k k, predictions ˆx k+1+j k and ẑ k+1+j k, previous input u k 1 current measurement y k 7 / 27

8 Linear Economic MPC min φ k = P i=1 N 1 φ i,k + j=0 s.t. ˆx i,k+j+1 k = A iˆx i,k+j k + B i û i,k+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 ˆ ρ k+j+1 kˆ s k+j+1 k 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 = i=1 C z,iˆx i,k+j+1 k 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 8 / 27

9 Current section Problem definition Economic MPC Decomposition techniques Dantzig-Wolfe Reduced Dantzig-Wolfe algorithm Computational results Performances: Dantzig-Wolfe Vs. reduced Dantzig-Wolfe Future works 9 / 27

10 Dantzig-Wolfe decomposition The control problem obtained has the following block-angular matrix: min {q i } M i=1 s.t. M φ = e iq i i=1 F 1 F 2... F M g G 1 q 1 G 2 q 2 h 1. h GM q M h M M = {1,..., P, P + 1} because ˆ s k+j+1 k are considered as an independent subsystem. 10 / 27

11 Theorem of convex combinations Q i = {q i G i q i h i }, V i q i = λ ij v j i V i j=1 j=1 λ ij 0 λ ij = 1 i = 1, 2,..., M j = 1, 2,..., V i M φ = e iq i i=1 i=1 j=1 i=1 i=1 j=1 M V i e iv j i λ ij M M V i F i q i g F i v j i λ ij g 11 / 27

12 Master problem (MP) or extremal problem Setting f ij = e i vj i and p ij = F i v j i min λ φ = s.t. M V i f ij λ ij i=1 j=1 M V i p ij λ ij g i=1 j=1 V i j=1 λ ij = 1, λ ij 0, yields to i = 1, 2,..., M, j = 1, 2,..., V i Compared to the block-angular problem: equivalent less number of constraints more variables 12 / 27

13 Reduced master problem (RMP) Reduced Master Problem is introduced as a MP but with l number of vertices, where l V i : min λ φ = s.t. M M i=1 j=1 i=1 j=1 l f ij λ ij l p ij λ ij g π l λ ij = 1, i = 1, 2,..., M ρ j=1 λ ij 0, in which l V i for all i {1, 2,..., M}. i = 1, 2,..., M, j = 1, 2,..., l 13 / 27

14 Pricing problem Master Problem (RMP) π, ρ 1 ψ 1, v 1 π, ρ 2 ψ 2, v 2 π, ρ M ψ M, v M Subproblem 1 Subproblem 2 Subproblem M DW utilizes dual variables π and ρ i to generates only the columns having the most negative reduced cost. In this way the algorithm does not built the full MP. This pricing problem is expressed in the subproblems i M. ψ i = min q i φ = [ e i F i π ] qi s.t. G i q i h i 14 / 27

15 Optimality condition Optimality: the optimal solution for the RMP is also optimal for the MP if ψ i = ρ i, i M. ψ i ρ i 0 for all i = 1, 2,..., M = all the reduced costs will be nonnegative q i = l v j i λ ij j=1 ψ i ρ i < 0 for some i {1, 2,..., M} i = 1, 2,..., M we augment the Reduced Master Problem with the new extreme points, vi l+1, obtained by solution of the subproblems. 15 / 27

16 Reduced Dantzig-Wolfe Optimality condition DW: ψ i ρ i 0, i = 1, 2,..., M Reduced DW: a subproblem i M satisfies the optimality condition ψ i ρ i 0, i S M No more extreme points are added to its polytope Q s. At the next iteration, the Dantzig-Wolfe algorithm solves every subproblem i M\S. 16 / 27

17 S = { }, L=1. while Converged == false do Solve the L th RMP. Solve subproblem i, for i M\S. if optimality condition is satisfied i M then Converged == true else if a subproblem s, s M, satisfies the optimality condition then S = {s}, S M p j+1 i = c i v j i h j+1 i = F i v j i, i M\S else Compute p j+1 i and h j+1 i, i M end if end if L = L + 1 end while 17 / 27

18 Current section Problem definition Economic MPC Decomposition techniques Dantzig-Wolfe Reduced Dantzig-Wolfe algorithm Computational results Performances: Dantzig-Wolfe Vs. reduced Dantzig-Wolfe Future works 18 / 27

19 Case study: energy systems and power units Energy systems consist of multiple power units. Power units are defined as Z i (s) = G i (s)u i (s) G i (s) = representing: central thermal power plants, diesel generators, gas turbines. τ is randomly generated. Simulations run in MATLAB. 1 (τ i s + 1) 3 19 / 27

20 Power production: 75 energy units Customers demand DW Reduced DW Time 20 / 27

21 Objective function: 75 energy units Obj. func.[euro] 10 5 DW Reduced DW Time 21 / 27

22 Computational time 10 5 CPU time [s] 10 4 DW Reduced DW Power units 22 / 27

23 Objective function optimal values 10 8 Obj. func.[euro] 10 6 DW Reduced DW Power units 23 / 27

24 Objective function optimal values [%] Comp. time decrease Obj. func. deterioration Power units 24 / 27

25 Current section Problem definition Economic MPC Decomposition techniques Dantzig-Wolfe Reduced Dantzig-Wolfe algorithm Computational results Performances: Dantzig-Wolfe Vs. reduced Dantzig-Wolfe Future works 25 / 27

26 Future Works C code for Dantzig-Wolfe decomposition (.mex file) + real parallel computing MATLAB toolbox for Economic MPC applied to large-scale scenario Journal paper about the toolbox performances PhD thesis 26 / 27

27 Opstarten af projektet har været udsat for en forsinkelse pga. a. sagsbehandlingstid i forbindelse med tilsagn b. ansættelse af og indhentning af visum til PhD studerende En anmodning om forlængelse frem til skal skrives til EBST - ansvarlige er MLL og RW 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 støtteberettigede omkostninger b. Den ønskede forlængelsen af projektet input fra alle sendes til MLL (igennem google.docs budgetfil, adgang rekvireres hos MLL) c. Danfoss reducerer Thank engagementet you i projektet MLL modificerer budgetfordelingen til at den svarer til den nye situation; finansiering af de 4 PhD studerende på AAU og DTU bliver uændret 3. Næste afrapportering er 1. marts For de fleste partnere er dette FØRSTE afrapportering. Derfor skal der allerede nu opbygges de nødvendige procedurer og indsamles de fornødne data. Startende med dokumentation for tidsforbrug. Alle vil blive kontaktet af Ole Sloth fra Lean 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 Næste møde Smart & Cool: d. 11 juni i Nordborg (Danfoss) /RW, MLL 1 27 / 27