Economic MPC for large and distributed energy systems

Economic MPC for large and distributed energy systems WP4 Laura Standardi, Niels Kjølstad Poulsen, John Bagterp Jørgensen August 15, 2014 1 / 34

Outline Introduction and motivation Problem definition Economic Model Predictive Control First & Second outcome Dantzig-Wolfe Vs. Centralized MPC Dantzig-Wolfe Vs. LP solvers Third & Forth outcome Early termination Reduced Dantzig-Wolfe Conclusion 2 / 34

Current section Introduction and motivation Problem definition Economic Model Predictive Control First & Second outcome Dantzig-Wolfe Vs. Centralized MPC Dantzig-Wolfe Vs. LP solvers Third & Forth outcome Early termination Reduced Dantzig-Wolfe Conclusion 3 / 34

Denmark and its energy future (*http://www.smartgriddenmark.com/) Wind Wind + other RE sources Non-wind Danish Climate Goals 2020 50% wind in electricity sector 2030 All coal power plants decommissioned 2035 2050 100% renewables in electricity & heating sector 100% renewables in all energy sectors 4 / 34

Future energy systems Actual energy scenario Old, not updated Centralized energy production Losses Future increase in energy consumption CO 2 emissions High oil prices Domino effects after fault Future energy scenario Embody ICT (Information and Communication Technology) DER (Distributed Energy Resources) Reduced losses Flexibility RES (Renewable Energy Sources) Less dependent of fossil fuel Higher reliability after fault 5 / 34

Large and distributed energy systems 6 / 34

Large and distributed energy systems Operation center: Control and coordinate energy units Minimize production costs and meet energy consumption 6 / 34

Energy units Linear state-space models x k+1 = Ax k + Bu k + Gw k y k = Cx k + v k z k = C z x k Linear constraints on input and output variables u min u k u max u min u k u max r min,k s k z k r max,k + s k s k 0 8 / 34

Linear control problem Linear production costs 1 φ = lim N N N c k u k + ρ k s k k=0 Linear models Linear constraints Linear production costs Linear control problem 9 / 34

Controller Energy demand Coordinator Economic MPC formulated as a LP. Power unit 1 Power unit 2 Minimize production costs. Meet customers consumption. 25 20 15 Power plant portfolio Power production 10 5 0-5 Time 11 / 34

Electricl grid: multiple energy units P = {1,..., P } energy units x i,k+1 = A i x i,k + B i u i,k + G i w 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 the total electric power provided to the grid is z k, ỹ k ỹ k = z k = P i=1 P i=1 C i x i,k + ṽ k C z,i x i,k 12 / 34

Linear Economic Model Predictive Control (MPC) min φ k = û k+j k,ŝ k+j+1 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, ŝ i,k+j+1 k 0 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 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 s k+j+1 k 0 ˆ z k+j+1 k ˆ s k+j+1 k ˆ r max,k+j+1 k, ˆ s k+j+1 k 0 13 / 34

Compute optimal input sequence Many variables and constraints High computation time Fast computation of the optimal input sequence. Reliable and efficient controller. If not optimal, is a feasible solution acceptable? 14 / 34

Decomposition techniques Control problem has a specific matrix constraint (block-angular). Coup. Constr. 1 Coup. Constr. 2... Dec. Constr. 1 Coup. Constr. M-1 Coup. Constr. M Dec. Constr. 2... Dec. Constr. M-1 Dec. Constr. M Decomposition technique applied to our control problem: Exploit such block-angular structure Reduce computational time Introduce scalability 16 / 34

Dantzig-Wolfe algorithm: key parts Dantzig-Wolfe decomposition applies to LPs that have block-angular constraints matrix. 1. Dantzig-Wolfe representation = RMP. 2. Subproblems. 3. Optimality condition. 17 / 34

Dantzig-Wolfe decomposition Setup RMP with initial feasible vertex via DW representation Solve RMP and send simplex multipliers to subproblems Update cost coefficients of each subproblem. Add one column to the RMP. Solve all subproblems. Optimal solution found. Yes Is the optimality condition satisfied? 18 / 34

First outcome Computational time (seconds) 10 3 10 2 10 1 Centralized MPC Dantzig Wolfe Parallel computing 10 0 5 20 40 80 150 500 750 1000 Number of power units Laura Standardi, Kristian Edlund, Niels Kjølstad Poulsen, John Bagterp Jørgensen, A Dantzig-Wolfe decomposition algorithm for linear Economic MPC of a Power Plant Portfolio, 10 th European Workshop on advanced control and diagnosis, ACD 2012. 19 / 34

CLE SecondIN outcome PRESS al. / Journal of Process Control xxx (2014) xxx xxx ce settings. 0K CPU @ 1 Pro opersolving the ating units. Wempc and he OCP for lgorithms. h accuracy, Table 4 is Fig. 7. CPU time for the different solvers as a function of the number of power generating units, M. Leo Emil Sokoler, Laura Standardi, Kristian Edlund, Niels Kjølstad Poulsen, John Bagterp Jørgensen, A Dantzig-Wolfe decomposition algorithm for linear economic model predictive control of dynamically decoupled systems, Journal of Process Control 2014. an attractive optimization algorithm for large scale dynamically decoupled energy management problems. Note from Fig. 6 that ADMMempc needs many more iterations to converge than DWempc for the high accuracy tolerance specification, (h). Table 5 further shows that the number of iterations increases 20 / 34

Suboptimal solution Suboptimal solutions but feasible. This brings a reduction in computation time. Extra costs might occur. Early termination Fix a maximum number of iterations to the DW algorithm. The solution is not optimal, but feasible. 22 / 34

Early termination Setup RMP with initial feasible vertex via DW representation Solve RMP and send simplex multipliers to subproblems Update cost coefficients of each subproblem. Solve all subproblems. Add one column to the RMP. Suboptimal solution found. Optimal solution found. Yes Yes Is the number of iterations a fixed number? No Is the optimality condition satisfied? 23 / 34

Early termination: set bound on iterations a priori First iteration: 10 vertices provides the optimal value but the algorithm stops after 32 extreme points. 10 8 Obj. Func. (time step 1) 10 6 0 5 10 15 20 25 30 35 10 4 Obj. Func. (time step 100) 10 2 0 5 10 15 20 25 30 35 Number of extreme points l 100 th iteration: the warm-start reduces the number of vertices required to 13 but 3 are enough to compute the optimal value. 24 / 34

Early termination performances: computational time Distributions based on 20 stochastic simulations. 8 8 Exact Dantzig-Wolfe algorithm. Early termination 15 vertices. Early termination 10 vertices. Early termination 5 vertices. 6 4 2 0 200 400 600 800 CPU time [s] 15 10 5 6 4 2 0 15 10 5 400 450 500 CPU time [s] 0 200 300 400 CPU time [s] 0 100 120 140 160 180 CPU time [s] 25 / 34

Early termination performances: costs Distributions based on 20 stochastic simulations. 8 6 Exact Dantzig-Wolfe algorithm. Early termination 15 vertices. Early termination 10 vertices. Early termination 5 vertices. 6 4 2 0 6 4 2 0.2 0.8 Costs [million euro] 4 2 0 8 6 4 2 0.2 0.8 Costs [million euro] 0 0.2 1 Costs [million euro] 0 0.6 1.4 Costs [million euro] 26 / 34

Third outcome: Early termination Laura Standardi, Leo Emil Sokoler, Niels Kjølstad Poulsen, John Bagterp Jørgensen, Computational Efficiency of Economic MPC for Power System Operation, 4 th IEEE PES Innovative Smart Grid Technologies Europe, 2013. 27 / 34

Reduced Dantzig-Wolfe decomposition No need of max. number of iterations set a priori. Reduce computation time. Improve applicability of the controller. 28 / 34

Reduced Dantzig-Wolfe Initial feasible vertex for RMP coeff. Solve RMP and send simplex multipliers to subproblems Update cost coefficients of each subproblem left. Solve subproblems left. Add one column to the RMP only for the subproblems left. Yes Is the optimality condition satisfied for any subproblem? These subproblem are not solved. All suproblems have satisfied opt. condition Subptimal solution found. 29 / 34

Forth outcome: reduced Dantzig-Wolfe 10 5 CPU time [s] 10 4 DW Reduced DW 10 3 20 40 60 80 100 120 140 160 Power units 10 8 Obj. func.[euro] 10 6 DW Reduced DW 10 4 20 40 60 80 100 120 140 160 Power units 30 / 34

Forth outcome: reduced Dantzig-Wolfe Laura Standardi, John Bagterp Jørgensen, Niels Kjølstad Poulsen, A reduced Dantzig-Wolfe decomposition for a suboptimal linear MPC, 19 th IFAC World Congress 2014. 31 / 34

Papers 1. A reduced Dantzig-Wolfe decomposition for a suboptimal linear MPC, L.Standardi, N. Kjølstad Poulsen, J. Bagterp Jørgensen, 19 th IFAC World Congress, 2014. 2. A Dantzig-Wolfe decomposition algorithm for linear economic model predictive control of dynamically decoupled subsystems, L. Emil Sokoler, L. Standardi, K. Edlund, J. Bagterp Jørgensen, N. Kjølstad Poulsen, Journal of Process Control, 2014. 3. Comptuational efficiency of Economic MPC for power systems operation, L.Standardi,L. Emil Sokoler, N. Kjølstad Poulsen, J. Bagterp Jørgensen, 4 th IEEE PES ISGT, 2013. 4. A decomposition algorithm for optimal control of distributed energy system, L. Emil Sokoler, L.Standardi, N. Kjølstad Poulsen, J. Bagterp Jørgensen, 4 th IEEE PES ISGT, 2013. 5. Early termination of Dantzig-Wolfe algorithm for economic MPC, L.Standardi, L. Emil Sokoler, N. Kjølstad Poulsen, J. Bagterp Jørgensen, 18 th NPCW, 2013. 6. A Dantzig-Wolfe decomposition algorithm for economic MPC of distributed energy systems, L. Emil Sokoler, L.Standardi, N. Kjølstad Poulsen, J. Bagterp Jørgensen, 18 th NPCW, 2013. 7. A Dantzig-Wolfe decomposition algorithm for linear economic MPC of a power plant portfolio, L.Standardi, L. Emil Sokoler, N. Kjølstad Poulsen, J. Bagterp Jørgensen, ACD 2012. 33 / 34

Conclusion Over the last three years we have: Speeded up controllers applied to smart grids, Increased scalability of the control systems, Analysed the consequences of suboptimal control sequences, Cooperated with DONG energy. 34 / 34