Verteilte modellprädiktive Regelung intelligenter Stromnetze

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1 Verteilte modellprädiktive Regelung intelligenter Stromnetze Institut für Mathematik Technische Universität Ilmenau in Zusammenarbeit mit Philipp Braun, Lars Grüne (U Bayreuth) und Christopher M. Kellett, Steven R. Weller (U Newcastle, Australien) Workshop Mathematische Systemtheorie, Elgersburg, 9. Februar 2016 Institut für Mathematik Seite 1 / 22

2 Outline Residential Energy System Setting and usage of batteries Simple control Institut für Mathematik Seite 2 / 22

3 Outline Residential Energy System Setting and usage of batteries Simple control Model predictive control (MPC) Decentralized MPC flexibility Institut für Mathematik Seite 2 / 22

4 Outline Residential Energy System Setting and usage of batteries Simple control Model predictive control (MPC) Decentralized MPC flexibility Centralized MPC performance Institut für Mathematik Seite 2 / 22

5 Outline Residential Energy System Setting and usage of batteries Simple control Model predictive control (MPC) Decentralized MPC flexibility Centralized MPC Distributed MPC performance Examples Synthetic energy profiles Australian data set Conclusions, Extensions, and & Outlook Institut für Mathematik Seite 2 / 22

6 Motivation Average power consumption of 100 households Power in [kw] Power in [kw] time in hours Power in [kw] time in hours Average power generation (using solar photovolatic panels) time in hours Power in [kw] time in hours (Dataset provided by electricity company Ausgrid. Sample of power consumption/generation of 100 households during one day/week) single household; average at specific time steps; average during the day/week Motivation Institut für Mathematik Seite 3 / 22

7 Motivation Average power consumption of 100 households Power in [kw] Power in [kw] time in hours Power in [kw] time in hours Average power generation (using solar photovolatic panels) time in hours Power in [kw] time in hours (Dataset provided by electricity company Ausgrid. Sample of power consumption/generation of 100 households during one day/week) single household; average at specific time steps; average during the day/week Question: do peaks of power generation/consumption match? Motivation Institut für Mathematik Seite 3 / 22

8 Motivation (continued) Average power demand Power in [kw] time in hours Power in [kw] Consumption 0 Generation Demand Average demand time in hours (Dataset provided by electricity company Ausgrid. Sample of power consumption/generation of 100 households during one day/week) Motivation Institut für Mathematik Seite 4 / 22

9 Motivation (continued) Average power demand Power in [kw] = = time in hours Power in [kw] Consumption 0 Generation Demand Average demand time in hours (Dataset provided by electricity company Ausgrid. Sample of power consumption/generation of 100 households during one day/week) Goal: flatten the average power demand Motivation Institut für Mathematik Seite 4 / 22

10 Motivation (continued) Average power demand Power in [kw] = = time in hours Power in [kw] Consumption 0 Generation Demand Average demand time in hours (Dataset provided by electricity company Ausgrid. Sample of power consumption/generation of 100 households during one day/week) Goal: flatten the average power demand Idea: use distributed energy storages Motivation Institut für Mathematik Seite 4 / 22

11 Electricity Network Connected to the Grid + - RES 1 RES G.O RES I G.O.: Grid operator RES i : Residential energy system i, system dynamics: x i (k + 1) = f i (x i (k), u i (k)) z i (k) = h i (u i (k), w i (k)) I: Number of RESs Network parameters: w i : Power demand (without storage device; known sequence) z i : x i : u i : Power demand (with storage device); communication variable State of the storage device Control input (charging/discharging of the storage device) Electricity Network Institut für Mathematik Seite 5 / 22

12 Residential Energy System System dynamics of RES i (i = 1,..., I): x i (k + 1) = α i x i (k) +T (β i u + i (k) + u i (k)) z i (k) = w i (k) + (u + i (k) + γ i u i (k)) with α i, β i, γ i (0, 1] and sampling interval T (T = 0.5 [h]) Electricity Network Institut für Mathematik Seite 6 / 22

13 Residential Energy System System dynamics of RES i (i = 1,..., I): x i (k + 1) = α i x i (k) +T (β i u + i (k) + u i (k)) z i (k) = w i (k) + (u + i (k) + γ i u i (k)) with α i, β i, γ i (0, 1] and sampling interval T (T = 0.5 [h]) Constraints: capacity of the battery; charging/discharging rates 0 x i (k) C i k N, ([kwh]) u i u i (k) 0 k N, ([kw]) 0 u + i (k) u i k N, ([kw]) 0 u i (k)/u i + u + i (k)/u i 1 k N, Electricity Network Institut für Mathematik Seite 6 / 22

14 Residential Energy System System dynamics of RES i (i = 1,..., I): x i (k + 1) = α i x i (k) +T (β i u + i (k) + u i (k)) z i (k) = w i (k) + (u + i (k) + γ i u i (k)) with α i, β i, γ i (0, 1] and sampling interval T (T = 0.5 [h]) Constraints: capacity of the battery; charging/discharging rates 0 x i (k) C i k N, ([kwh]) u i u i (k) 0 k N, ([kw]) 0 u + i (k) u i k N, ([kw]) 0 u i (k)/u i + u + i (k)/u i 1 k N, Observations: Subsystems are physically decoupled. Electricity Network Institut für Mathematik Seite 6 / 22

15 Residential Energy System System dynamics of RES i (i = 1,..., I): x i (k + 1) = α i x i (k) +T (β i u + i (k) + u i (k)) z i (k) = w i (k) + (u + i (k) + γ i u i (k)) with α i, β i, γ i (0, 1] and sampling interval T (T = 0.5 [h]) Constraints: capacity of the battery; charging/discharging rates 0 x i (k) C i k N, ([kwh]) u i u i (k) 0 k N, ([kw]) 0 u + i (k) u i k N, ([kw]) 0 u i (k)/u i + u + i (k)/u i 1 k N, Observations: Subsystems are physically decoupled. Constraint satisfaction independent of other subsystems Electricity Network Institut für Mathematik Seite 6 / 22

16 Residential Energy System System dynamics of RES i (i = 1,..., I): x i (k + 1) = α i x i (k) +T (β i u + i (k) + u i (k)) z i (k) = w i (k) + (u + i (k) + γ i u i (k)) with α i, β i, γ i (0, 1] and sampling interval T (T = 0.5 [h]) Constraints: capacity of the battery; charging/discharging rates 0 x i (k) C i k N, ([kwh]) u i u i (k) 0 k N, ([kw]) 0 u + i (k) u i k N, ([kw]) 0 u i (k)/u i + u + i (k)/u i 1 k N, Observations: Subsystems are physically decoupled. Constraint satisfaction independent of other subsystems Conclusion: decentralized control possible (no communication). Electricity Network Institut für Mathematik Seite 6 / 22

17 Three Synthetic Energy Profiles Time varying power demand (w i (k)) k N0 of RES i (i = 1, 2,..., I). 4 Periodic energy profile 3 2 Power in [KW] RES 1: no battery time in hours Electricity Network Institut für Mathematik Seite 7 / 22

18 Three Synthetic Energy Profiles Time varying power demand (w i (k)) k N0 of RES i (i = 1, 2,..., I). Power in [KW] Periodic energy profile Simple strategy: If (generation > load): charge battery If (load > generation): discharge battery 3 RES 1: no battery 4 RES 1: with battery time in hours Electricity Network Institut für Mathematik Seite 7 / 22

19 Three Synthetic Energy Profiles Time varying power demand (w i (k)) k N0 of RES i (i = 1, 2,..., I). Power in [KW] RES 1: no battery 4 RES 1: with battery time in hours Periodic energy profile Simple strategy: If (generation > load): charge battery If (load > generation): discharge battery Initial state: 0.5 [KWh] C i = 4, u i = 0.5, ū i = 0.5 Electricity Network Institut für Mathematik Seite 7 / 22

20 Three Synthetic Energy Profiles Time varying power demand (w i (k)) k N0 of RES i (i = 1, 2,..., I). Power in [KW] RES 1 4 RES time in hours Periodic energy profile Simple strategy: If (generation > load): charge battery If (load > generation): discharge battery Initial state: 0.5 [KWh] C i = 4, u i = 0.5, ū i = 0.5 Electricity Network Institut für Mathematik Seite 7 / 22

21 Three Synthetic Energy Profiles Time varying power demand (w i (k)) k N0 of RES i (i = 1, 2,..., I). Power in [KW] RES 1 RES 2 4 RES time in hours Periodic energy profile Simple strategy: If (generation > load): charge battery If (load > generation): discharge battery Initial state: 0.5 [KWh] C i = 4, u i = 0.5, ū i = 0.5 Electricity Network Institut für Mathematik Seite 7 / 22

22 Three Synthetic Energy Profiles Time varying power demand (w i (k)) k N0 of RES i (i = 1, 2,..., I). Power in [KW] Periodic energy profile Simple strategy: If (generation > load): charge battery If (load > generation): discharge battery RES 1 3 RES 2 RES 3 4 average time in hours Grid operator s point of view: average Initial state: 0.5 [KWh] C i = 4, u i = 0.5, ū i = 0.5 Electricity Network Institut für Mathematik Seite 7 / 22

23 Three Synthetic Energy Profiles Time varying power demand (w i (k)) k N0 of RES i (i = 1, 2,..., I). Power in [KW] Periodic energy profile Simple strategy: If (generation > load): charge battery If (load > generation): discharge battery RES 1 3 RES 2 RES 3 4 average time in hours Grid operator s point of view: average Particularly interesting: consumption peaks Simple strategy performs poorly Initial state: 0.5 [KWh] C i = 4, u i = 0.5, ū i = 0.5 Electricity Network Institut für Mathematik Seite 7 / 22

24 Three Synthetic Energy Profiles Time varying power demand (w i (k)) k N0 of RES i (i = 1, 2,..., I). Power in [KW] Periodic energy profile Simple strategy: If (generation > load): charge battery If (load > generation): discharge battery RES 1 3 RES 2 RES 3 4 average time in hours Initial state: 0.5 [KWh] C i = 4, u i = 0.5, ū i = 0.5 Grid operator s point of view: average Particularly interesting: consumption peaks Simple strategy performs poorly Idea: take forecast into accout model predictive control Electricity Network Institut für Mathematik Seite 7 / 22

25 Principle of Model Predictive Control Idea: receding horizon control (MPC) 1. Obtain state measurement Model Predictive Control Institut für Mathematik Seite 8 / 22

26 Principle of Model Predictive Control Idea: receding horizon control (MPC) 1. Obtain state measurement 2. Predict system and optimize input Model Predictive Control Institut für Mathematik Seite 8 / 22

27 Principle of Model Predictive Control Idea: receding horizon control (MPC) 1. Obtain state measurement 2. Predict system and optimize input 3. Apply optimal input value Model Predictive Control Institut für Mathematik Seite 8 / 22

28 Principle of Model Predictive Control Idea: receding horizon control (MPC) 1. Obtain state measurement 2. Predict system and optimize input 3. Apply optimal input value Model Predictive Control Institut für Mathematik Seite 8 / 22

29 Principle of Model Predictive Control Idea: receding horizon control (MPC) 1. Obtain state measurement 2. Predict system and optimize input 3. Apply optimal input value Model Predictive Control Institut für Mathematik Seite 8 / 22

30 Principle of Model Predictive Control Idea: receding horizon control (MPC) 1. Obtain state measurement 2. Predict system and optimize input 3. Apply optimal input value Model Predictive Control Institut für Mathematik Seite 8 / 22

31 Principle of Model Predictive Control Idea: receding horizon control (MPC) 1. Obtain state measurement 2. Predict system and optimize input 3. Apply optimal input value Model Predictive Control Institut für Mathematik Seite 8 / 22

32 Principle of Model Predictive Control Idea: receding horizon control (MPC) Control via repeated prediction & optimization 1. Obtain state measurement 2. Predict system and optimize input 3. Apply optimal input value Institut für Mathematik Seite 8 / 22 Thanks to B. Kern, OVG Universität Magdeburg. Model Predictive Control

33 Model Predictive Control: Algorithm Basic Idea: optimize on a finite horizon N < Model Predictive Control Institut für Mathematik Seite 9 / 22

34 Model Predictive Control: Algorithm Basic Idea: optimize on a finite horizon N < Algorithm: 1. Measure current state x(k). Model Predictive Control Institut für Mathematik Seite 9 / 22

35 Model Predictive Control: Algorithm Basic Idea: optimize on a finite horizon N < Algorithm: 1. Measure current state x(k). 2. Minimize N 1 n=0 l(n + k, ˆx û(n; x(k)), û(n)) subject to system dynamics control & state constraints optimal control û = (û (0), û (1),..., û (N 1)) Model Predictive Control Institut für Mathematik Seite 9 / 22

36 Model Predictive Control: Algorithm Basic Idea: optimize on a finite horizon N < Algorithm: 1. Measure current state x(k). 2. Minimize N 1 n=0 l(n + k, ˆx û(n; x(k)), û(n)) subject to system dynamics control & state constraints optimal control û = (û (0), û (1),..., û (N 1)) 3. Apply first control value û (0) an Model Predictive Control Institut für Mathematik Seite 9 / 22

37 Model Predictive Control: Algorithm Basic Idea: optimize on a finite horizon N < Algorithm: 1. Measure current state x(k). 2. Minimize N 1 n=0 l(n + k, ˆx û(n; x(k)), û(n)) subject to system dynamics control & state constraints optimal control û = (û (0), û (1),..., û (N 1)) 3. Apply first control value û (0) an Define (static) state feedback µ N mittels µ N (ˆx) := û (0). Model Predictive Control Institut für Mathematik Seite 9 / 22

38 Model Predictive Control: Algorithm Basic Idea: optimize on a finite horizon N < Algorithm: 1. Measure current state x(k). 2. Minimize N 1 n=0 l(n + k, ˆx û(n; x(k)), û(n)) subject to system dynamics control & state constraints optimal control û = (û (0), û (1),..., û (N 1)) 3. Apply first control value û (0) an Define (static) state feedback µ N mittels µ N (ˆx) := û (0). Observation: running costs l needed (degree of freedom) Model Predictive Control Institut für Mathematik Seite 9 / 22

39 MPC from the Trajectory Point of View Past State N=4 Future x Control N=4 Model Predictive Control Institut für Mathematik Seite 10 / 22

40 MPC from the Trajectory Point of View Past Future State N=4 x Control N=4 u (0) u (1) u (2) u (3) Model Predictive Control Institut für Mathematik Seite 10 / 22

41 MPC from the Trajectory Point of View Past Future State N=4 x Control N=4 u (0) u (1) u (2) u (3) Model Predictive Control Institut für Mathematik Seite 10 / 22

42 MPC from the Trajectory Point of View Past State x N=4 Future Control N=4 Model Predictive Control Institut für Mathematik Seite 10 / 22

43 MPC from the Trajectory Point of View Past Future State x N=4 N=4 Control u (0) u (1) u (2) u (3) Model Predictive Control Institut für Mathematik Seite 10 / 22

44 MPC from the Trajectory Point of View Past Future State x N=4 N=4 Control u (0) u (1) u (2) u (3) Model Predictive Control Institut für Mathematik Seite 10 / 22

45 MPC from the Trajectory Point of View Past Future State x N=4 N=4 Control u (0) u (1) u (2) u (3) Model Predictive Control Institut für Mathematik Seite 10 / 22

46 MPC from the Trajectory Point of View Past State x N=4 Future Control N=4 u (0) Model Predictive Control Institut für Mathematik Seite 10 / 22

47 Decentralized and Centralized MPC Opt. criterion: minimize the deviation from a given reference Decentralized MPC decoupled objective Local reference value for RES i : ζ i (k) := 1 N 1 N n=0 w i(k + n) minimize N 1 n=0 ẑ i(n) ζ i (k) 2 subject to local constraints Optimization Algorithms Institut für Mathematik Seite 11 / 22

48 Decentralized and Centralized MPC Opt. criterion: minimize the deviation from a given reference Decentralized MPC decoupled objective Local reference value for RES i : ζ i (k) := 1 N 1 N n=0 w i(k + n) minimize N 1 n=0 ẑ i(n) ζ i (k) 2 subject to local constraints Centralized MPC coupled objective Global reference value: ζ(k) := 1 I I i=1 ζ i(k) minimize N 1 n=0 1 I I i=1 ẑi(n) ζ(k) 2 subject to global constraints Optimization Algorithms Institut für Mathematik Seite 11 / 22

49 Decentralized and Centralized MPC Opt. criterion: minimize the deviation from a given reference Decentralized MPC decoupled objective Local reference value for RES i : ζ i (k) := 1 N 1 N n=0 w i(k + n) minimize N 1 n=0 ẑ i(n) ζ i (k) 2 subject to local constraints Centralized MPC coupled objective Global reference value: ζ(k) := 1 I I i=1 ζ i(k) minimize N 1 n=0 1 I I i=1 ẑi(n) ζ(k) 2 subject to global constraints Trade-off: Flexibility versus Performance (Optimality) Optimization Algorithms Institut für Mathematik Seite 11 / 22

50 Closed Loop Performance Performance (G.O.) z in [kw] Uncontrolled Decentralized MPC Centralized MPC x in [kwh] Average Battery Profiles Time in hours Time in hours Setting: 100 RES; 1 week simulation length prediction horizon 24[h]; sampling time 0.5[h] maximal charging/discharging rates per hour: 0.3[kWh] no loss in the battery model Optimization Algorithms Institut für Mathematik Seite 12 / 22

51 Hierarchical Distributed Optimization Scheme RES i : measure x i (k), define and transmit ẑ 1 i = ẑ 1 i = w i. Grid operator: compute and broadcast (global) average demand ζ For l = 1, 2,..., l max (maximal iteration number) Grid operator Step size θ [0, 1]: minimize I (θẑ l i + (1 θ)ẑi l ) 1 ζ I i=0 ẑ l+1 = θẑ l + (1 θ)ẑ l I i=1 ẑl+1 Compute Π l (n) = 1 I i (n) (predicted average demand) Evaluate N 1 n=0 ( ζ Π l (n)) 2 (performance index) Broadcast Π l and θ l 2 Compute ẑ l+1 i RES i = θẑ l i + (1 θ)ẑ l i Minimize ( ) Π l ẑl+1 i ẑ i 1 ζ I subject to the local constraints (unique) minimizer ẑ l+1 i Transmit ẑ l+1 i 2 Optimization Algorithms Institut für Mathematik Seite 13 / 22

52 Hierarchical Distributed Optimization Scheme Properties: Size of local minimization problems is independent of the number of RESs N 1 n=0 ( ζ Π l (n) + I 1 ẑ l+1 i (n) I 1 ẑ i (n)) 2 Optimization Algorithms Institut für Mathematik Seite 14 / 22

53 Hierarchical Distributed Optimization Scheme Properties: Size of local minimization problems is independent of the number of RESs N 1 n=0 ( ζ Π l (n) + I 1 ẑ l+1 i (n) I 1 ẑ i (n)) 2 Grid operator solves a minimization problem in one variable Grid operator needs no knowledge on system dynamics and constraints of the RESs privacy of the RESs is maintained Optimization Algorithms Institut für Mathematik Seite 14 / 22

54 Hierarchical Distributed Optimization Scheme Properties: Size of local minimization problems is independent of the number of RESs N 1 n=0 ( ζ Π l (n) + I 1 ẑ l+1 i (n) I 1 ẑ i (n)) 2 Grid operator solves a minimization problem in one variable Grid operator needs no knowledge on system dynamics and constraints of the RESs privacy of the RESs is maintained Number of communication variables independent of the number of RESs connected to the grid Convergence to the solution of the centralized problem (works for linear systems with convex objective function w.r.t. convex, compact, and decoupled constraints) Optimization Algorithms Institut für Mathematik Seite 14 / 22

55 Sketch of the Convergence Proof (1/2) Performance index V l decreases in each iteration: V l+1 = N 1 2 ( ζ Π (j)) l j=0 = N 1 j=0 N 1 I j=0 i=0 ( ζ Π l 1 (j) + 1 I I i=1 θl (ẑi l ( 1 I 1 I I N 1 i=1 ( ζ Π l 1 (j) + 1 I (ẑl i ) 2 (j) ẑ l i (j)) )) 2 (j) ẑ l i (j)) ( ζ Π l 1 (j) + 1 ) 2 j=0 I (ẑl i (j) ẑ l i (j)) }{{} = N 1 ) 2 ( ζ Π l 1 (j) = V l j=0 N 1 j=0 ( ζ Π l 1 (j)) 2 Since V is positive, this implies convergence of V l. Optimization Algorithms Institut für Mathematik Seite 15 / 22

56 Sketch of the Convergence Proof (2/2) Convergence to the centralized solution V. Assume V > V. Feasible set compact = feasible z : V (z ) = V Optimization Algorithms Institut für Mathematik Seite 16 / 22

57 Sketch of the Convergence Proof (2/2) Convergence to the centralized solution V. Assume V > V. Feasible set compact = feasible z : V (z ) = V Show that V ( z + ) < V ( z) holds for all z with V ( z) > V Assume V ( z + ) = V ( z) = z + = z. Consider F (η) := N 1 j=0 ( ζ 1 I I i=1 ( (1 η i ) z i (j) + η i z i (j) ) ) 2 with F(1 I ) = V < V ( z) = F(0 I ) F convex = 0 > grad F(0 I ), 1 I = i : F η i (0 I ) < 0 = i-th RES updates z i Optimization Algorithms Institut für Mathematik Seite 16 / 22

58 Sketch of the Convergence Proof (2/2) Convergence to the centralized solution V. Assume V > V. Feasible set compact = feasible z : V (z ) = V Show that V ( z + ) < V ( z) holds for all z with V ( z) > V Assume V ( z + ) = V ( z) = z + = z. Consider F (η) := N 1 j=0 ( ζ 1 I I i=1 ( (1 η i ) z i (j) + η i z i (j) ) ) 2 with F(1 I ) = V < V ( z) = F(0 I ) F convex = 0 > grad F(0 I ), 1 I = i : F η i (0 I ) < 0 = i-th RES updates z i Continuity of V = accumulation point z : V (z ) = V = V ( z + ) < V z B ε (z ) (local costs are cont. w.r.t. parameters) = contradiction to monotonicity Optimization Algorithms Institut für Mathematik Seite 16 / 22

59 Numerical Experiments Number of iterations to obtain an error of less than ε = 10 1, 10 2,..., 10 5 in comparison to the centralized solution. without warm-start with warm-start Number of iterations Number of iterations Time index k Time index k Setting: 20 RESs; prediction horizon 24[h]; sampling time 0.5[h] Numerical Experiments (1/4) Institut für Mathematik Seite 17 / 22

60 Numerical Experiments (2/4) Number of iterations needed to obtain an error below 10 2 depending on the number of RESs 80 Average number of iterations Number of RES without warm-start; with warm-start Setting: average out of 144 minimization problems Numerical Experiments (1/4) Institut für Mathematik Seite 18 / 22

61 Numerical Experiments (3/4) Closed loop performance with incomplete optimization Centralized MPC Distributed MPC Centralized MPC Distributed MPC z in [kw] z in [kw] Time in hours Time in hours with maximal iteration number l max = 1 (left) and l max = 3 (right) Setting: 100 RESs and simulation length of one week Numerical Experiments (1/4) Institut für Mathematik Seite 19 / 22

62 Numerical Experiments (4/4) Robustness of model predictive control normally distributed disturbance (ρ i (k)) k N0 (SD: 0.1) Disturbed prediction sequences leads to (j = 0,..., N 1) ẑ i (n) = û i (n) + w i (k + n) + n m=0 ρ i(k + m) Monte Carlo method: 1000 realization of disturbance sequences z in [kw] j m=0 ρi(m) Time in hours Time in hours Setting: 20 RESs Numerical Experiments (1/4) Institut für Mathematik Seite 20 / 22

63 Summary & Outlook Distributed optimization Flexibility due to local controller Iterative solution of the minimization problem Good performance with few iterations Optimal (Centralized) solution is obtained Privacy of the RESs is maintained Conclusions & Outlook Institut für Mathematik Seite 21 / 22

64 Summary & Outlook Distributed optimization Flexibility due to local controller Iterative solution of the minimization problem Good performance with few iterations Optimal (Centralized) solution is obtained Privacy of the RESs is maintained Current work Comparison with dual decomposition methods (ADMM) Development of cost functions covering real Energy Prices benefit for individual RESs by minimizing energy bill Maximal Islanding Time Conclusions & Outlook Institut für Mathematik Seite 21 / 22

65 Literatur (Auswahl) P. Braun, L. Grüne, C.M. Kellett, S.R. Weller, and K. Worthmann. A Real-Time Pricing Scheme for Residential Energy Systems Using a Market Maker, Proceedings of the Australian Control Conference (AUCC 2015), Gold Coast, Australia, P. Braun, L. Grüne, C.M. Kellett, S.R. Weller, and K. Worthmann. Model Predictive Control of Residential Energy Systems Using Energy Storage & Controllable Loads, 18th European Conference on Mathematics for Industry (ECMI 2014), Taormina, Italy. P. Braun, L. Grüne, C.M. Kellett, S.R. Weller, and K. Worthmann. A Distributed Optimization Algorithm for the Predictive Control of Smart Grids, IEEE Transactions on Automatic Control, to appear. P. Braun, L. Grüne, C.M. Kellett, S.R. Weller, and K. Worthmann. Predictive Control of a Smart Grid: A Distributed Optimization Algorithm with Centralized Performance Properties, Proceedings of the 54th IEEE Conference on Decision and Control (CDC 2015), Osaka, Japan, K. Worthmann, C.M. Kellett, P. Braun, L. Grüne, S.R. Weller. Distributed and decentralized Control of Residential Energy Systems incorporating Battery Storage, IEEE Transactions on Smart Grid 6(4), , K. Worthmann, C.M. Kellett, L. Grüne, and S.R. Weller. Distributed Control of Residential Energy Systems Using a Market Maker, Proceedings of the 19th IFAC World Congress 2014, Cape Town, South Africa, Literatur Institut für Mathematik Seite 22 / 22