Optimal Tap Settings for Voltage Regulation Transformers in Distribution Networks

Transcription

1 Optimal Tap Settings for Voltage Regulation Transformers in Distribution Networks Brett A. Robbins Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign May 9, 2014

2 Introduction Distribution systems are undergoing significant transformations in structure and functionality advanced communication, sensing, and control variable generation, e.g., photovoltaics storage-capable loads, e.g., plug-in electric vehicles 1/17

3 Introduction Distribution systems are undergoing significant transformations in structure and functionality advanced communication, sensing, and control variable generation, e.g., photovoltaics storage-capable loads, e.g., plug-in electric vehicles Incorporating DERs has many engineering challenges potential voltage rise and reversal of active power flow additional power demand variability introduced could impact component lifetimes 1/17

4 Controlled Behavior 1.05 Voltage [p.u.] Bus 2/17

5 Controlled Behavior 1.05 Voltage [p.u.] Bus 2/17

6 Controlled Behavior 1.05 Voltage [p.u.] Bus 2/17

7 Controlled Behavior 1.05 Tap Settings Voltage [p.u.] Reactive Power Support Bus 2/17

8 Controlled Behavior 1.05 This Presentation Voltage [p.u.] Different Presentation Bus 2/17

9 Overview Introduction Problem Derivation Transformer Model Semidefinite Programming Problem Statement Distributed vs. Centralized Solvers Distributed Solver Case Studies Concluding Remarks

10 Transformer Notation + V pt I pt a t : 1 + V s t y pts t I st + V st For transformer t we define p t is the primary-side bus s t is the secondary-side bus s t is the secondary-side virtual bus a t is the turns ratio 3/17

11 Transformer Notation V 1 y 1pt V pt V s t y pt s t V st S pt S pt s t S pt s t For transformer t we define p t is the primary-side bus s t is the secondary-side bus s t is the secondary-side virtual bus a t is the turns ratio S pts t is the power transferred through the transformer 3/17

12 Transformer Notation V 1 y 1pt V pt z t V s t y pt s t V st S pt S pt s t S pt s t For transformer t we define p t is the primary-side bus s t is the secondary-side bus s t is the secondary-side virtual bus a t is the turns ratio S pts t is the power transferred through the transformer 3/17

13 Set Notation Consider an n-bus power system with r transformers T := {1, 2,..., r} is the set of transformers N p := {p t t T } are buses incident to the primary-side N s := {s t t T } are buses incident to the secondary-side N s := {s t t T } are the secondary-side virtual buses N b := {1, 2,..., m} are standard system buses The set of physical buses will be and the augmented network is N = N b N p N s N a = N N s 4/17

14 Overview Introduction Problem Derivation Transformer Model Semidefinite Programming Problem Statement Distributed vs. Centralized Solvers Distributed Solver Case Studies Concluding Remarks

15 Semidefinite Programming Consider the positive semidefinite matrix W = VV H = V1 2 V 1 Vn+r..... V1 V n+r Vn+r 2, where V, S C (n+r) and W C (n+r) (n+r) has a rank of 1. 5/17

16 Semidefinite Programming Consider the positive semidefinite matrix W = VV H = V1 2 V 1 Vn+r..... V1 V n+r Vn+r 2, where V, S C (n+r) and W C (n+r) (n+r) has a rank of 1. The power flow equations S i = V i [Y ] ik Vk k H i can be rewritten linearly in W as S i = Tr (H i W ), H i C (n+r) (n+r) 5/17

17 Semidefinite Programming convex relaxation The graphical relationship between the original non-convex problem and the SDP relaxation rank(w ) = 1 rank(w ) > 1 Line Loss Limit P ik P ik Pki Pki 6/17

18 Overview Introduction Problem Derivation Transformer Model Semidefinite Programming Problem Statement Distributed vs. Centralized Solvers Distributed Solver Case Studies Concluding Remarks

19 Problem Statement The minimization problem will have the form Minimize Cost Function C( ) such that and Power Flow Power Flow on the Primary-Side Power Flow on the Virtual Secondary-Side Primary- and Secondary-Side Voltage Voltage and Tap Limits 7/17

20 Problem Statement such that [Y ] ptk min C (V, a) V, a, S ps [Y ] ik Vi Vk e jθ ik Si = 0, i N \N p k H i V pt V k e jθ pt k Spt + S pts t = 0, t T and k H p t k H s t [Y ] s t k V s t V k e jθ s t k S pts t = 0, t T V pt a t V s t = 0, t T V V i V, i N a a t a, t T 8/17

21 Problem Statement SDP Approach such that and min C (W ) W 0, S ps Tr (H i W ) S i = 0, i N \N p Tr (H pt W ) S pt + S pts t = 0, t T Tr ( H s t W ) S pts t = 0, t T V 2 [W ] ii V 2, i N a 2 [W ] ptp t [W ] s t s t a2 [W ] ptp t, t T and rank(w ) = 1 9/17

22 Problem Statement Convex Relaxation such that and min C (W ) W 0, S ps Tr (H i W ) S i = 0, i N \N p Tr (H pt W ) S pt + S pts t = 0, t T Tr ( H s t W ) S pts t = 0, t T V 2 [W ] ii V 2, i N a 2 [W ] ptp t [W ] s t s t a2 [W ] ptp t, t T 9/17

23 Overview Introduction Problem Derivation Transformer Model Semidefinite Programming Problem Statement Distributed vs. Centralized Solvers Distributed Solver Case Studies Concluding Remarks

24 Distributed Solver Our goal is to solve the optimization problem distributively computed with either a centralized or distributed system architecture the complexity scales with size of the partitions rather than the network size W = W = VV H 10/17

25 Distributed Solver Our goal is to solve the optimization problem distributively computed with either a centralized or distributed system architecture the complexity scales with size of the partitions rather than the network size W (1) W = 10/17

26 Distributed Solver Our goal is to solve the optimization problem distributively computed with either a centralized or distributed system architecture the complexity scales with size of the partitions rather than the network size W (1) W = W (2) 10/17

27 Distributed Solver Our goal is to solve the optimization problem distributively computed with either a centralized or distributed system architecture the complexity scales with size of the partitions rather than the network size W (1) W = W (2,1) W (2) 10/17

28 Distributed Solver Our goal is to solve the optimization problem distributively computed with either a centralized or distributed system architecture the complexity scales with size of the partitions rather than the network size W (1) W = W (1,2) W (2) 10/17

29 Boundary Conditions W (1) W (1,2) = W (2,1) W (2) The power flow equations are not separable due to the line (2, 3) introduce auxiliary variables E (1,2) = E (2,1) R 2 2 and F (1,2) = F (2,1) R 2 2 enforce in each area Re { W (1,2)} = E (1,2) Re { W (2,1)} = E (2,1) Im { W (1,2)} = F (1,2) Im { W (2,1)} = F (2,1) 11/17

30 Distributed Solver The boundary conditions enable us to reformulation the minimization problem separable per area Alternating Direction Method of Multipliers (ADMM) uses the Augmented Lagrangian L c (x, y) = f (x) + y T (Ax b) + c 2 Ax b 2 2, separate minimizations for local power flow and boundary variables solve using the dual ascent 12/17

31 Overview Introduction Problem Derivation Transformer Model Semidefinite Programming Problem Statement Distributed vs. Centralized Solvers Distributed Solver Case Studies Concluding Remarks

32 Partitions 15-Bus Distribution System The system is divided into two partitions: W (1) and W (2) The centralized problem will have 1763 optimization variables W (1) Feeder The distributed solution will have 1379 variables total 21.8% reduction same solution as the centralized scheme W (2) 13/17

33 Three-Phase Results 15-Bus Distribution System Tap Ratio t a t b tc C(W ) C(W ) C (1) (W (1) ) C (2) (W (2) ) ,000 Iteration ,000 Iteration Power [p.u.] 10 5 S pas a S pbs b S pc s c ,000 Iteration 14/17

34 IEEE 123-Bus Distribution System # # #3 # Feeder 15/17

35 IEEE 123-Bus Distribution System Comprehensive system that includes mostly unbalanced loads overhead/underground transmission lines with various phasing four voltage regulators required to solve for 259 phases divided into 6 areas The centralized solution vs. the distributed solution 269,854 vs. 59,436 optimization variables the objective function 6 ( C(W ) = vs C (i) W (i)) = i=1 16/17

36 Concluding Remarks Slow Time-Scale Control (Hardware Dispatch) improve distributed solver incorporate voltage regulation Fast Time-Scale Control (Reactive Power Support) time-varying active power injections/loads explore possible solvers Time domain simulations 17/17