A New Predictor-Corrector Method for Optimal Power Flow
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1 1 st Brazilian Workshop on Interior Point Methods Federal Univesity of Technology Paraná (UTFPR) A New Predictor-Corrector Method for Optimal Power Flow Roy Wilhelm Probst April, Campinas, Brazil Roy Wilhelm Probst A New PC Method for Optimal Power Flow 1 / 20
2 Linear Programming Nonlinear Programming Complete PC (CPC) Affine-Scaling and Central Path Methods min s.t. c t x Ax = b x 0 A A t I A A t I x y z x y z or = = b Ax c A t z XZu b Ax c A t z µu XZu Roy Wilhelm Probst A New PC Method for Optimal Power Flow 2 / 20
3 Linear Programming Nonlinear Programming Complete PC (CPC) Affine-Scaling and Central Path Methods min s.t. c t x Ax = b x 0 A A t I A A t I x y z x y z or = = b Ax c A t z XZu b Ax c A t z µu XZu Roy Wilhelm Probst A New PC Method for Optimal Power Flow 2 / 20
4 Predictor-Corrector Method Linear Programming Nonlinear Programming Complete PC (CPC) min s.t. c t x Ax = b x 0 A A t I x ỹ z = r 1 r 2 r 3 A A t I ˆx ŷ ẑ = 0 0 µu X Zu Roy Wilhelm Probst A New PC Method for Optimal Power Flow 3 / 20
5 Linear Programming Nonlinear Programming Complete PC (CPC) PC Method for Nonlinear Programming min f (x) s.t. g(x) = 0 x 0 x g(x) xxl(x, y, z) x g(x) t I x ỹ z = r 1 r 2 r 3 x g(x) xxl(x, y, z) x g(x) t I x ỹ z = 0 0 µu X Zu Roy Wilhelm Probst A New PC Method for Optimal Power Flow 4 / 20
6 Linear Programming Nonlinear Programming Complete PC (CPC) PC Method for Nonlinear Programming min f (x) s.t. g(x) = 0 x 0 x g(x) xxl(x, y, z) x g(x) t I x ỹ z = r 1 r 2 r 3 x g(x) xxl(x, y, z) x g(x) t I x ỹ z = 0 0 µu X Zu Roy Wilhelm Probst A New PC Method for Optimal Power Flow 4 / 20
7 Complete PC Corrections Linear Programming Nonlinear Programming Complete PC (CPC) min s.t. x t Hx XAx = b x 0 XZu = 0 Z x + X z = XZu (X + X )(Z + Z)u = X Zu Roy Wilhelm Probst A New PC Method for Optimal Power Flow 5 / 20
8 Complete PC Corrections Linear Programming Nonlinear Programming Complete PC (CPC) min s.t. x t Hx XAx = b x 0 XZu = 0 Z x + X z = XZu (X + X )(Z + Z)u = X Zu Roy Wilhelm Probst A New PC Method for Optimal Power Flow 5 / 20
9 Primal Corrections Interior-Point Methods Linear Programming Nonlinear Programming Complete PC (CPC) min s.t. x t Hx XAx = b x 0 XAx b = 0 (XA + diag(ax)) x = b XAx (X + X )A(x + x) b = XA x Roy Wilhelm Probst A New PC Method for Optimal Power Flow 6 / 20
10 Dual Corrections Interior-Point Methods Linear Programming Nonlinear Programming Complete PC (CPC) min s.t. x t Hx XAx = b x 0 2Hx + A t Yx + YAx = 0 (2H + A t Y + YA) x + (A t X + diag(ax)) y = 2Hx A t Yx YAx = A t Y x + YA x Roy Wilhelm Probst A New PC Method for Optimal Power Flow 7 / 20
11 Linear Programming Nonlinear Programming Complete PC (CPC) Complete Predictor-Corrector Method min s.t. x t Hx XAx = b x 0 XA + diag(ax) 0 0 2H + YA + A t Y A t X + diag(ax) I x ỹ z = r 1 r 2 r I ˆx ŷ ẑ = X A X u Ỹ A X u A t Ỹ X u µu X Zu Roy Wilhelm Probst A New PC Method for Optimal Power Flow 8 / 20
12 IEEE 30 bus system Interior-Point Methods Example Model T k = e k + jf k e k : Voltage Real part f k : Voltage Complex part 3 6 P k : Active Power liquid injection Barras de Geração Barras de Carga Q k : Reactive Power liquid injection Roy Wilhelm Probst A New PC Method for Optimal Power Flow 9 / 20
13 Model Interior-Point Methods Example Model min e t Ge + f t Gf s.t. P k (e, f ) + P Ck P Gk = 0 k C Q k (e, f ) + Q Ck Q Gk = 0 k C (vk min ) 2 V k (e, f ) (vk max ) 2 k N Pk min P k (e, f ) Pk max k G Qk min Q k (e, f ) Qk max k R where: P = EGe + FGf + FBe EBf Q = FGe EGf EBe FBf V = Ee + Ff Roy Wilhelm Probst A New PC Method for Optimal Power Flow 10 / 20
14 CPC Corrections Interior-Point Methods Example Model [ x P( x) ˆr 1 = t ỹ p + x P( x) t z 1p x P( x) t z 2p x Q( x) t ỹ q + x Q( x) t z 1q x Q( x) t z 2q ] ˆr 2 = ˆr 4 = S 1v z 1v µu S 1p z 1p µu S 1q z 1q µu V ( x) P( x) Q( x) ˆr 5 = ˆr 3 = V ( x) P( x) Q( x) S 2v z 2v µu S 2p z 2p µu S 2q z 2q µu ˆr 6 = V ( x) P( x) Q( x) Roy Wilhelm Probst A New PC Method for Optimal Power Flow 11 / 20
15 CPC Corrections (cont.) Example Model where: P( x) = ẼG ẽ + F G f + F B ẽ ẼB f Q( x) = F G ẽ ẼG f ẼB ẽ F B f V ( x) = Ẽ ẽ + F f [ ] ẼG + F B + diag(g ẽ B f ) x P( x) = F G ẼB + diag(g f + B ẽ) [ ] F G ẼB diag(g f + B ẽ) x Q( x) = B + diag(g ẽ B f ) Roy Wilhelm Probst A New PC Method for Optimal Power Flow 12 / 20
16 Example Model OPF using Cartesian coordinates Quadratic objective function Quadratic constraints Constant Hessian Nonlinear corrections in all optimality conditions Roy Wilhelm Probst A New PC Method for Optimal Power Flow 13 / 20
17 Test cases Results MATLAB 7.8 (R2009a) Windows XP Intel Core 2 Quad Q9550 2,83 GHz 3,23 GB of RAM Roy Wilhelm Probst A New PC Method for Optimal Power Flow 14 / 20
18 Power systems test cases Test cases Results Buses and lines System N A R C M IEEE IEEE IEEE BRAZIL Roy Wilhelm Probst A New PC Method for Optimal Power Flow 15 / 20
19 Iteration count and CPU time Test cases Results v k [0.90, 1.10] Iterations CPU Time System PD PC CPC PD PC CPC IEEE IEEE IEEE Roy Wilhelm Probst A New PC Method for Optimal Power Flow 16 / 20
20 Iteration count and CPU time Test cases Results v k [0.95, 1.05] Iterations CPU Time System PD PC CPC PD PC CPC IEEE IEEE IEEE Roy Wilhelm Probst A New PC Method for Optimal Power Flow 17 / 20
21 Test cases Results Active constraints v k at optimal solution v k [0.90, 1.10] v k [0.95, 1.05] System v max v min v max v min IEEE IEEE IEEE Roy Wilhelm Probst A New PC Method for Optimal Power Flow 18 / 20
22 Iteration count and CPU time Test cases Results BRAZIL system P max / P C Iterations CPU Time PD PC CPC PD PC CPC Roy Wilhelm Probst A New PC Method for Optimal Power Flow 19 / 20
23 Interior-Point Methods Quadratic objective function and constraints OPF Problem Complete Predictor-Corrector Nonlinear Corrections Robustness and Speed Heuristic Roy Wilhelm Probst A New PC Method for Optimal Power Flow 20 / 20
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