AC Power Flows and their Derivatives using Complex Matrix Notation and Cartesian Coordinate Voltages

Transcription

1 AC Power Flows and their Derivatives using Complex Matrix Notation and Cartesian Coordinate Voltages Baljinnyam Sereeter Ray D. Zimmerman April 2, 2018 Matpower echnical Note 4 c 2008, 2010, 2011, 2017, 2018 Power Systems Engineering Research Center (Pserc All Rights Reserved Revision 1 October 25, See Section 8 for revision history details.

2 CONENS CONENS Contents 1 Notation 4 2 Introduction 5 3 Voltages Bus Voltages First Derivatives Second Derivatives Branch Voltages First Derivatives Reference Bus Voltage Angles First Derivatives Second Derivatives Bus Voltage Magnitude Limits First Derivatives Second Derivatives Branch Angle Difference Limits First Derivatives Second Derivatives Bus Injections Complex Current Injections First Derivatives Second Derivatives Complex Power Injections First Derivatives Second Derivatives Branch Flows Complex Currents First Derivatives Second Derivatives Complex Power Flows First Derivatives Second Derivatives Squared Current Magnitudes Squared Apparent Power Magnitudes

3 CONENS CONENS 5.5 Squared Real Power Magnitudes Generalized AC OPF Costs 23 7 Lagrangian of the AC OPF Nodal Current Balance First Derivatives Second Derivatives Nodal Power Balance First Derivatives Second Derivatives Revision History 29 Appendix A Scalar Polar Coordinate Derivatives 30 A.1 First Derivatives A.2 Second Derivatives References 32 3

4 1 NOAION 1 Notation n b, n g, n l, n r u i, w i number of buses, generators, branches, reference buses, respectively real and imaginary parts of bus voltage at bus i v i, θ i bus voltage magnitude and angle at bus i v i complex bus voltage at bus i, that is v i e jθ i or u i + jw i, W n b 1 vectors of real and imaginary parts of bus voltage V, Θ n b 1 vectors of bus voltage magnitudes and angles V I bus I f, I t S bus S f, S t S g n b 1 vector of complex bus voltages v i, + jw n b 1 vector of complex bus current injections n l 1 vectors of complex branch current injections, from and to ends n b 1 vector of complex bus power injections n l 1 vectors of complex branch power flows, from and to ends n g 1 vector of generator complex power injections P, Q real and reactive power flows/injections, S = P + jq M, N real and imaginary parts of current flows/injections, I = M + jn Y bus Y f, Y t C g C f, C t C ref [A] A A A b n b n b system bus admittance matrix n l n b system branch admittance matrices, from and to ends n b n g generator connection matrix (i, j th element is 1 if generator j is located at bus i, 0 otherwise n l n b branch connection matrices, from and to ends, (i, j th element is 1 if from end, or to end, respectively, of branch i is connected to bus j, 0 otherwise n r n b reference bus indicator matrix (i, j th element is 1 if bus j is i th reference bus, 0 otherwise diagonal matrix with vector A on the diagonal (non-conjugate transpose of matrix A complex conjugate of A matrix exponent for matrix A, or element-wise exponent for vector A 1 n, [1 n ] n 1 vector of all ones, n n identity matrix 0 appropriately-sized vector or matrix of all zeros 4

5 2 INRODCION 2 Introduction his document is a companion to Matpower echnical Note 2 [1] and Matpower echnical Note 3 [2]. he purpose of these documents is to show how the AC power balance and flow equations used in power flow and optimal power flow computations can be expressed in terms of complex matrices, and how their first and second derivatives can be computed efficiently using complex sparse matrix manipulations. he relevant code in Matpower [3,4] is based on the formulas found in these three notes. Matpower echnical Note 2 presents a standard formulation based on complex power flows and nodal power balances using a polar representation of bus voltages, Matpower echnical Note 3 adds the formulas needed for nodal current balances, and this note presents versions of both based on a cartesian coordinate representation of bus voltages. We will be looking at complex functions of the real valued vector X = W P g Q g. (1 For a complex scalar function f : R n C of a real vector X = [ x 1 x 2 x n ], we use the following notation for the first derivatives (transpose of the gradient f X = f [ X = f f x 1 x 2 he matrix of second partial derivatives, the Hessian of f, is f XX = 2 f X = ( 2 f x f 2 1 =. 2 X X f x n x 1 ] f x n. (2 2 f x 1 x n 2 f x 2 n For a complex vector function F : R n C m of a vector X, where. (3 F (X = [ f 1 (X f 2 (X f m (X ], (4 the first derivatives form the Jacobian matrix, where row i is the transpose of the gradient of f i. f 1 f F X = F x 1 1 x n X =..... (5 f m x 1 5 f m x n

6 3 VOLAGES In these derivations, the full 3-dimensional set of second partial derivatives of F will not be computed. Instead a matrix of partial derivatives will be formed by computing the Jacobian of the vector function obtained by multiplying the transpose of the Jacobian of F by a constant vector λ, using the following notation. F XX (α = ( ( FX λ X λ=α (6 Just to clarify the notation, if Y and Z are subvectors of X, then F Y Z (α = ( ( FY λ. (7 Z λ=α One common operation encountered in these derivations is the element-wise multiplication of a vector A by a vector B to form a new vector C of the same dimension, which can be expressed in either of the following forms C = [A] B = [B] A (8 It is useful to note that the derivative of such a vector can be calculated by the chain rule as 3 Voltages C X = C X 3.1 Bus Voltages = [A] B X + [B] A X = [A] B X + [B] A X (9 V is the n b 1 vector of complex bus voltages. he element for bus i is v i = u i +jw i. and W are the vectors of real and imaginary parts of the bus voltages. Consider also the vector of inverses of bus voltages 1 v i, denoted by Λ. Note that 1 v i = 1 u i + jw i = u i jw i u 2 i + w2 i = v i v i 2 (10 Λ = V 1 = [V] 2 V (11 Θ = tan 1 ( [] 1 W (12 V = ( 2 + W (13 6

7 3.1 Bus Voltages 3 VOLAGES First Derivatives V = V = [1 n b ] (14 V W = V = j [1 n b ] (15 Λ = Λ = [Λ]2 (16 Λ W = Λ = j [Λ]2 (17 he following could also be useful for implementing certain constraints on voltage magnitude or angles. For the derivations, see the scalar versions found in Appendix A Second Derivatives Θ = Θ = [V] 2 [W ] (18 Θ W = Θ = [V] 2 [] (19 V = V = [V] 1 [] (20 V W = V = [V] 1 [W ] (21 For the derivations, see the scalar versions found in Appendix A. Θ (λ = ( Θ λ (22 = 2 [λ] [V] 4 [] [W ] (23 Θ W (λ = ( Θ λ (24 7

8 3.2 Branch Voltages 3 VOLAGES = [λ] [V] 4 ( [W ] 2 [] 2 (25 Θ W (λ = ( ΘW λ (26 = [λ] [V] 4 ( [W ] 2 [] 2 (27 Θ W W (λ = ( ΘW λ (28 = 2 [λ] [V] 4 [] [W ] (29 V (λ = ( V λ (30 = [λ] [V] 3 [W ] 2 (31 V W (λ = ( V λ (32 = [λ] [V] 3 [] [W ] (33 V W (λ = ( VW λ (34 = [λ] [V] 3 [] [W ] ( Branch Voltages V W W (λ = ( VW λ (36 = [λ] [V] 3 [] 2 (37 he n l 1 vectors of complex voltages at the from and to ends of all branches are, respectively V f = C f V (38 V t = C t V (39 8

9 3.3 Reference Bus Voltage Angles 3 VOLAGES First Derivatives 3.3 Reference Bus Voltage Angles V f = C V f = C f (40 V f = C V f = jc f (41 he n r 1 vector of complex voltages at reference buses is V ref = C ref V (42 he equality constraint on voltage angles at reference buses is First Derivatives G ref (X = C ref Θ Θ specified ref (43 G ref G ref W = Gref = C refθ = C ref [V] 2 [W ] (44 = Gref = C refθ W = C ref [V] 2 [] ( Second Derivatives G ref (λ = C ref Θ (λ = 2C ref [λ] [V] 4 [] [W ] (46 G ref W (λ = C ref Θ W (λ = C ref [λ] [V] 4 ( [W ] 2 [] 2 (47 G ref W (λ = C ref Θ W (λ = C ref [λ] [V] 4 ( [W ] 2 [] 2 (48 G ref W W (λ = C ref Θ W W (λ = 2C ref [λ] [V] 4 [] [W ] (49 9

10 3.4 Bus Voltage Magnitude Limits 3 VOLAGES 3.4 Bus Voltage Magnitude Limits pper and lower bounds on bus voltage magnitudes are the n b 1 vectors H Vmax (X = V V max (50 H Vmin (X = V min V ( First Derivatives H Vmax = V = [V] 1 [] (52 H Vmax W = V W = [V] 1 [W ] (53 H Vmin H Vmin W = H Vmax (54 = H Vmax W ( Second Derivatives H Vmax (λ = V (λ = [λ] [V] 3 [W ] 2 (56 H Vmax W (λ = V W (λ = [λ] [V] 3 [] [W ] (57 H Vmax W (λ = V W (λ = [λ] [V] 3 [] [W ] (58 H Vmax W W (λ = V W W (λ = [λ] [V] 3 [] 2 (59 H Vmin (λ = H Vmax (λ (60 H Vmin W (λ = H Vmax W (λ (61 H Vmin W (λ = H Vmax W (λ (62 H Vmin W W (λ = H Vmax W W (λ (63 10

11 3.5 Branch Angle Difference Limits 3 VOLAGES 3.5 Branch Angle Difference Limits pper and lower bounds on branch voltage angle differences are the n l 1 vectors First Derivatives H Θmax (X = (C f C t Θ Θ max ft (64 H Θmin (X = Θ min ft (C f C t Θ (65 H Θmax = (C f C t Θ = (C f C t [V] 2 [W ] (66 H Θmax W = Θ W = (C f C t [V] 2 [] (67 H Θmin H Θmin W = H Θmax (68 = H Θmax W ( Second Derivatives H Θmax (λ = (C f C t Θ (λ = 2(C f C t [λ] [V] 4 [] [W ] (70 H Θmax W (λ = (C f C t Θ W (λ = (C f C t [λ] [V] 4 ( [W ] 2 [] 2 (71 H Θmax W (λ = (C f C t Θ W (λ = (C f C t [λ] [V] 4 ( [W ] 2 [] 2 (72 H Θmax W W (λ = (C f C t Θ W W (λ = 2(C f C t [λ] [V] 4 [] [W ] (73 H Θmin (λ = H Θmax (λ (74 HW Θmin (λ = HW Θmax (λ (75 H Θmin W (λ = H Θmax W (λ (76 H Θmin W W (λ = H Θmax W W (λ (77 11

12 4 BS INJECIONS 4 Bus Injections 4.1 Complex Current Injections Consider the complex current balance equation, G c (X = 0, where G c (X = I bus + (78 and I bus = Y bus V (79 = [S d C g S g ] Λ ( First Derivatives I bus X = Ibus X = [ I bus I bus W 0 0 ] (81 I bus I bus W = Ibus = Ibus = Y V bus = Y bus (82 = Y V bus = jy bus (83 X = Idg X = [ W Idg P g Q g ] (84 = Idg = [S d C g S g ] [Λ ] 2 (85 W = Idg = j[s d C g S g ] [Λ ] 2 (86 P g = Idg P g = [Λ ] C g (87 Q g = Idg Q g = j [Λ ] C g (88 12

13 4.1 Complex Current Injections 4 BS INJECIONS G c X = Gc X = [ G c Gc W Gc P g G c Q g ] (89 G c = Gc = Ibus + = Y bus [S d C g S g ] [Λ ] 2 (90 G c W = Gc = Ibus W + W = j ( Y bus + [S d C g S g ] [Λ ] 2 (91 G c P g G c Q g = Gc P g = P g = [Λ ] C g (92 = Gc Q g = Q g = j [Λ ] C g ( Second Derivatives I bus XX(λ = ( X I bus X λ = 0 (94 XX (λ = ( X λ X (λ Idg W (λ Idg P g (λ Q g (λ W = (λ Idg W W (λ Idg W P g (λ W Q g (λ P (λ g Idg P (λ 0 0 gw Q (λ g Idg Q (λ 0 0 gw = (λ = C jc D jd jc C jd D D jd 0 0 jd D 0 0 ( = λ (95 (96 (97 (98 ( [Sd C g S g ] [Λ ] 2 λ (99 13

14 4.1 Complex Current Injections 4 BS INJECIONS W (λ = P g (λ = = 2[S d C g S g ] [λ] [Λ ] Λ (100 = 2[S d C g S g ] [λ] [Λ ] 3 (101 = C (102 ( W λ (103 = ( j[sd C g S g ] [Λ ] 2 λ (104 = 2j[S d C g S g ] [λ] [Λ ] Λ (105 = 2j[S d C g S g ] [λ] [Λ ] 3 (106 = jc (107 ( P g λ (108 = ( Cg [Λ ] λ (109 = C g [λ] Λ (110 = C g [λ] [Λ ] 2 (111 = D (112 Q (λ = ( g Q g λ (113 = ( jcg [Λ ] λ (114 = jc g [λ] Λ (115 W (λ = = jc g [λ] [Λ ] 2 (116 = jd (117 ( λ (118 = ( [Sd C g S g ] [Λ ] 2 λ (119 = 2[S d C g S g ] [λ] [Λ ] Λ W (120 14

15 4.1 Complex Current Injections 4 BS INJECIONS W W (λ = P gw (λ = Q gw (λ = P g (λ = = 2j[S d C g S g ] [λ] [Λ ] 3 (121 = W (λ = jc (122 ( W λ (123 = ( j[sd C g S g ] [Λ ] 2 λ (124 = 2j[S d C g S g ] [λ] [Λ ] Λ W (125 = 2[S d C g S g ] [λ] [Λ ] 3 (126 = C (127 ( P g λ (128 ( Cg [Λ ] λ (129 = = C g [λ] Λ W (130 = jc g [λ] [Λ ] 2 (131 = jd (132 ( Q g λ (133 ( jcg [Λ ] λ (134 = = jc g [λ] Λ W (135 = C g [λ] [Λ ] 2 (136 = D (137 ( P g λ (138 = P g ( [Sd C g S g ] [Λ ] 2 λ (139 = [Λ ] 2 [λ] C g (140 15

16 4.1 Complex Current Injections 4 BS INJECIONS = P g (λ = D (141 W P g (λ = Q g (λ = W Q g (λ = ( P g W λ (142 = P g ( j[sd C g S g ] [Λ ] 2 λ (143 = j [Λ ] 2 [λ] C g (144 = P gw (λ = jd ( Q g λ (145 (146 = Q g ( [Sd C g S g ] [Λ ] 2 λ (147 = j [Λ ] 2 [λ] C g (148 = Q g (λ = jd ( Q g W λ (149 (150 = Q g ( j[sd C g S g ] [Λ ] 2 λ (151 = [Λ ] 2 [λ] C g (152 = Q gw (λ = D (153 G c XX(λ = ( G c X X λ (154 G c (λ Gc W (λ Gc P g (λ G c Q g (λ = G c W (λ Gc W W (λ Gc W P g (λ G c W Q g (λ G c P (λ g Gc P (λ 0 0 (155 gw G c Q (λ g Gc Q (λ 0 0 gw = XX (λ (156 16

17 4.2 Complex Power Injections 4 BS INJECIONS = C jc D jd jc C jd D D jd 0 0 jd D 0 0 (157 Computational savings can be achieved by storing and reusing certain intermediate terms during the computation of these second derivatives, as follows: A = [Λ ] (158 B = [λ] A 2 (159 C = 2[S d C g S g ] AB (160 D = C g B (161 G c (λ = C (162 G c W (λ = jc (163 G c P g(λ = D (164 G c Q g(λ = jd (165 G c W W (λ = C (166 G c P gw (λ = jd (167 G c Q gw (λ = D (168 G c W (λ = G c W (λ (169 G c P g (λ = G c P g (λ (170 G c W P g (λ = G c P gw (λ (171 G c Q g (λ = G c Q g (λ (172 G c W Q g (λ = G c Q gw (λ ( Complex Power Injections Consider the complex power balance equation, G s (X = 0, where G s (X = S bus + S d C g S g (174 and S bus = [V ] I bus (175 17

18 4.2 Complex Power Injections 4 BS INJECIONS First Derivatives G s X = Gs X = [ G s Gs W Gs P g G s Q g ] G s = Sbus = [ = I bus ] V [ I bus ] + [V ] Y bus + [V ] Ibus [ G s W = Sbus = I bus ] V Ibus + [V ] ([ = j I bus ] [V ] Y bus (176 (177 (178 (179 (180 G s P g = C g (181 G s Q g = jc g ( Second Derivatives G s XX(λ = ( G s X X λ (183 G s (λ Gs W (λ 0 0 = G s W (λ Gs W W (λ ( G s (λ = G s W (λ = ( G s λ (185 (([ λ = I bus ] + Y bus [V ] (186 = ( [λ] Y bus V + Y bus [λ] V (187 = [λ] Y bus + Y bus [λ] (188 = F (189 ( G s W λ (190 18

19 4.2 Complex Power Injections 4 BS INJECIONS = ( ([ j I bus ] Y bus [V ] λ = ( ( j [λ] Y bus V Y bus [λ] V ( = j [λ] Y bus Y bus [λ] (191 (192 (193 = G (194 G s W (λ = ( G s λ (195 = (([ I bus ] + Y bus [V ] λ (196 = ( [λ] Y bus V + Y bus [λ] V (197 ( = j Y bus [λ] [λ] Y bus (198 = G s W (λ = G (199 G s W W (λ = ( G s W λ (200 = ( ([ j I bus ] Y bus [V ] λ (201 = ( ( j [λ] Y bus V Y bus [λ] V (202 = [λ] Y bus + Y bus [λ] (203 = F (204 Computational savings can be achieved by storing and reusing certain intermediate terms during the computation of these second derivatives, as follows: E = [λ] Y bus (205 F = E + E (206 G = j ( E E (207 G s (λ = F (208 G s W (λ = G (209 G s W (λ = G (210 G s W W (λ = F (211 19

20 5 BRANCH FLOWS 5 Branch Flows Consider the line flow constraints of the form H(X < 0. his section examines 3 variations based on the square of the magnitude of the current, apparent power and real power, respectively. he relationships are derived first for the complex flows at the from ends of the branches. Derivations for the to end are identical (i.e. just replace all f sub/super-scripts with t. 5.1 Complex Currents I f = Y f V (212 I t = Y t V ( First Derivatives I f X = If X = [ I f If W If P g I f Q g ] (214 I f = If = Y f (215 I f W = If = jy f (216 I f P g = If P g = 0 (217 I f Q g = If Q g = 0 ( Second Derivatives I f XX (µ = 5.2 Complex Power Flows ( I f X µ = 0 (219 X S f = [V f ] I f (220 S t = [V t ] I t (221 20

21 5.2 Complex Power Flows 5 BRANCH FLOWS First Derivatives S f X = Sf X = [ = S f Sf W Sf P g S f Q g ] [ I f ] V f X + [V f] If X [ S f = I f ] V f + [V f] If [ = I f ] C f + [V f ] Y f [ S f W = I f ] V f + [V f] If ([ = j I f ] C f [V f ] Y f (222 (223 (224 (225 (226 (227 S f P g = 0 (228 S f Q g = 0 ( Second Derivatives S f XX (µ = X = S f (µ = ( S f X µ S f (µ Sf W (µ 0 0 S f W (µ Sf W W (µ ( = S f µ (( C f [ I f ] + Y f [V f ] µ (230 (231 (232 (233 = C f [µ] If + Y f [µ] V f (234 = C f [µ] Y f + Y f [µ] C f (235 21

22 5.2 Complex Power Flows 5 BRANCH FLOWS = B f (236 S f W (µ = ( = ( j S f W (µ = = S f W ( µ C f [ I f ] Y f [V f ] µ Y f [µ] V f = j (C f [µ] If = j (C f [µ] Y f Y f [µ] C f (237 (238 (239 (240 = D f (241 ( S f µ (( C f [ I f ] + Y f [V f ] µ = C f [µ] If + Y f [µ] V f = j (C f [µ] Y f Y f [µ] C f (242 (243 (244 (245 = S f W (µ = Df (246 S f W W (µ = ( S f W µ = ( ( [ j C f I f ] Y f [V f ] µ = j (C f [µ] If Y f [µ] V f = j (C f [µ] ( jy f Y f [µ] (jc f (247 (248 (249 (250 = C f [µ] Y f + Y f [µ] C f (251 = B f (252 Computational savings can be achieved by storing and reusing certain intermediate terms during the computation of these second derivatives, as follows: A f = C f [µ] Y f (253 22

23 5.3 Squared Current Magnitudes 6 GENERALIZED AC OPF COSS B f = A f + A f 5.3 Squared Current Magnitudes (254 D f = j ( A f A f (255 S f (µ = B f (256 S f W (µ = D f (257 S f W (µ = Sf W (µ = Df (258 S f W W (µ = B f (259 See the corresponding section in Matpower echnical Note Squared Apparent Power Magnitudes See the corresponding section in Matpower echnical Note Squared Real Power Magnitudes See the corresponding section in Matpower echnical Note 2. 6 Generalized AC OPF Costs Let X be defined as X = W P g Q g Y Z (260 where Y is the n y 1 vector of cost variables associated with piecewise linear generator costs and Z is an n z 1 vector of additional linearly constrained user variables. See the corresponding section in Matpower echnical Note 2 for additional details. 23

24 7 LAGRANGIAN OF HE AC OPF 7 Lagrangian of the AC OPF Consider the following AC OPF problem formulation, where X is defined as in (260, f is the cost function, and X represents the reduced form of X, consisting of only, W, P g and Q g, without Y and Z. subject to where and G(X = min f(x (261 X G(X = 0 (262 H(X 0 (263 H(X = R{G b (X } I{G b (X } G ref (X A E X B E H f (X H t (X H Vmax (X H Vmin (X H Θmax (X H Θmin (X A I X B I (264 (265 and G b is the nodal balance function, equal to either G c for current balance or to G s for power balance. Partitioning the corresponding multipliers λ and µ similarly, λ = λ P λ Q λ ref λ E, µ = the Lagrangian for this problem can be written as µ f µ t µ V max µ V min µ Θ max µ Θ min µ I (266 L(X, λ, µ = f(x + λ G(X + µ H(X (267 24

25 7.1 Nodal Current Balance 7 LAGRANGIAN OF HE AC OPF 7.1 Nodal Current Balance Let the nodal balance function G b be the nodal complex current balance G c First Derivatives where and G X = R{G c X } 0 0 I{G c X } 0 0 G ref X 0 0 A E H X = H Vmax H Vmin H Θmax H Θmin L X (X, λ, µ = f X + λ G X + µ H X (268 L λ (X, λ, µ = G (X (269 L µ (X, λ, µ = H (X (270 = A I Second Derivatives H f X 0 0 HX t 0 0 X 0 0 X 0 0 = X 0 0 X 0 0 R{G c } R{Gc W } R{Gc P g } R{G c Q g } 0 0 I{G c } I{Gc W } I{Gc P g } I{G c Q g } 0 0 G ref G ref W H Vmax H Vmin H Θmax H Θmin A E H f H f W H t HW t HW Vmax HW Vmin HW Θmax HW Θmin A I (271 (272 where L XX (X, λ, µ = f XX + G XX (λ + H XX (µ (273 G XX (λ = G X X (λ (274 25

26 7.1 Nodal Current Balance 7 LAGRANGIAN OF HE AC OPF G X X (λ = R{G c X X (λ P } + I{G c X X (λ Q } + G ref X X (λ ref (275 = R + I + G c (λ P G c W (λ P G c P g (λ P G c Q g (λ P G c W (λ P G c W W (λ P G c W P g (λ P G c W Q g (λ P G c P (λ g P G c P (λ gw P 0 0 G c Q (λ g P G c Q (λ gw P 0 0 G c (λ Q G c W (λ Q G c P g (λ Q G c Q g (λ Q G c W (λ Q G c W W (λ Q G c W P g (λ Q G c W Q g (λ Q G c P (λ g Q G c P (λ gw Q 0 0 G c Q (λ g Q G c Q (λ gw Q 0 0 G ref (λ ref G ref W (λ ref G ref W (λ ref 0 0 G ref W W (λ ref (276 and H XX (µ = H X X (µ (277 H X X (µ = H (µ H W (µ 0 0 H W (µ H W W (µ (278 H (µ = H f (µ f + H t (µ t + H Vmax (µ V max + H Vmin (µ V min + H Θmax (µ Θ max + H Θmin (µ Θ min (279 H W (µ = H f W (µ f + H t W (µ t + HW Vmax (µ V max + HW Vmin (µ V min + HW Θmax (µ Θ max + HW Θmin (µ Θ min (280 H W (µ = H f W (µ f + H t W (µ t + HW Vmax (µ V max + HW Vmin (µ V min 26

27 7.2 Nodal Power Balance 7 LAGRANGIAN OF HE AC OPF 7.2 Nodal Power Balance + HW Θmax (µ Θ max + HW Θmin (µ Θ min (281 H W W (µ = H f W W (µ f + H t W W (µ t + H Vmax W W (µ V max + H Vmin W W (µ V min + H Θmax W W (µ Θ max + H Θmin W W (µ Θ min (282 Let the nodal balance function G b be the nodal complex power balance G s First Derivatives where G X = R{G s X } 0 0 I{G s X } 0 0 G ref X 0 0 A E L X (X, λ, µ = f X + λ G X + µ H X (283 L λ (X, λ, µ = G (X (284 L µ (X, λ, µ = H (X (285 = R{G s } R{Gs W } C g I{G s } I{Gs W } 0 C g 0 0 G ref W G ref and H X is the same as for nodal current balance in ( Second Derivatives A E (286 where L XX (X, λ, µ = f XX + G XX (λ + H XX (µ (287 G XX (λ = G X X (λ (288 27

28 7.2 Nodal Power Balance 7 LAGRANGIAN OF HE AC OPF G X X (λ = R{G s X X (λ P } + I{G s X X (λ Q } + G ref X X (λ ref (289 G s (λ P G s W (λ P 0 0 = R G s W (λ P G s W W (λ P G s (λ Q G s W (λ Q I G s W (λ Q G s W W (λ Q G ref (λ ref G ref W (λ ref G ref W (λ ref G ref W W (λ ref ( and H XX (µ is the same as for nodal current balance in (277 (

29 8 REVISION HISORY 8 Revision History Revision 1 (October 25, 2018 Added missing equality constraint for reference voltage angles. See Sections 3.3 and 7. Added missing inequality constraints for bus voltage magnitude limits. See Sections 3.4 and 7. Added missing inequality constraints for branch voltage angle difference limits. See Sections 3.5 and 7. Initial version (April 2, 2018 Published as Matpower echnical Note 4. 29

30 A SCALAR POLAR COORDINAE DERIVAIVES Appendix A Scalar Polar Coordinate Derivatives When using cartesian coordinates for the voltages, the voltage magnitudes and angles are now functions of the cartesian coordinates. Constraints on these functions require their derivatives as well. Consider a scalar complex voltage v that can be expressed in polar coordinates v and θ or cartesian coordinates u and w as: We also have A.1 First Derivatives Given that we have v = v e jθ (291 = u + jw (292 θ = tan 1 w u (293 v 2 = u 2 + w 2 (294 tan 1 (y x θ u = 1 (u 1 w 1 + u 2 w 2 u θ w = 1 (u 1 w 1 + u 2 w 2 w = = = y 2 y x ( u 2 w 2 ( u 2 w = w (296 v u 2 w 2 u 1 = u (297 v 2 v u = v v 2 v 2 u v w = v v 2 v 2 w = 1 2 ( v 2 1 u 2 (2u = v = 1 2 ( v 2 1 w 2 (2w = v (298 (299 30

31 A.2 Second Derivatives A SCALAR POLAR COORDINAE DERIVAIVES A.2 Second Derivatives 2 θ u = ( v 2 w 2 u 2 θ w u = ( v 2 u u 2 θ u w = ( v 2 w w = w( 2 v 3 u v = 2uw v 4 (300 = 1 v 2 + u = w2 u 2 v 4 = 2 θ w u 2 θ w = ( v 2 u 2 w ( 2 v 3 = 1 v 2 w u v = v 2 2u 2 v 4 = w2 u 2 v 4 (301 ( 2 v 3 = u( 2 v 3 w v = 2uw v 4 w v = v 2 + 2w 2 v 4 (302 (303 = 2 θ u 2 (304 2 v u = ( v 1 u 2 u 2 v w u = ( v 1 w u = v 1 + u( v 2 u v = v 2 u 2 v 3 = w2 v 3 (305 = w( v 2 u v = uw v 3 (306 2 v u w = ( v 1 u w = u( v 2 w v = uw v 3 = 2 v w u (307 2 v w = ( v 1 w 2 w = v 1 + w( v 2 w v = v 2 w 2 v 3 = u2 v 3 (308 31

32 REFERENCES REFERENCES References [1] R. D. Zimmerman, AC Power Flows, Generalized OPF Costs and their Derivatives using Complex Matrix Notation, Matpower echnical Note 2, February [Online]. Available: N2-OPF-Derivatives.pdf 2 [2] B. Sereeter and R. D. Zimmerman, Addendum to AC Power Flows and their Derivatives using Complex Matrix Notation: Nodal Current Balance, Matpower echnical Note 3, April [Online]. Available: cornell.edu/matpower/n3-more-opf-derivatives.pdf 2 [3] R. D. Zimmerman, C. E. Murillo-Sánchez, and R. J. homas, Matpower: Steady-State Operations, Planning and Analysis ools for Power Systems Research and Education, Power Systems, IEEE ransactions on, vol. 26, no. 1, pp , Feb DOI: /PWRS [4] Matpower. [Online]. Available: matpower/. 2 32