AC Power Flows, Generalized OPF Costs and their Derivatives using Complex Matrix Notation

Transcription

1 AC Power Flows, Generalized OPF Costs and their Derivatives using Complex Matrix Notation Ray D. Zimmerman February 24, 2010 Matpower Technical Note 2 c 2008, 2010, 2011, 2017, 2018 Power Systems Engineering Research Center Pserc All Rights Reserved Revision 5 April 2, See Section 8 for revision history details.

2 CONTENTS CONTENTS Contents 1 Notation 4 2 Introduction 5 3 Voltages Bus Voltages First Derivatives Second Derivatives Branch Voltages First Derivatives Bus Injections Complex Current Injections First 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 First Derivatives Second Derivatives Squared Apparent Power Magnitudes First Derivatives Second Derivatives Squared Real Power Magnitudes First Derivatives Second Derivatives

3 CONTENTS CONTENTS 6 Generalized AC OPF Costs Polynomial Generator Costs First Derivatives Second Derivatives Piecewise Linear Generator Costs First Derivatives Second Derivatives General Cost Term First Derivatives Second Derivatives Full Cost Function First Derivatives Second Derivatives Lagrangian of the AC OPF Nodal Current Balance Nodal Power Balance First Derivatives Second Derivatives Revision History 27 References 28 3

4 1 NOTATION 1 Notation n b, n g, n l number of buses, generators, branches, respectively 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 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 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 A A T 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 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 INTRODUCTION 2 Introduction The purpose of this document 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. Similarly, the derivatives of the generalized AC OPF cost function used by Matpower 1, 2 and the corresponding OPF Lagrangian function are developed. The relevant code in Matpower is based on the formulas found in this note, in which nodal balances are expressed in terms of complex power and voltages are represented in polar coordinates, and in the companion Matpower Technical Note 2 3 and Matpower Technical Note 3 4, which present formulas for variations based on nodal current balances and cartesian coordinate voltages, respectively. We will be looking at complex functions of the real valued vector X = Θ V 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 T, we use the following notation for the first derivatives transpose of the gradient f X = f = f x 1 f x 2 The matrix of second partial derivatives, the Hessian of f, is f XX = 2 f = 2 f T x f 2 1 = 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 T, 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 = f m x 1 5 f m x n

6 3 VOLTAGES 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 = T FX λ λ=α 6 Just to clarify the notation, if Y and Z are subvectors of X, then F Y Z α = T 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 3.1 Bus Voltages = A B + B A = A B X + B A X 9 V is the n b 1 vector of complex bus voltages. The element for bus i is v i = v i e jθ i. V and Θ are the vectors of bus voltage magnitudes and angles. Let First Derivatives V Θ = E = V 1 V 10 = j V 11 V V = = V V 1 = E 12 6

7 3.2 Branch Voltages 4 BUS INJECTIONS Second Derivatives E Θ = E = j E 13 E V = E = 0 14 It may be useful in later derivations to note that V VV λ = T λ = λ E V = Branch Voltages The n l 1 vectors of complex voltages at the from and to ends of all branches are, respectively V f = C f V 16 V t = C t V First Derivatives 4 Bus Injections 4.1 Complex Current Injections f = C f = jc f V 18 f = C f = C f V V 1 = C f E 19 I bus = Y bus V 20 7

8 4.2 Complex Power Injections 4 BUS INJECTIONS First Derivatives I bus X = Ibus = I bus I bus I bus Θ I bus V = Ibus = Ibus = Y bus = jy bus V 22 = Y bus = Y bus V V 1 = Y bus E Complex Power Injections Consider the complex power balance equation, G s X = 0, where and First Derivatives G s X = S bus + S d C g S g 24 S bus = V I bus 25 G s X = Gs = G s Θ Gs V G s P g G s Q g G s Θ = Sbus = I bus = = j V G s V = Sbus = I bus = = V + V Ibus I bus j V + V jy bus V 28 I bus Y bus V 29 + V Ibus 30 I bus E + V Y bus E 31 I bus + Y bus V V 1 32 G s P g = C g 33 8

9 4.2 Complex Power Injections 4 BUS INJECTIONS Second Derivatives G s Q g = jc g 34 G s XXλ = G s T X λ 35 G s ΘΘ λ Gs ΘV λ 0 0 = G s VΘ λ Gs VV λ G s ΘΘλ = G s VΘλ = G s T Θ λ 37 = = j j I bus V T Y bus V λ I bus V λ V Y T bus V λ = j V λ jy bus V } {{ } I bus V Y T bus λ j V = V + λ V Y T bus V λ Y bus V + I bus λ j V Y T bus V λ j V 40 Y T bus V λ I bus 41 = E + F 42 G s T V λ 43 = E I bus λ + E Y T bus V λ I bus λ j E E = E λ jy bus V + I bus 9 44

10 4.2 Complex Power Injections 4 BUS INJECTIONS + E Y bus T λ j V + Y T bus V λ j E E 45 = j E Y T bus V λ Y T bus V λ λ E Y bus V I bus 46 = j V 1 V Y T bus V λ Y T bus V λ λ V Y bus V I bus 47 = jge F 48 G s ΘVλ = G s T Θ λ 49 = j λ V Y bus Y T bus V λ V V Y T bus I bus V λ V 1 50 = G s VΘT λ 51 G s VVλ = G s T V λ 52 = E = E λ Y bus E + I bus I bus λ + E Y T bus V λ I bus λ 0 E + E Y bus T λ E + Y T bus V λ 0 E = V 1 λ V Y bus V + V Y T bus V λ V 1 55 = GC + C T G 56 Computational savings can be achieved by storing and reusing certain intermediate terms during the computation of these second derivatives, as follows: A = λ V 57 B = Y bus V 58 10

11 5 BRANCH FLOWS 5 Branch Flows C = AB 59 D = Y bus T V 60 E = V D λ Dλ 61 F = C A I bus = j λ G s Θ 62 G = V 1 63 G s ΘΘλ = E + F 64 G s VΘλ = jge F 65 G s ΘVλ = G s VΘT λ 66 G s VVλ = GC + C T G 67 Consider the line flow constraints of the form HX < 0. This section examines 3 variations based on the square of the magnitude of the current, apparent power and real power, respectively. The 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 First Derivatives I f X = If = I f = Y f V 68 I t = Y t V 69 I f Θ If V I f P g I f Q g 70 I f Θ = Y f = jy f V 71 I f V = Y f = Y f V V 1 = Y f E 72 I f P g = 0 73 I f Q g =

12 5.1 Complex Currents 5 BRANCH FLOWS Second Derivatives I f XX µ = = I f T X µ I f ΘΘ µ If ΘV µ 0 0 I f VΘ µ If VV µ I f ΘΘ µ = I f T Θ µ 77 = j V T Yf µ 78 = Y T f µ V 79 I f VΘ µ = I f T V µ 80 = E T Yf µ 81 = j Y T f µ E 82 = ji f ΘΘ µ V 1 83 I f ΘV µ = I f T Θ µ 84 = j V T Yf µ 85 = j Y T f µ E 86 = I f VΘ µ 87 I f VV µ = I f T V µ 88 = E T Yf µ 89 =

13 5.2 Complex Power Flows 5 BRANCH FLOWS 5.2 Complex Power Flows First Derivatives S f X = Sf = Second Derivatives S f XX µ = = = S f = V f I f 91 S t = V t I t 92 S f Θ Sf V S f P g S f Q g I f f + V f If S f Θ = I f f + V f If 95 = I f jc f V + C f V jy f V 96 = j I f C f V C f V Y f V 97 S f V = I f f + V f If 98 = I f C f E + C f V Y f E 99 S f P g = S f Q g = S f T X µ S f ΘΘ µ Sf ΘV µ 0 0 S f VΘ µ Sf VV µ

14 5.2 Complex Power Flows 5 BRANCH FLOWS S f ΘΘ µ = = = j S f Θ T µ V C f T j T V C f = j V C f T µ jy f V I f V Y T f C f V I f µ V Y f T C f V µ } {{ } I f V Y f T µ C f j V µ T + C f I f µ j V Y T f C f V µ j V 107 = V Y T f µ C f V + V C T f µ Y f V Y T f µ C f V V C T f µ Y f V V 108 = F f D f E f 109 S f VΘ µ = S f T V µ = T E C f I f µ + E Y T f C f V µ = E C T f µ jy f V T + C f I f µ I f + E Y f T µ C f j V j E E + Y T f C f V µ j E E = j E Y T f µ C f V E C T f µ Y f V Y T f µ C f V E + C T f µ Y f V E = j V 1 V Y T f µ C f V V C T f µ Y f V Y T f µ C f V V + C T f µ Y f V V = jgb f B f T D f + E f

15 5.2 Complex Power Flows 5 BRANCH FLOWS S f ΘV µ = S f T Θ µ 116 = j V C T f µ Y f V V Y T f µ C f V Y T f µ C f V V + C T f µ Y f V V V = S f VΘT µ 118 S f VV µ = S f V T µ = T E C f I f µ + E Y T f C f V µ = E C T f µ Y f E T + C f I f µ 0 I f E + E Y f T µ C f E + Y T f C f V µ 0 E = V 1 V Y T f µ C f V + V C T f µ Y f V V = GF f G 123 Computational savings can be achieved by storing and reusing certain intermediate terms during the computation of these second derivatives, as follows: A f = Y T f µ C f 124 B f = V A f V 125 D f = A f V V 126 E f = A T f V V 127 T F f = B f + B f 128 G = V S f ΘΘ µ = F f D f E f 130 S f VΘ µ = jgb f B T f D f + E f 131 S f ΘV µ = VΘT Sf µ 132 S f VV µ = GF fg

16 5.3 Squared Current Magnitudes 5 BRANCH FLOWS 5.3 Squared Current Magnitudes Let Imax 2 denote the vector of the squares of the current magnitude limits. Then the flow constraint function HX can be defined in terms of the square of the current magnitudes as follows: H f X = I f I f Imax where I f = M f + jn f First Derivatives H f X I = f I f X + I f I f X I f I fx + = = 2 R { I f I f X = M f M f + N f N f I 2 max I f IX f 137 } 138 = M f jn f M f X + jn f X + M f + jn f M f X jn f X 139 M f = 2 M f X + N f N f X 140 = 2 R { I f} { } R I f X + I { I f} { } I I f X Second Derivatives H f XX µ = = H f T X µ H f ΘΘ µ Hf ΘV µ 0 0 H f VΘ µ Hf VV µ

17 5.4 Squared Apparent Power Magnitudes 5 BRANCH FLOWS H f XX µ = = H f X T µ I f T X I f µ + I f T X I f µ = I f XX I f µ + I f X = 2 R T µ I f X + I f XX I f µ + I f T X µ I f X 146 {I fxx I f } µ + I f T X µ I f X 147 H f ΘΘ µ = 2 R { I f ΘΘ I f µ + I f Θ H f VΘ µ = 2 R { I f VΘ I f µ + I f V H f ΘV µ = 2 R { I f ΘV I f µ + I f Θ H f VV µ = 2 R { I f VV I f µ + I f V 5.4 Squared Apparent Power Magnitudes } T µ I f Θ 148 } T µ I f Θ 149 } T µ I f V 150 } T µ I f V 151 Let Smax 2 denote the vector of the squares of the apparent power flow limits. Then the flow constraint function HX can be defined in terms of the square of the apparent power flows as follows: H f X = S f S f Smax where S f = P f + jq f First Derivatives H f X S = f S f X + S f S f X S f S fx + = = 2 R { S f S f X = P f P f + Q f Q f S 2 max S f SX f 155 } 156 = P f jq f P f X + jqf X + P f + jq f P f X jqf X

18 5.5 Squared Real Power Magnitudes 5 BRANCH FLOWS P f = 2 P f = 2 R { S f} R X + Q f Q f X { } S f X + I { S f} I { } S f X Second Derivatives H f XX µ = = H f XX µ = H f X T µ = H f T X µ H f ΘΘ µ Hf ΘV µ 0 0 H f VΘ µ Hf VV µ S f T X S f µ + S f T X S f µ = S f XX S f µ + S f X = 2 R T µ S f X + S f XX S f µ + S f T X µ S f X 164 {S fxx S f } µ + S f T X µ S f X 165 H f ΘΘ µ = 2 R { S f ΘΘ S f µ + S f Θ H f VΘ µ = 2 R { S f VΘ S f µ + S f V H f ΘV µ = 2 R { S f ΘV S f µ + S f Θ H f VV µ = 2 R { S f VV S f µ + S f V 5.5 Squared Real Power Magnitudes } T µ S f Θ 166 } T µ S f Θ 167 } T µ S f V 168 } T µ S f V 169 Let P 2 max denote the vector of the squares of the real power flow limits. Then the flow constraint function HX can be defined in terms of the square of the real power flows as follows: H f X = R { S f} R { S f} P 2 max 170 = P f P f P 2 max

19 6 GENERALIZED AC OPF COSTS First Derivatives H f X = 2 P f P f X 172 = 2 R { S f} { } R S f X Second Derivatives H f XX µ = = H f T X µ H f ΘΘ µ Hf ΘV µ 0 0 H f VΘ µ Hf VV µ H f XX µ = H f T X µ = 2P f T X P f µ = 2 = 2 P f XX P f µ + P f T X µ P f X { S f XX R { S f} } µ + R R { } { } S f T X µ R S f X { H f ΘΘ R µ = 2 S f ΘΘ R { S f} } { } { } µ + R SΘT f µ R S f Θ { H f VΘ R µ = 2 S f VΘ R { S f} } { } { } µ + R S f T V µ R S f Θ { H f ΘV R µ = 2 S f ΘV R { S f} } { } { } µ + R SΘT f µ R S f V { H f VV R µ = 2 S f VV R { S f} } { } { } µ + R S f T V µ R S f V Generalized AC OPF Costs The generalized cost function for the AC OPF consists of three parts, fx = f a X + f b X + f c X

20 6.1 Polynomial Generator Costs 6 GENERALIZED AC OPF COSTS expressed as functions of the full set of optimization variables. Θ V X = P g Q g Y Z 185 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. 6.1 Polynomial Generator Costs Let fp i pi g and fq i qi g be polynomial cost functions for real and reactive power for generator i and F P and F Q be the n g 1 vectors of these costs First Derivatives fp 1 F P p1 g P g =. f ng P png g fq 1 F Q q1 g Q g =. f ng Q qng g f a X = 1 T n g F P P g + F Q Q g 188 We will use F P and F P to represent the vectors of first and second derivatives of each of these real power cost functions with respect to the corresponding generator output. Likewise for the reactive power costs. fx a = f a = fθ a f V a fp a g fq a g fy a fz a = 0 0 F P T F Q T

21 6.2 Piecewise Linear Generator Costs 6 GENERALIZED AC OPF COSTS Second Derivatives fxx a = f X a T = 0 0 fp a gp g fq a gq g where fp a gp g = F P 194 fq a gq g = F Q Piecewise Linear Generator Costs First Derivatives Second Derivatives 6.3 General Cost Term f b X = 1 T n y Y 196 fx b = f b = fθ b f V b fp b g fq b g fy b fz b = T n y f b XX = Let the general cost be defined in terms of the n w n w matrix H w and n w 1 vector C w of coefficients and the parameters specified in the n w n x matrix N and the n w 1 vectors D, R, K, and M. The parameters N and R provide a linear 21

22 6.3 General Cost Term 6 GENERALIZED AC OPF COSTS transformation and shift to the full set of optimization variables X, resulting in a new set of variables R. R = N X R 201 Each element of K specifies the size of a dead zone in which the cost is zero for the corresponding element of R. The elements k i are used to define n w 1 vectors Ū, K and R, where { 0, ki r ū i = i k i 202 1, otherwise k i, r i < k i k i = 0, k i r i k i 203 k i, r i > k i The dead zone costs are zeroed by multiplying by Ū. The remaining elements are shifted toward zero by the size of the dead zone by adding K, before applying a cost. R = R + K 204 Each element of D specifies whether to apply a linear or quadratic function to the corresponding element of R. This can be done via two more n w 1 vectors, D L and D Q, defined as follows { 1, d L di = 1 i = 205 0, otherwise { d Q 1, di = 2 i = 206 0, otherwise The result is scaled by the corresponding element of M to form a new n w 1 vector where W = M Ū D L + D Q R R 207 = D L + D Q R R 208 D L = M Ū D L 209 D Q = M Ū D Q 210 The full general cost term is then expressed as a quadratic function of W as follows f c X = 1 2 WT H w W + C wt W

23 6.4 Full Cost Function 6 GENERALIZED AC OPF COSTS First Derivatives For simplicity of derivation and computation, we defined A and B as follows A = W R = W R = D L + 2 D Q R Second Derivatives f c XX = B = f c W = f c W = WT H w + C wt 213 R X = R = N 214 W X = W = W R R X 215 = AN 216 fx c = f c = f W c W X 217 = BAN 218 f c T X 219 N T AB T 220 = 6.4 Full Cost Function = N T A Hw W + C w + 2 D Q B T R = N T AH w W X + 2 D Q B T RX = N T AH w A + 2 D Q B T N 223 fx = f a X + f b X + f c X 224 = 1 T n g F P P g + F Q Q g + 1 T n y Y WT H w W + C wt W

24 7 LAGRANGIAN OF THE AC OPF First Derivatives f X = f = f X a + fx b + fx c 226 = 0 0 F P T F Q T 1 T n y 0 + BAN Second Derivatives f XX = 2 f = f a 2 XX + fxx b + fxx c F P = F Q Lagrangian of the AC OPF + N T AH w A + 2 D Q B T N 229 Consider the following AC OPF problem formulation, where X is defined as in 185, f is the generalized cost function described above, and X represents the reduced form of X, consisting of only Θ, V, P g and Q g, without Y and Z. subject to min fx 230 X GX = HX where GX = R{G s X } I{G s X } A E X B E

25 7.1 Nodal Current Balance 7 LAGRANGIAN OF THE AC OPF and HX = H f X H t X A I X B I 234 Partitioning the corresponding multipliers λ and µ similarly, λ P λ = λ Q λ E, µ f µ = µ t µ I 235 the Lagrangian for this problem can be written as 7.1 Nodal Current Balance LX, λ, µ = fx + λ T GX + µ T HX 236 See the corresponding section in Matpower Technical Note Nodal Power Balance First Derivatives where G X = R{G s X } 0 0 I{G s X } 0 0 A E L X X, λ, µ = f X + λ T G X + µ T H X 237 L λ X, λ, µ = G T X 238 L µ X, λ, µ = H T X 239 = R{G s Θ } R{Gs V } C g I{G s Θ } I{Gs V } 0 C g 0 0 A E 240 and H X = Hf X 0 0 H t X 0 0 A I = Hf Θ Hf V H t Θ Ht V A I

26 7.2 Nodal Power Balance 7 LAGRANGIAN OF THE AC OPF Second Derivatives where G XX λ = L XX X, λ, µ = f XX + G XX λ + H XX µ 242 = R R{G s X X λ P } + I{G s X X λ Q} I G s ΘΘ λ P G s ΘV λ P G s VΘ λ P G s VV λ P G s ΘΘ λ Q G s ΘV λ Q G s VΘ λ Q G s VV λ Q and H XX µ = = Hf X X µ f + H t X X µ t H f ΘΘ µ f + HΘΘ t µ t H f ΘV µ f + HΘV t µ t H f VΘ µ f + HVΘ t µ t H f VV µ f + HVV t µ t

27 8 REVISION HISTORY 8 Revision History Revision 5 April 2, 2018 Added References section and mention of Matpower Technical Note 3 and Matpower Technical Note 4. Updated some L A TEX syntax for consistency across Matpower Tech Notes 2, 3, and 4. Also, the following notation changes were made to allow U and W to be used for cartesian coordinates in Matpower Technical Note 4: u ū, U Ū, W W. Revision 4 January 22, 2018 Clarified second derivative notation in 6 and 7 to make explicit that, even if input argument is a function of X, λ is a constant in the context of the derivative. Added this revision history section. Thanks to Baljinnyam Sereeter. Revision 3 September 25, 2017 Corrected I bus to I bus in 50. Thanks to Salman Zaferanlouei. Revision 2 March 14, 2011 Corrected dimensions transpose mistake in the first derivative of the Lagrangian expression in 237. Thanks to Ali Mehrizi-Sani. Revision 1 February 24, 2010 Swapped g and h and G and H in notation to match convention used in previous publications. Published as Matpower Technical Note 2. Initial draft February 29,

28 REFERENCES REFERENCES References 1 R. D. Zimmerman, C. E. Murillo-Sánchez, and R. J. Thomas, Matpower: Steady-State Operations, Planning and Analysis Tools for Power Systems Research and Education, Power Systems, IEEE Transactions on, vol. 26, no. 1, pp , Feb DOI: /TPWRS Matpower. Online. Available: matpower/. 2 3 B. Sereeter and R. D. Zimmerman, Addendum to AC Power Flows and their Derivatives using Complex Matrix Notation: Nodal Current Balance, Matpower Technical Note 3, April Online. Available: cornell.edu/matpower/tn3-more-opf-derivatives.pdf 2 4 B. Sereeter and R. D. Zimmerman, AC Power Flows and their Derivatives using Complex Matrix Notation and Cartesian Coordinate Voltages, Matpower Technical Note 4, April Online. Available: edu/matpower/tn4-opf-derivatives-cartesian.pdf 2 28