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Parallel reduced gradient optimal power flow solution
Electric Power Systems Research, 1 7 (1989) 229 - 237
229
Parallel Reduced Gradient Optimal Power Flow Solution LU WANG, NIANDE XIANG, SHIYING WANG and MEI HUANG Department of Electrical Engineering, Tsinghua University, 100084 Beijing (People's Republic of China) (Received May 30, 1989)
ABSTRACT
T w o algorithms for the parallel solution o f the reduced gradient optimal p o w e r flow by two-level computer networks are presented for the first time. The optimization m e t h o d is the steepest descent m e t h o d with an augmented Lagrangian function. In order to minimize the a m o u n t o f information transmitted between the computers o f the twolevel computer networks, a new type o f bus, a PQVO separating bus, is introduced into the parallel reduced gradient optimal p o w e r flow calculation. Generalized p o w e r flow solution m e t h o d s are used to solve the p o w e r flow problem. There are no approximations between the lower-level parallel computation and the higher-level coordination computation in the two parallel reduced gradient optimal p o w e r flow calculation algorithms. The proposed two algorithms have high parallelism and only a very small a m o u n t o f data needs to be transmitted between the higherlevel computer and the lower-level computers. Numerical results for a 30-bus two-area system are also presented.
INTRODUCTION Optimal power flow is the power flow of an electrical power system which gives the lowest generation cost when the system meets the demands of consumers and is in a secure operation state. One-line optimal power flow can maintain the system in the most economic operation state in real time, and thus improve the system profitability. With the trends of increased size and complexity of electrical power systems, the a m o u n t of computation required for real-time analysis and control is rising quickly, and computers, with their limited speed and 0378-7796/89/$3.50
storage, are becoming unable to cope with the tremendous a m o u n t of data. One way to circumvent this problem is to use piecewise or parallel calculation algorithms. Modern electrical power systems are generally multiarea systems; t h e y have the property of being hierarchical structures and are suited to parallel computation. With the development of electrical power system computerization, many two-level computer networks have been set up, as shown in Fig. 1. These networks are capable of parallel processing and support parallel or piecewise calculation algorithms. Since the late 1960s, many algorithms for solving multi-area electrical power system problems have been reported. Most of them are suitable for parallel computation, such as piecewise power flow algorithms [1, 2], hierarchical economic dispatch algorithms [3, 4] and parallel static-state estimation algorithms [5, 6]. But to the authors' knowledge, no literature on parallel optimal power flow calculation algorithms has been published. Two algorithms for the parallel solution of reduced gradient optimal power flow are presented for the first time. The algorithms are based on the steepest descent m e t h o d combined with an augmented Lagrangian function. According to the characteristics of the reduced gradient optimal power flow, PQVO separating buses are introduced into the reduced gradient optimal power flow calculation, and consequently other types of buses are also introduced to correspond with the PQVO buses. Generalized power flow calculation methods (see Appendix) are applied to solve the power flow problem
7 larea co.p,,tor I
to.purer I I roa co.p,,tor i
Fig. 1. A two-level computer network. © Elsevier Sequoia/Printed in The Netherlands
230
in the two algorithms. A parallel computation mode is developed strictly from an exact optimal power flow model by use of the generalized N e w t o n - R a p h s o n power flow (GNRPF) calculation method; no approximation exists between the lower-level parallel computations (area computers) and the higher-level coordination computation (central computer), so this algorithm is an exact one. Another parallel computation mode is also developed using the generalized fast decoupled p o w e r flow (GFDPF) calculation method; although no approximation exists between the lower-level parallel computations (area computers) and the higher-level coordination computation (central computer), this algorithm is an approximate one because of the approximation in the Lagrangian multipliers. The proposed two algorithms are tested on a 30-bus two-area system. Numerical results and their analyses are given.
MODEL OF THE PARALLEL REDUCED GRADIENT OPTIMAL POWER FLOW
The two algorithms for the P R G O P F calculation are designed for the two-level computer network shown in Fig. 1, and ought to transmit as small an amount of data as possible between the computers. The two algorithms are based on generalized power flow claculation methods (see Appendix for details).
Decomposition of the power flow calculation Figure 2 shows a three-area system; separating buses connect the three areas together. In order to solve the p o w e r flow of each area independently, voltages, angles, real and reactive power flows transmitted through the separating buses from one area to its neighboring area must be known, that is, they must be control variables in the p o w e r flow calculation. So each separating bus is 'torn'
s e p a b a t i ng
s e p a r a t i ng
s e p a r a t lng bus Fig. 2. A t h r e e - a r e a s y s t e m .
PI+JQ~~P3~'JQ3 I -PI-JQI
P3*JQ3
-P2-JQ2 Fig. 3. Decomposition of the powerflow caJcu]ation. P2*aQ2
into two buses, which are set to be PQVO buses and are called PQVO separating buses, as shown in Fig. 3. The two PQVO separating buses belong to t w o different areas, have the same voltage value and angle, and have the same real and reactive power injections (corresponding to the real and reactive power flows transmitted through the separating bus from one area to its neighboring area) in opposite directions. If the real and reactive p o w e r injections, voltage values and angles of the PQVO separating buses are known, the p o w e r flow of different areas can be solved independently when generalized power flow calculation methods are used, and the p o w e r flow solutions o f all the areas make an exact p o w e r flow solution of the whole system if they are combined together. In the decomposed power flow calculations, P, Q, V and 0 of each PQVO separating bus must be known, and this is easy in the reduced gradient optimal power flow calculation. The control variables in the reduced gradient optimal power flow calculation are known when the power flow is solved, so the four bus variables (P, Q, V, 0) of each PQVO separating bus are set to be control variables, that is, a pair of PQVO separating buses, corresponding to a separating bus, has four control variables. N o w the whole system's p o w e r flow calculation is decomposed into absolutely independent area power flow calculations. As a consequence of the PQVO buses, some other types of buses have to be set in every area, such as U buses, P buses, etc., but the power flow of every area must be solvable (see Appendix for details).
Development of the PRGOPF model For the three-area system shown in Figs. 2 and 3, the following symbols are defined:
231
inner state variable vector of area k (k = A, B, C) u~ inner control variable vector of area k (k = A, B: C) control variable vector of U all PQVO separating buses G~(X~, U~, U)= 0 power flow equilibrium equations of area k (k = A, B, C) H~(X~, U~, U) >1 0 inequality constraints of area k (k = A, B, C) penalty term of area k W~(X~, Uk, U) (k = A, B, C) cost function of area k F~(X~, U~, U) (k = A, B, C) X~ Lagrangian multiplier vector of power flow equations of area k (k = A, B, C) The P R G O P F model is
Xk
~_, F~(X~, U~, U)
min
(I)
k= A,B,C
Gk(Xk, Uk, U)
0
k = A, B, C
H~,(Xk, Vk, U) >1 0
k = A, B, C
=
k = A, B, C
Ukm~ <~Uk <. U ~ x
k = A, B, C
Um~ < U <. U~x and the augmented Lagrangian function is
~ k=
[Fk(X~, Uh, U)
-
0U~
(}Gkt +
~Uk
.
a wk
-X~+
- 0
OUk
~Uk
(5)
k = A, B, C aL "~U
-
~
\ ~U +
k = A,B,C
Xh + au
=0 --~-/
(6)
where every area of (3), (4), and (5) is indep e n d e n t of the others, and every area of (6) is separable.
DEVELOPMENT ALGORITHMS
OF PRGOPF CALCULATION
Two, one exact and one approximate, P R G O P F calculation algorithms can be developed from the P R G O P F model.
From the optimal conditions {3), (4), (5), and (6), an exact algorithm for the P R G O P F calculation scheme by using the G N R P F method is developed in the following steps: Step 1. Initialization, IT = 0. Step 2. IT = IT + 1, parallel solution by area computers o f power flow equations Gk(X~, Uk, U) = 0 using the G N R P F calculation method. Step 3. Parallel solution by area computers of (4) for area Lagrangian multipliers:
(2)
Model (1) gives the differences between P R G O P F and S R G O P F (serial reduced gradient optimal p o w e r flow) [7]: in P R G O P F , the cost function, p o w e r flow equations, and all the inequality constraints are in separable form expressed by inner control variables, inner state variables, and control variables of PQVO separating buses, the coupling between every area being implied by the control variables of the PQVO separating buses, while in S R G O P F it is not. The optimal conditions of (2) are:
a~,k
~Fk
A,B,C
+ ~,ktGk{Xk, Uk, U) + Wk(Xk, Vk, U)]
~L
(4)
k = A, B, C
i)L
Exact calculation algorithm of PRGOPF with GNRPF
such that
L=
~L - OFk + OGkt"ha + OWk - 0 0Xh axk 0X~ OXk
- Gk(Xk, Uk, U) = 0
k = A, B, C
(3)
The inverse, multiplying its following vector, of the transposed Jacobian matrix is performed by forward and backward substitutions using the triangulated factors of the Jacobian matrix formed in the latest iteration of the G N R P F calculation. Step 4. Area computers calculate in parallel the gradient c o m p o n e n t s with respect to Uk and part of the gradient c o m p o n e n t s with respect to U from (5) and (6):
guk
~L
~F k
aGff
~ Wk
OUk
0U~
a Uk
0 Uk k=A,B,C
232
OLk
g~]
-
~Fk -
aU
OGkt +
'
~U
bU
~Wk X
k
+
-
-
~U
k = A, B, C Step 5. Transmit each area's g ~ (k = A, B, C) to the central c o m p u t e r to form the full gradient c o m p o n e n t s with respect to U: g u = k= A~,B,cg~f
Step 6. Form a search direction and do a linear optimal search to get the new inner control variables Uk (k = A, B, C) and the PQVO separating control variables U. Step 7. Parallel solution by area computers of the power flow equations Gk(Xk, Uk, U) = 0 using the G N R P F calculation method. Step 8. Converge? Yes, stop. No, go to step 3. In the above procedure, steps 1 - 4 and step 7 are processed in parallel; step 5 takes little execution time because of the small amount of g~]. The a m o u n t of transmitted data in the exact P R G O P F is largely dependent on step 6. When different optimization methods are used to form the search direction, the amounts of transmitted data are different. The minus gradient direction is used as the search direction in this paper, so a very small a m o u n t of data needs to be transmitted. The parallel computation mode of step 6 will be illustrated in the next section. This P R G O P F calculation algorithm is strictly developed, so it is an exact algorithm; its high parallelism is formed naturally when the PQVO separating buses are introduced into the P R G O P F calculation. Approximate algorithm of PRGOPF with GFDPF An approximate calculation scheme for P R G O P F is developed if, in the exact P R G O P F calculation scheme, p o w e r flow is solved using the G F D P F calculation m e t h o d instead of the G N R P F , and Lagrangian multipliers are calculated approximately in step 3 by ~kk = --
[ OV-I([B']t)-IV-1 0 1 V_l{ [B"lt} -1 I~Fk
X
LD--X~ +
}
~Wk DXk]
k=A,B,C
The inverses, multiplying their following vectors, of the transposed B' and B" matrices are performed by forward and backward substitutions using the already triangulated factors of B' and B" respectively. In the scheme of this approximate calculation algorithm, all steps except step 3 are the same as those of the scheme of the exact PRGOPF calculation algorithm because the G N R P F calculation and the GFDPF calculation produce the same power flow solution. This P R G O P F calculation algorithm is approximate because the multiplier vector is calculated approximately in step 3. But there is no approximation between the parallel computation (area computers) and the coordination computation (central computer) in this algorithm.
IMPLEMENTATION OF THE PRGOPF In order to accelerate the P R G O P F to convergence, the generations of all the generators of the whole electrical power system are predispatched in step 1 by use of the classical economic dispatch method without consideration of the loss modification, and also in step 1 a parallel power flow calculation is performed by using the method presented b y E1-Marsatawy et al. [2], then the initial values of the inner control variables and control variables of the PQVO separating buses are determined. If the initial voltage control variables are over the limits, they are set at their limits. In order to coordinate the changes of all the control variables, each c o m p o n e n t of the minus gradient is multiplied by a corresponding factor, the modified minus gradient being used for the optimal linear search. Factor ri for the jth component, --gvj, of the minus gradient is determined as follows:
I Uj Ujmin if -gvi < 0 rj = U j m a x _ _ Uj if --gvj >~ 0 (7) where Ui may be any control variable. It is obvious that ri is always greater than zero. Only the voltage value of the four control variables of each pair of PQVO separating buses has limit constraints, so three factors are chosen by experience for the unlimited control variables. The penalty terms are "
-
-
233
Wk(Xk, =~
U) 1
onjk(Xk, Uk, U ) ]
<[max(O, P j k k
--ttsk 2 }
~jk = max[O, P j k -
2
f~ = akt 3 + b~t 2 + cht + dk
(IHjk(gk, Uk, U)] k
A, B, C
=
(9)
where o is the penalty factor, and pj~ the multiplier of the inequality constraints Hyk(Xk, Uk, U) >1 0 which is updated in every PRGOPF iteration (constraints of X ~ < Xk ~< Xhmax are implied by Hyk(Xk, Uk, U) >1 0 in (8)). The linear search direction is the minus gradient direction modified by the factors in (7). The optimal step length is caluclated as follows. Step I. Each area computer calculates its third-order interpolation polynomial of the
f = ata + bt2 + ct + d
in which k= A,B,C
parallelcomputation
centralcompuLer
coordination computation
'
I[
l
{
l
constraints detection, update ~k partial derivatives calculatioH Lagrangian multiplier calculation
transmit gUk
i
and
1 transmit gu
F f o r , searchl direction ) ~
I
jI calculato
,transmit a.,b.,cL calculate area's three order polynomial I [ K K ylth r espect_______t?step l e n g t h __1 ! ] transmit T . | ]update control varlables I = [°P~alculate i["~- +--~ -
I
I
[power flow c a l c u l a t i o n l
Top ~
"
[
transmit area's convergence information l area'sconvergence detection [ [ l ~
no~ ~
whole ~
system converges;
~ ~
Fig. 4. Flowchart of the PRGOPF calculation.
k = A,B,C
Top~ has two values, tl and t2, and is determined by
l Initialization ] [power flow calculation]
k= A,B,C
The optimal step is solved by --2b + (4b 2 -- 12ac) 1/2 Topt = 6a
I[
area computers
k = A, B, C
and transmits ak, bk, and ck to the central computer. Step 2. The central computer forms the third-order interpolation polynomial of the whole system cost function
(8)
A, B, C
=
area cost function with respect to step length t:
whole system's l c°nvergence detect ion
transmit whole system's , . . convergence xnformat lon
234 Top t =
min(ti[ti > 0)
i=1,2
Topt is transmitted to every area computer. Step 3. Area computers update inner control variables and PQVO separating control variables. In the above procedure, steps 1 and 3 are processed in parallel by area computers. The amount of transmitted data is very small, each area having only three data to be transmitted to the central computer. The central c o m p u t e r has only one datum to transmit to each area computer, that is, the optimal step Topt. In step 3, if any control variable is over the limit after updating, it is set at its limit. Figure 4 gives the flowchart of the P R G O P F ; it is suitable for both the exact and the approximate P R G O P F calculation algorithms. The flowchart supports these remarks: the a m o u n t of transmitted data is very small, the central c o m p u t e r has little computation effort, most computations being performed by the area computers, so the two algorithms have very high parallelism.
NUMERICAL EXAMPLE
A 30-bus two-area system, shown in Fig. 5, is used to test the proposed two algorithms. The system has 12 generators, and is separated into two areas by three separating buses. The generators have a quadratic cost
j/
Fig. 5. A 30-bus two-area s y s t e m .
function. At the initial point, the voltages of four buses and the power flow of one line are over the limit. The proposed two parallel algorithms were simulated on a serial microcomputer IBM-PCAT. Two SRGOPF calculation algorithms (an exact one [7] and an approximate one [8], just like the P R G O P F ) were also programmed for comparison. Table 1 gives the results. The tabular results show that the exact P R G O P F calculation algorithm needs more iterations to converge than the exact S R G O P F calculation algorithm, but the
TABLE 1 N u m e r i c a l results: serial c l o c k i n g Algorithm
Exact PRGOPF
Approx. PRGOPF
Exact SRGOPF
Approx. SRGOPF
Initialization Initial cost 1 0 t h i t e r a t i o n cost CPU t i m e for 10 i t e r a t i o n s 1 5 t h i t e r a t i o n cost Total iterations O p t i m a l cost CPU t i m e p e r i t e r a t i o n Total CPU iteration time CPU o u t p u t t i m e T o t a l CPU t i m e
9.00 13.270 13.432 26.95 13.285 15 13.285 2.67 40.04 5.17 54.21
9.12 13.270 a a a 17 13.521 4.26 67.83 5.22 82.17
9.06 13.270 13.425 28.45 13.392 10 13.425 2.85 28.45 3.51 41.02
9.22 13.270 13.429 32.19 13.390 10 13.429 3.20 32.19 3.46 44.87
a C o n s t r a i n t v i o l a t i o n s exist. Notes: (1) Precise o p t i m a l c o s t = 1 3 . 2 2 7 . (2) C o n v e r g e n c e criteria: cost decrease is less t h a n 0 . 0 1 5 a n d v i o l a t i o n o f a n y c o n s t r a i n t is less t h a n 0.005. (3) T i m e u n i t : second.
235 f o r m e r ' s o p t i m a l c o s t is less t h a n t h a t o f t h e latter, a n d t h e f o r m e r ' s average CPU t i m e p e r i t e r a t i o n is also less t h a n t h a t o f t h e latter. T h e s e t w o a l g o r i t h m s cause all c o n s t r a i n t v i o l a t i o n s to be r e m o v e d a f t e r t h e 5 t h iterat i o n . T h e e x a c t P R G O P F c a l c u l a t i o n algor i t h m , a f t e r i t e r a t i o n 10, h a s t a k e n 2 6 . 9 5 s f o r t h e first 10 i t e r a t i o n s at a c o s t o f 1 3 . 4 3 2 (no c o n s t r a i n t v i o l a t i o n ) w h i c h is v e r y close to t h e o p t i m a l c o s t o f t h e e x a c t S R G O P F . I t c o n t i n u e s to i t e r a t e b e c a u s e o f t h e g o o d d e c r e a s e in cost; a f t e r t h e 1 5 t h i t e r a t i o n , it satisfies t h e c o n v e r g e n c e criteria a n d converges. M e a n w h i l e , t h e e x a c t S R G O P F calc u l a t i o n a l g o r i t h m , a f t e r i t e r a t i o n 10 ( c o s t 1 3 . 4 2 5 ) , has t a k e n 2 8 . 4 5 s f o r t h e 10 iterat i o n s , satisfies t h e c o n v e r g e n c e criteria, a n d c o n v e r g e s b e c a u s e o f t h e p o o r d e c r e a s e in c o s t ; if it h a d c o n t i n u e d t o t h e 1 5 t h iterat i o n it w o u l d h a v e c o s t 1 3 . 3 9 2 (no c o n s t r a i n t violation). T h u s , t h e e x a c t P R G O P F calculat i o n a l g o r i t h m has a smaller average CPU time per iteration than the exact SRGOPF c a l c u l a t i o n a l g o r i t h m u n d e r serial simulation, and under comparable conditions, the c o n v e r g e n c e a n d o p t i m a l cost o f t h e f o r m e r are n o t w o r s e t h a n t h o s e o f t h e latter. The approximate PRGOPF calculation a l g o r i t h m d o e s n o t p e r f o r m well. It is far w o r s e t h a n t h e a p p r o x i m a t e S R G O P F calculation a l g o r i t h m . T h e r e a s o n is t h a t t h e r e are v e r y large errors in t h e L a g r a n g i a n m u l t i p l i e r o f t h e r e a c t i v e p o w e r f l o w e q u a t i o n s , so t h e c o s t decreases slowly a n d t h e c o n s t r a i n t v i o l a t i o n s are r e m o v e d w i t h d i f f i c u l t y in t h e i t e r a t i o n . This a p p r o x i m a t e P R G O P F algor i t h m is n o t c o m p e t i t i v e .
CONCLUSION T w o P R G O P F c a l c u l a t i o n a l g o r i t h m s are p r e s e n t e d f o r t h e first t i m e in this p a p e r . I t is possible t o i n t r o d u c e P Q V O s e p a r a t i n g buses a n d generalized p o w e r f l o w c a l c u l a t i o n m e t h o d s into t h e P R G O P F c a l c u l a t i o n algorithms. The exact PRGOPF calculation a l g o r t h i m is v e r y successful; it has v e r y high parallelism, o n l y a v e r y small a m o u n t o f d a t a n e e d s t o b e t r a n s m i t t e d , a n d t h e r e is n o a p p r o x i m a t i o n b e t w e e n t h e lower-level c o m p u t a t i o n (area c o m p u t e r s ) a n d higher-level computation (central computer). This exact a l g o r i t h m is reliable a n d c o m p e t i t i v e , its
speed, c o n v e r g e n c e a n d o p t i m a l c o s t being no worse than those of the exact SRGOPF c a l c u l a t i o n a l g o r i t h m w h e n parallel c o m p u t a t i o n is s i m u l a t e d on a serial p r o c e s s i n g c o m p u t e r . G o o d savings in CPU t i m e can be m a d e if t h e a l g o r i t h m is p e r f o r m e d on a t w o level c o m p u t e r n e t w o r k . The approximate PRGOPF calculation a l g o r i t h m p e r f o r m s badly. I t can o n l y be r e g a r d e d as an a t t e m p t in t h e search f o r n e w parallel c a l c u l a t i o n a l g o r i t h m s .
NOMENCLATURE GNRPF GFDPF PRGOPF SRGOPF
generalized N e w t o n - R a p h s o n power flow generalized fast d e c o u p l e d p o w e r flow parallel r e d u c e d g r a d i e n t o p t i m a l power flow serial r e d u c e d g r a d i e n t o p t i m a l power flow
REFERENCES 1 R. Kasturi and M. S. N. Potti, Piecewise NewtonRaphson load flow -- an exact method using ordered elimination, IEEE Trans., PAS-95 (1976) 1244 - 1253. 2 M. El-Marsatawy, R. W. Menzies and R. M. Mathur, A new, exact, diakoptic, fast decoupled load flow technique for very large power systems, IEEE PES Summer Meeting, 1979, Paper No. A79 440-9. 3 C. E. Lin and G. L. Viviani, Hierarchical economic dispatch for piecewise quadratic cost functions, IEEE Trans., PAS-103 (1984) 1170 - 1175. 4 K. Aoki and T. Satoh, Robust algorithm for economic dispatch of interconnected systems, IEEE Trans., PAS-104 (1985) 266 - 272. 5 K. Seidu and H. Mukai, Parallel multi-area static state estimation, IEEE Trans., PAS-104 (1985) 1026 - 1034. 6 H. Sasaki, K. Aoki and R. Yokoyama, A parallel computation algorithm for static state estimation, IEEE Trans., PWRS-2 (1987) 624 - 632. 7 H. W. Dommel and W. F. Tinney, Optimal power flow solution, IEEE Trans., PAS-87 (1968) 1866 - 1876. 8 O. Alsac and B. Stott, Decoupled algorithms in optimal load flow calculations, IEEE PES Summer Meeting, 1975, Paper No. A75 545-4. 9 G. R. Krumpholz, K. A. Clements and P. W. Davis, Power system observability: a practical algorithm using network topology, IEEE Trans., PAS-99 (1980) 1534 - 1542.
236 APPENDIX
Generalized power flow calculation There are four variables at a bus, P, Q, V, and 0 (real and reactive power injections, voltage value and angle}; these four variables are called the bus variables. In a c o m m o n power flow calculation, two of the bus variables are control variables, and there are only three types of buses, the PQ bus (real and reactive power injections are control variables}, the PV bus (real power injection and voltage value are control variables}, and the slack bus (voltage value and angle are control variables)• In generalized power flow calculations, however, the control variables of a bus can be any combination of the bus variables P, Q, V, and 0, so the types of buses are generalized, the number of possible combinations of the four bus variables is fifteen, and a bus at which none of the four bus variables is a control variable is also a type of bus, the total number of bus types being sixteen. In generalized power flow calculation, real and/or reactive power flows of transmission lines can be set to be control variables, that is, real and/or reactive power flows of some lines can be specified, and the specified line real and/or reactive power flows are used as power flow equilibrium equations.
cause it makes the parallel optimal power flow calculation possible• The power flow equilibrium equations in the generalized power flow calculation are (i) the real power equilibrium equations of the PQ, P, PV and PQVO buses, and the line real power flow specification equations, and (ii) the reactive power equilibrium equations of the PQ and PQ VO buses, and the line reactive power flow specification equations• The state variables to be solved are (i) the angles of the PQ, P, U, V, and PV buses, and (ii} the voltages of the PQ, P, and U buses• The number of state variables to be solved must be equal to the total number of power flow equilibrium equations. Because of the P-Q decomposition characteristics of electrical power systems, the number of real power equilibrium equations must be equal to the number of u n k n o w n bus angles, and the number of reactive power equilibrium equations must be equal to the number of unknown bus voltage values. The modified equations of the GNRPF calculation are AP1
A01
AQ:
AV
•
°
AP~
A0~
AQ,
=J
A V,
(A-l)
GNRPF solution and GFDPF solution For simplicity, only seven bus types are introduced in the generalized power flow calculation of this paper; t h e y are the PQ, P, U, V, PV, PQ VO and the slack bus and they are sufficient for the present PRGOPF calculation. The symbols for the first six buses indicate the bus control variables of the corresponding buses; for example, the P bus means that the bus real power injection is the control variable, and the V bus means that the bus voltage value is the control variable, but the U bus means that none of the four bus variables is a control variable (U means unknown}. It is obvious that only generator buses can be set as P, U, V, PV, or slack buses because of their adjustable
P,Q, and V. The PQVO bus is the bus at which all four bus
variables are control variables. The PQVO bus is the key bus of this paper, be-
AP~ 1
A0~
AP~
A0~
where s is the number of u n k n o w n bus angles and r the number of u n k n o w n bus voltage values• The modified equations of the GFDPF calculation are
= VB'V ~0~
(A-2)
= VB" A V~
(A-3)
8
E'°I
237 In eqn. (A-l), the structure of J is not symmetrical, and in eqns. (A-2) and (A-3), B' and B" are n o t symmetrical. The calculation scheme of the GNRPF is the same as that of the c o m m o n NRPF calculation, and the calculation scheme of the GFDPF is the same as that of the comm o n FDPF calculation.
Solvability of the generalized power flow calculation If the power flow equilibrium equations in the generalized power flow calculation are taken as measurement equations w i t h o u t redundancy in the static-state estimation, the solvability of the generalized power flow calculation is the same as the observability of the static-state estimation. The algorithms for the observability analysis of the staticstate estimation can be used for the solvability analysis of the generalized power flow
calculation. The algorithm for the observability analysis of the static-state estimation (also t h a t for the solvability analysis of the generalized power flow calculation) will not be presented here; details are reported in ref. 9.
Concluding remarks The results of the generalized power flow calculation methods on the test systems are not listed, but some remarks are given here. (i) Many types of buses can be introduced into the generalized power flow calculation. Generalized power flow calculation methods not only can be used for c o m m o n power flow calculation, but also for some special power flow calculations. (ii) The GNRPF calculation has nearly the same convergence and speed as the c o m m o n NRPF calculation. {iii) The GFDPF calculation has a convergence and speed close to the c o m m o n FDPF calculation.
229
Parallel Reduced Gradient Optimal Power Flow Solution LU WANG, NIANDE XIANG, SHIYING WANG and MEI HUANG Department of Electrical Engineering, Tsinghua University, 100084 Beijing (People's Republic of China) (Received May 30, 1989)
ABSTRACT
T w o algorithms for the parallel solution o f the reduced gradient optimal p o w e r flow by two-level computer networks are presented for the first time. The optimization m e t h o d is the steepest descent m e t h o d with an augmented Lagrangian function. In order to minimize the a m o u n t o f information transmitted between the computers o f the twolevel computer networks, a new type o f bus, a PQVO separating bus, is introduced into the parallel reduced gradient optimal p o w e r flow calculation. Generalized p o w e r flow solution m e t h o d s are used to solve the p o w e r flow problem. There are no approximations between the lower-level parallel computation and the higher-level coordination computation in the two parallel reduced gradient optimal p o w e r flow calculation algorithms. The proposed two algorithms have high parallelism and only a very small a m o u n t o f data needs to be transmitted between the higherlevel computer and the lower-level computers. Numerical results for a 30-bus two-area system are also presented.
INTRODUCTION Optimal power flow is the power flow of an electrical power system which gives the lowest generation cost when the system meets the demands of consumers and is in a secure operation state. One-line optimal power flow can maintain the system in the most economic operation state in real time, and thus improve the system profitability. With the trends of increased size and complexity of electrical power systems, the a m o u n t of computation required for real-time analysis and control is rising quickly, and computers, with their limited speed and 0378-7796/89/$3.50
storage, are becoming unable to cope with the tremendous a m o u n t of data. One way to circumvent this problem is to use piecewise or parallel calculation algorithms. Modern electrical power systems are generally multiarea systems; t h e y have the property of being hierarchical structures and are suited to parallel computation. With the development of electrical power system computerization, many two-level computer networks have been set up, as shown in Fig. 1. These networks are capable of parallel processing and support parallel or piecewise calculation algorithms. Since the late 1960s, many algorithms for solving multi-area electrical power system problems have been reported. Most of them are suitable for parallel computation, such as piecewise power flow algorithms [1, 2], hierarchical economic dispatch algorithms [3, 4] and parallel static-state estimation algorithms [5, 6]. But to the authors' knowledge, no literature on parallel optimal power flow calculation algorithms has been published. Two algorithms for the parallel solution of reduced gradient optimal power flow are presented for the first time. The algorithms are based on the steepest descent m e t h o d combined with an augmented Lagrangian function. According to the characteristics of the reduced gradient optimal power flow, PQVO separating buses are introduced into the reduced gradient optimal power flow calculation, and consequently other types of buses are also introduced to correspond with the PQVO buses. Generalized power flow calculation methods (see Appendix) are applied to solve the power flow problem
7 larea co.p,,tor I
to.purer I I roa co.p,,tor i
Fig. 1. A two-level computer network. © Elsevier Sequoia/Printed in The Netherlands
230
in the two algorithms. A parallel computation mode is developed strictly from an exact optimal power flow model by use of the generalized N e w t o n - R a p h s o n power flow (GNRPF) calculation method; no approximation exists between the lower-level parallel computations (area computers) and the higher-level coordination computation (central computer), so this algorithm is an exact one. Another parallel computation mode is also developed using the generalized fast decoupled p o w e r flow (GFDPF) calculation method; although no approximation exists between the lower-level parallel computations (area computers) and the higher-level coordination computation (central computer), this algorithm is an approximate one because of the approximation in the Lagrangian multipliers. The proposed two algorithms are tested on a 30-bus two-area system. Numerical results and their analyses are given.
MODEL OF THE PARALLEL REDUCED GRADIENT OPTIMAL POWER FLOW
The two algorithms for the P R G O P F calculation are designed for the two-level computer network shown in Fig. 1, and ought to transmit as small an amount of data as possible between the computers. The two algorithms are based on generalized power flow claculation methods (see Appendix for details).
Decomposition of the power flow calculation Figure 2 shows a three-area system; separating buses connect the three areas together. In order to solve the p o w e r flow of each area independently, voltages, angles, real and reactive power flows transmitted through the separating buses from one area to its neighboring area must be known, that is, they must be control variables in the p o w e r flow calculation. So each separating bus is 'torn'
s e p a b a t i ng
s e p a r a t i ng
s e p a r a t lng bus Fig. 2. A t h r e e - a r e a s y s t e m .
PI+JQ~~P3~'JQ3 I -PI-JQI
P3*JQ3
-P2-JQ2 Fig. 3. Decomposition of the powerflow caJcu]ation. P2*aQ2
into two buses, which are set to be PQVO buses and are called PQVO separating buses, as shown in Fig. 3. The two PQVO separating buses belong to t w o different areas, have the same voltage value and angle, and have the same real and reactive power injections (corresponding to the real and reactive power flows transmitted through the separating bus from one area to its neighboring area) in opposite directions. If the real and reactive p o w e r injections, voltage values and angles of the PQVO separating buses are known, the p o w e r flow of different areas can be solved independently when generalized power flow calculation methods are used, and the p o w e r flow solutions o f all the areas make an exact p o w e r flow solution of the whole system if they are combined together. In the decomposed power flow calculations, P, Q, V and 0 of each PQVO separating bus must be known, and this is easy in the reduced gradient optimal power flow calculation. The control variables in the reduced gradient optimal power flow calculation are known when the power flow is solved, so the four bus variables (P, Q, V, 0) of each PQVO separating bus are set to be control variables, that is, a pair of PQVO separating buses, corresponding to a separating bus, has four control variables. N o w the whole system's p o w e r flow calculation is decomposed into absolutely independent area power flow calculations. As a consequence of the PQVO buses, some other types of buses have to be set in every area, such as U buses, P buses, etc., but the power flow of every area must be solvable (see Appendix for details).
Development of the PRGOPF model For the three-area system shown in Figs. 2 and 3, the following symbols are defined:
231
inner state variable vector of area k (k = A, B, C) u~ inner control variable vector of area k (k = A, B: C) control variable vector of U all PQVO separating buses G~(X~, U~, U)= 0 power flow equilibrium equations of area k (k = A, B, C) H~(X~, U~, U) >1 0 inequality constraints of area k (k = A, B, C) penalty term of area k W~(X~, Uk, U) (k = A, B, C) cost function of area k F~(X~, U~, U) (k = A, B, C) X~ Lagrangian multiplier vector of power flow equations of area k (k = A, B, C) The P R G O P F model is
Xk
~_, F~(X~, U~, U)
min
(I)
k= A,B,C
Gk(Xk, Uk, U)
0
k = A, B, C
H~,(Xk, Vk, U) >1 0
k = A, B, C
=
k = A, B, C
Ukm~ <~Uk <. U ~ x
k = A, B, C
Um~ < U <. U~x and the augmented Lagrangian function is
~ k=
[Fk(X~, Uh, U)
-
0U~
(}Gkt +
~Uk
.
a wk
-X~+
- 0
OUk
~Uk
(5)
k = A, B, C aL "~U
-
~
\ ~U +
k = A,B,C
Xh + au
=0 --~-/
(6)
where every area of (3), (4), and (5) is indep e n d e n t of the others, and every area of (6) is separable.
DEVELOPMENT ALGORITHMS
OF PRGOPF CALCULATION
Two, one exact and one approximate, P R G O P F calculation algorithms can be developed from the P R G O P F model.
From the optimal conditions {3), (4), (5), and (6), an exact algorithm for the P R G O P F calculation scheme by using the G N R P F method is developed in the following steps: Step 1. Initialization, IT = 0. Step 2. IT = IT + 1, parallel solution by area computers o f power flow equations Gk(X~, Uk, U) = 0 using the G N R P F calculation method. Step 3. Parallel solution by area computers of (4) for area Lagrangian multipliers:
(2)
Model (1) gives the differences between P R G O P F and S R G O P F (serial reduced gradient optimal p o w e r flow) [7]: in P R G O P F , the cost function, p o w e r flow equations, and all the inequality constraints are in separable form expressed by inner control variables, inner state variables, and control variables of PQVO separating buses, the coupling between every area being implied by the control variables of the PQVO separating buses, while in S R G O P F it is not. The optimal conditions of (2) are:
a~,k
~Fk
A,B,C
+ ~,ktGk{Xk, Uk, U) + Wk(Xk, Vk, U)]
~L
(4)
k = A, B, C
i)L
Exact calculation algorithm of PRGOPF with GNRPF
such that
L=
~L - OFk + OGkt"ha + OWk - 0 0Xh axk 0X~ OXk
- Gk(Xk, Uk, U) = 0
k = A, B, C
(3)
The inverse, multiplying its following vector, of the transposed Jacobian matrix is performed by forward and backward substitutions using the triangulated factors of the Jacobian matrix formed in the latest iteration of the G N R P F calculation. Step 4. Area computers calculate in parallel the gradient c o m p o n e n t s with respect to Uk and part of the gradient c o m p o n e n t s with respect to U from (5) and (6):
guk
~L
~F k
aGff
~ Wk
OUk
0U~
a Uk
0 Uk k=A,B,C
232
OLk
g~]
-
~Fk -
aU
OGkt +
'
~U
bU
~Wk X
k
+
-
-
~U
k = A, B, C Step 5. Transmit each area's g ~ (k = A, B, C) to the central c o m p u t e r to form the full gradient c o m p o n e n t s with respect to U: g u = k= A~,B,cg~f
Step 6. Form a search direction and do a linear optimal search to get the new inner control variables Uk (k = A, B, C) and the PQVO separating control variables U. Step 7. Parallel solution by area computers of the power flow equations Gk(Xk, Uk, U) = 0 using the G N R P F calculation method. Step 8. Converge? Yes, stop. No, go to step 3. In the above procedure, steps 1 - 4 and step 7 are processed in parallel; step 5 takes little execution time because of the small amount of g~]. The a m o u n t of transmitted data in the exact P R G O P F is largely dependent on step 6. When different optimization methods are used to form the search direction, the amounts of transmitted data are different. The minus gradient direction is used as the search direction in this paper, so a very small a m o u n t of data needs to be transmitted. The parallel computation mode of step 6 will be illustrated in the next section. This P R G O P F calculation algorithm is strictly developed, so it is an exact algorithm; its high parallelism is formed naturally when the PQVO separating buses are introduced into the P R G O P F calculation. Approximate algorithm of PRGOPF with GFDPF An approximate calculation scheme for P R G O P F is developed if, in the exact P R G O P F calculation scheme, p o w e r flow is solved using the G F D P F calculation m e t h o d instead of the G N R P F , and Lagrangian multipliers are calculated approximately in step 3 by ~kk = --
[ OV-I([B']t)-IV-1 0 1 V_l{ [B"lt} -1 I~Fk
X
LD--X~ +
}
~Wk DXk]
k=A,B,C
The inverses, multiplying their following vectors, of the transposed B' and B" matrices are performed by forward and backward substitutions using the already triangulated factors of B' and B" respectively. In the scheme of this approximate calculation algorithm, all steps except step 3 are the same as those of the scheme of the exact PRGOPF calculation algorithm because the G N R P F calculation and the GFDPF calculation produce the same power flow solution. This P R G O P F calculation algorithm is approximate because the multiplier vector is calculated approximately in step 3. But there is no approximation between the parallel computation (area computers) and the coordination computation (central computer) in this algorithm.
IMPLEMENTATION OF THE PRGOPF In order to accelerate the P R G O P F to convergence, the generations of all the generators of the whole electrical power system are predispatched in step 1 by use of the classical economic dispatch method without consideration of the loss modification, and also in step 1 a parallel power flow calculation is performed by using the method presented b y E1-Marsatawy et al. [2], then the initial values of the inner control variables and control variables of the PQVO separating buses are determined. If the initial voltage control variables are over the limits, they are set at their limits. In order to coordinate the changes of all the control variables, each c o m p o n e n t of the minus gradient is multiplied by a corresponding factor, the modified minus gradient being used for the optimal linear search. Factor ri for the jth component, --gvj, of the minus gradient is determined as follows:
I Uj Ujmin if -gvi < 0 rj = U j m a x _ _ Uj if --gvj >~ 0 (7) where Ui may be any control variable. It is obvious that ri is always greater than zero. Only the voltage value of the four control variables of each pair of PQVO separating buses has limit constraints, so three factors are chosen by experience for the unlimited control variables. The penalty terms are "
-
-
233
Wk(Xk, =~
U) 1
onjk(Xk, Uk, U ) ]
<[max(O, P j k k
--ttsk 2 }
~jk = max[O, P j k -
2
f~ = akt 3 + b~t 2 + cht + dk
(IHjk(gk, Uk, U)] k
A, B, C
=
(9)
where o is the penalty factor, and pj~ the multiplier of the inequality constraints Hyk(Xk, Uk, U) >1 0 which is updated in every PRGOPF iteration (constraints of X ~ < Xk ~< Xhmax are implied by Hyk(Xk, Uk, U) >1 0 in (8)). The linear search direction is the minus gradient direction modified by the factors in (7). The optimal step length is caluclated as follows. Step I. Each area computer calculates its third-order interpolation polynomial of the
f = ata + bt2 + ct + d
in which k= A,B,C
parallelcomputation
centralcompuLer
coordination computation
'
I[
l
{
l
constraints detection, update ~k partial derivatives calculatioH Lagrangian multiplier calculation
transmit gUk
i
and
1 transmit gu
F f o r , searchl direction ) ~
I
jI calculato
,transmit a.,b.,cL calculate area's three order polynomial I [ K K ylth r espect_______t?step l e n g t h __1 ! ] transmit T . | ]update control varlables I = [°P~alculate i["~- +--~ -
I
I
[power flow c a l c u l a t i o n l
Top ~
"
[
transmit area's convergence information l area'sconvergence detection [ [ l ~
no~ ~
whole ~
system converges;
~ ~
Fig. 4. Flowchart of the PRGOPF calculation.
k = A,B,C
Top~ has two values, tl and t2, and is determined by
l Initialization ] [power flow calculation]
k= A,B,C
The optimal step is solved by --2b + (4b 2 -- 12ac) 1/2 Topt = 6a
I[
area computers
k = A, B, C
and transmits ak, bk, and ck to the central computer. Step 2. The central computer forms the third-order interpolation polynomial of the whole system cost function
(8)
A, B, C
=
area cost function with respect to step length t:
whole system's l c°nvergence detect ion
transmit whole system's , . . convergence xnformat lon
234 Top t =
min(ti[ti > 0)
i=1,2
Topt is transmitted to every area computer. Step 3. Area computers update inner control variables and PQVO separating control variables. In the above procedure, steps 1 and 3 are processed in parallel by area computers. The amount of transmitted data is very small, each area having only three data to be transmitted to the central computer. The central c o m p u t e r has only one datum to transmit to each area computer, that is, the optimal step Topt. In step 3, if any control variable is over the limit after updating, it is set at its limit. Figure 4 gives the flowchart of the P R G O P F ; it is suitable for both the exact and the approximate P R G O P F calculation algorithms. The flowchart supports these remarks: the a m o u n t of transmitted data is very small, the central c o m p u t e r has little computation effort, most computations being performed by the area computers, so the two algorithms have very high parallelism.
NUMERICAL EXAMPLE
A 30-bus two-area system, shown in Fig. 5, is used to test the proposed two algorithms. The system has 12 generators, and is separated into two areas by three separating buses. The generators have a quadratic cost
j/
Fig. 5. A 30-bus two-area s y s t e m .
function. At the initial point, the voltages of four buses and the power flow of one line are over the limit. The proposed two parallel algorithms were simulated on a serial microcomputer IBM-PCAT. Two SRGOPF calculation algorithms (an exact one [7] and an approximate one [8], just like the P R G O P F ) were also programmed for comparison. Table 1 gives the results. The tabular results show that the exact P R G O P F calculation algorithm needs more iterations to converge than the exact S R G O P F calculation algorithm, but the
TABLE 1 N u m e r i c a l results: serial c l o c k i n g Algorithm
Exact PRGOPF
Approx. PRGOPF
Exact SRGOPF
Approx. SRGOPF
Initialization Initial cost 1 0 t h i t e r a t i o n cost CPU t i m e for 10 i t e r a t i o n s 1 5 t h i t e r a t i o n cost Total iterations O p t i m a l cost CPU t i m e p e r i t e r a t i o n Total CPU iteration time CPU o u t p u t t i m e T o t a l CPU t i m e
9.00 13.270 13.432 26.95 13.285 15 13.285 2.67 40.04 5.17 54.21
9.12 13.270 a a a 17 13.521 4.26 67.83 5.22 82.17
9.06 13.270 13.425 28.45 13.392 10 13.425 2.85 28.45 3.51 41.02
9.22 13.270 13.429 32.19 13.390 10 13.429 3.20 32.19 3.46 44.87
a C o n s t r a i n t v i o l a t i o n s exist. Notes: (1) Precise o p t i m a l c o s t = 1 3 . 2 2 7 . (2) C o n v e r g e n c e criteria: cost decrease is less t h a n 0 . 0 1 5 a n d v i o l a t i o n o f a n y c o n s t r a i n t is less t h a n 0.005. (3) T i m e u n i t : second.
235 f o r m e r ' s o p t i m a l c o s t is less t h a n t h a t o f t h e latter, a n d t h e f o r m e r ' s average CPU t i m e p e r i t e r a t i o n is also less t h a n t h a t o f t h e latter. T h e s e t w o a l g o r i t h m s cause all c o n s t r a i n t v i o l a t i o n s to be r e m o v e d a f t e r t h e 5 t h iterat i o n . T h e e x a c t P R G O P F c a l c u l a t i o n algor i t h m , a f t e r i t e r a t i o n 10, h a s t a k e n 2 6 . 9 5 s f o r t h e first 10 i t e r a t i o n s at a c o s t o f 1 3 . 4 3 2 (no c o n s t r a i n t v i o l a t i o n ) w h i c h is v e r y close to t h e o p t i m a l c o s t o f t h e e x a c t S R G O P F . I t c o n t i n u e s to i t e r a t e b e c a u s e o f t h e g o o d d e c r e a s e in cost; a f t e r t h e 1 5 t h i t e r a t i o n , it satisfies t h e c o n v e r g e n c e criteria a n d converges. M e a n w h i l e , t h e e x a c t S R G O P F calc u l a t i o n a l g o r i t h m , a f t e r i t e r a t i o n 10 ( c o s t 1 3 . 4 2 5 ) , has t a k e n 2 8 . 4 5 s f o r t h e 10 iterat i o n s , satisfies t h e c o n v e r g e n c e criteria, a n d c o n v e r g e s b e c a u s e o f t h e p o o r d e c r e a s e in c o s t ; if it h a d c o n t i n u e d t o t h e 1 5 t h iterat i o n it w o u l d h a v e c o s t 1 3 . 3 9 2 (no c o n s t r a i n t violation). T h u s , t h e e x a c t P R G O P F calculat i o n a l g o r i t h m has a smaller average CPU time per iteration than the exact SRGOPF c a l c u l a t i o n a l g o r i t h m u n d e r serial simulation, and under comparable conditions, the c o n v e r g e n c e a n d o p t i m a l cost o f t h e f o r m e r are n o t w o r s e t h a n t h o s e o f t h e latter. The approximate PRGOPF calculation a l g o r i t h m d o e s n o t p e r f o r m well. It is far w o r s e t h a n t h e a p p r o x i m a t e S R G O P F calculation a l g o r i t h m . T h e r e a s o n is t h a t t h e r e are v e r y large errors in t h e L a g r a n g i a n m u l t i p l i e r o f t h e r e a c t i v e p o w e r f l o w e q u a t i o n s , so t h e c o s t decreases slowly a n d t h e c o n s t r a i n t v i o l a t i o n s are r e m o v e d w i t h d i f f i c u l t y in t h e i t e r a t i o n . This a p p r o x i m a t e P R G O P F algor i t h m is n o t c o m p e t i t i v e .
CONCLUSION T w o P R G O P F c a l c u l a t i o n a l g o r i t h m s are p r e s e n t e d f o r t h e first t i m e in this p a p e r . I t is possible t o i n t r o d u c e P Q V O s e p a r a t i n g buses a n d generalized p o w e r f l o w c a l c u l a t i o n m e t h o d s into t h e P R G O P F c a l c u l a t i o n algorithms. The exact PRGOPF calculation a l g o r t h i m is v e r y successful; it has v e r y high parallelism, o n l y a v e r y small a m o u n t o f d a t a n e e d s t o b e t r a n s m i t t e d , a n d t h e r e is n o a p p r o x i m a t i o n b e t w e e n t h e lower-level c o m p u t a t i o n (area c o m p u t e r s ) a n d higher-level computation (central computer). This exact a l g o r i t h m is reliable a n d c o m p e t i t i v e , its
speed, c o n v e r g e n c e a n d o p t i m a l c o s t being no worse than those of the exact SRGOPF c a l c u l a t i o n a l g o r i t h m w h e n parallel c o m p u t a t i o n is s i m u l a t e d on a serial p r o c e s s i n g c o m p u t e r . G o o d savings in CPU t i m e can be m a d e if t h e a l g o r i t h m is p e r f o r m e d on a t w o level c o m p u t e r n e t w o r k . The approximate PRGOPF calculation a l g o r i t h m p e r f o r m s badly. I t can o n l y be r e g a r d e d as an a t t e m p t in t h e search f o r n e w parallel c a l c u l a t i o n a l g o r i t h m s .
NOMENCLATURE GNRPF GFDPF PRGOPF SRGOPF
generalized N e w t o n - R a p h s o n power flow generalized fast d e c o u p l e d p o w e r flow parallel r e d u c e d g r a d i e n t o p t i m a l power flow serial r e d u c e d g r a d i e n t o p t i m a l power flow
REFERENCES 1 R. Kasturi and M. S. N. Potti, Piecewise NewtonRaphson load flow -- an exact method using ordered elimination, IEEE Trans., PAS-95 (1976) 1244 - 1253. 2 M. El-Marsatawy, R. W. Menzies and R. M. Mathur, A new, exact, diakoptic, fast decoupled load flow technique for very large power systems, IEEE PES Summer Meeting, 1979, Paper No. A79 440-9. 3 C. E. Lin and G. L. Viviani, Hierarchical economic dispatch for piecewise quadratic cost functions, IEEE Trans., PAS-103 (1984) 1170 - 1175. 4 K. Aoki and T. Satoh, Robust algorithm for economic dispatch of interconnected systems, IEEE Trans., PAS-104 (1985) 266 - 272. 5 K. Seidu and H. Mukai, Parallel multi-area static state estimation, IEEE Trans., PAS-104 (1985) 1026 - 1034. 6 H. Sasaki, K. Aoki and R. Yokoyama, A parallel computation algorithm for static state estimation, IEEE Trans., PWRS-2 (1987) 624 - 632. 7 H. W. Dommel and W. F. Tinney, Optimal power flow solution, IEEE Trans., PAS-87 (1968) 1866 - 1876. 8 O. Alsac and B. Stott, Decoupled algorithms in optimal load flow calculations, IEEE PES Summer Meeting, 1975, Paper No. A75 545-4. 9 G. R. Krumpholz, K. A. Clements and P. W. Davis, Power system observability: a practical algorithm using network topology, IEEE Trans., PAS-99 (1980) 1534 - 1542.
236 APPENDIX
Generalized power flow calculation There are four variables at a bus, P, Q, V, and 0 (real and reactive power injections, voltage value and angle}; these four variables are called the bus variables. In a c o m m o n power flow calculation, two of the bus variables are control variables, and there are only three types of buses, the PQ bus (real and reactive power injections are control variables}, the PV bus (real power injection and voltage value are control variables}, and the slack bus (voltage value and angle are control variables)• In generalized power flow calculations, however, the control variables of a bus can be any combination of the bus variables P, Q, V, and 0, so the types of buses are generalized, the number of possible combinations of the four bus variables is fifteen, and a bus at which none of the four bus variables is a control variable is also a type of bus, the total number of bus types being sixteen. In generalized power flow calculation, real and/or reactive power flows of transmission lines can be set to be control variables, that is, real and/or reactive power flows of some lines can be specified, and the specified line real and/or reactive power flows are used as power flow equilibrium equations.
cause it makes the parallel optimal power flow calculation possible• The power flow equilibrium equations in the generalized power flow calculation are (i) the real power equilibrium equations of the PQ, P, PV and PQVO buses, and the line real power flow specification equations, and (ii) the reactive power equilibrium equations of the PQ and PQ VO buses, and the line reactive power flow specification equations• The state variables to be solved are (i) the angles of the PQ, P, U, V, and PV buses, and (ii} the voltages of the PQ, P, and U buses• The number of state variables to be solved must be equal to the total number of power flow equilibrium equations. Because of the P-Q decomposition characteristics of electrical power systems, the number of real power equilibrium equations must be equal to the number of u n k n o w n bus angles, and the number of reactive power equilibrium equations must be equal to the number of unknown bus voltage values. The modified equations of the GNRPF calculation are AP1
A01
AQ:
AV
•
°
AP~
A0~
AQ,
=J
A V,
(A-l)
GNRPF solution and GFDPF solution For simplicity, only seven bus types are introduced in the generalized power flow calculation of this paper; t h e y are the PQ, P, U, V, PV, PQ VO and the slack bus and they are sufficient for the present PRGOPF calculation. The symbols for the first six buses indicate the bus control variables of the corresponding buses; for example, the P bus means that the bus real power injection is the control variable, and the V bus means that the bus voltage value is the control variable, but the U bus means that none of the four bus variables is a control variable (U means unknown}. It is obvious that only generator buses can be set as P, U, V, PV, or slack buses because of their adjustable
P,Q, and V. The PQVO bus is the bus at which all four bus
variables are control variables. The PQVO bus is the key bus of this paper, be-
AP~ 1
A0~
AP~
A0~
where s is the number of u n k n o w n bus angles and r the number of u n k n o w n bus voltage values• The modified equations of the GFDPF calculation are
= VB'V ~0~
(A-2)
= VB" A V~
(A-3)
8
E'°I
237 In eqn. (A-l), the structure of J is not symmetrical, and in eqns. (A-2) and (A-3), B' and B" are n o t symmetrical. The calculation scheme of the GNRPF is the same as that of the c o m m o n NRPF calculation, and the calculation scheme of the GFDPF is the same as that of the comm o n FDPF calculation.
Solvability of the generalized power flow calculation If the power flow equilibrium equations in the generalized power flow calculation are taken as measurement equations w i t h o u t redundancy in the static-state estimation, the solvability of the generalized power flow calculation is the same as the observability of the static-state estimation. The algorithms for the observability analysis of the staticstate estimation can be used for the solvability analysis of the generalized power flow
calculation. The algorithm for the observability analysis of the static-state estimation (also t h a t for the solvability analysis of the generalized power flow calculation) will not be presented here; details are reported in ref. 9.
Concluding remarks The results of the generalized power flow calculation methods on the test systems are not listed, but some remarks are given here. (i) Many types of buses can be introduced into the generalized power flow calculation. Generalized power flow calculation methods not only can be used for c o m m o n power flow calculation, but also for some special power flow calculations. (ii) The GNRPF calculation has nearly the same convergence and speed as the c o m m o n NRPF calculation. {iii) The GFDPF calculation has a convergence and speed close to the c o m m o n FDPF calculation.