A Distributed Method for Solving Nonlinear Equations Applying the Power Load Flow Calculation Minetada Osano & Miriam A. M. Capretz School of Computer Science & Engineering University of Aizu Aizu-Wakamatsu City Fukushima, 965-80 Japan
[email protected] [email protected]
Abstract A new approach for distributed power load flow calculation using nonlinear equations is presented. This new approach, which is similar to the Newton Raphson’s simple method, uses an inverse Jacobian matrix of initial states for the iteration process. Moreover, nonlinear quadratic equations have been used as they are more appropriate for the distributed power load flow calculation. This paper describes and compares the new approach with the Newton Raphson method [1-6]. It shows that such an approach is more suitable for distributed power load flow calculation as well as it discusses some of its applications.
1. Introduction Nowadays, large and complex power systems are demanding higher speed processing control. One way of solving such a problem is by dividing these systems into subsystems so that control and operation can be applied to each of such subsystems individually. Thus, the subsystems are independent of each other but cooperation among them should exist in order to assure safety and security of the entire system. Research so far conducted in the field of power systems applies central load flow for distributed systems[7-10]. Power flow techniques that deal with distributed load flow calculation have not been seen in the literature yet. Central load flow for distributed systems causes an unbalance in the system as it requires the control of the total system structure. Therefore, distributed load flow calculation for distributed systems should be more appropriate. This paper presents a new technique of distributed load
flow calculation using nonlinear equations along with its application. Parallel processing is performed on this distributed power load flow calculation method. Such a technique is useful for operational processes such as stability control and power supply within local areas. The remaining of this paper is organized as follows. The next section presents the theory of the Constant Jacobian Gauss method which has so far been applied to non-parallel and non-distributed systems. An application of the power load flow calculation to non-distributed systems is also presented. In section 3, the modification of the Jacobian Matrix in order to be applied to distributed systems as well as an application of the power load flow calculation to distributed systems are described. Section 4 describes the complexity involved with the power load calculation methods for distributed and non-distributed systems. Finally, section 5 presents the conclusion of this paper.
2. Constant Jacobian Gauss Method 2.1. Theory The proposed technique uses only one Constant Jacobian matrix during the iteration process for solving nonlinear equations. In general, the Taylor’s theorem of the nonlinear equation xo + dx is seen as follows:
x0 + dx =
x0 +
0
x0 dx + G x0 dx
(1)
where x0 is the initial value and dx is the mismatch of the initial value x0 . If this equation is satisfied with the condition of x0 + dx = 0, then the equation can be rewritten as follows:
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J x0 dx = x0 G x0 dx (2) x0 , and x0 + dx is the approximate where J x0 = 0
solution for this equation. As finding such a solution is not straightforward, the technique of the iteration method uses the following two iterative equations at the k + 1th step:
G x0 dx
k
x0 + dx
=
k
+
Replacing the initial value of G Eq.(2) is rewritten as follows:
dx
k
+1
J x0
=
1
G x0 dx
k
x0
1
:
(3)
with
x0 , the
G x0 dx
(4)
k
where a solution can be obtained by the convergence of the mismatch dxk+1 after each iteration process. In the above equation, only function G xo dxk of Eq.(3) has to be calculated in order to obtain a better approximate mismatch dxk+1 . This dxk+1 is used in G x0 dxk+1 in the k+2th iteration step in order to obtain a closer approximate solution. Such an iteration process is executed until the approximate calculated solution is in conformity with the limited converged condition.
The above basic equations are the quadratic nonlinear equations with ej and fj . As the quadratic equations in G x0 dx = dx, the Eq.(1) in Taylor’s Theorem can be rewritten as follows:
F x0 + dx = F x0 + J x0 dx + F dx
where J x0 = F x0 . Thus, the calculation of G function in each iterative step in Eq.(3) and (4) can be simplified. Moreover, it shows that the proposed method is capable of solving the above equations as well as the conventional NR method. 0
Example of Power Load Calculation A simple example is given below where a power system is constructed with four nodes (node 1 is of standard type, 2 and 3 are of load type and 4 is of generation type). The power load flow calculation is executed in such a system using the Constant Jacobian Gaussian (CJG) and the Newton Raphson (NR) methods. At first, the parameters are set as:
2.2. Application of Power Load Calculation
"
Functions and variables for the power load flow calculation are defined in [11] as
F x = 2 '2 3 '3 :: ' ::: ' and x = e2 f2 e3 f3 :: e f ::::e f respectively ; and v = e + jf is a voltage vector of the ith node. e , and f are the real and imaginary parts of the voltage vector and each and ' are given as follows: i
i
i
i
i
n
i
n
n
i
i
i
i
i
- for the load node: n X
x = P 0 i
i
j
+
g ee ij
b ef ij
j
n X
' x = Q 0 i
=1
i
j
=1
j
ij
j
g ef
b ee j
j
i
ff j
i
j
i
f f = 0 j
(5)
fe
i
i
(6)
- for the generation node:
' x = v20 i
i
e2 + f 2 = 0 i
Q 20
:
Q 30
:
:
v
:
j
:
2
v4
0
= 1 21 :
and the admittance matrix is denoted as: 0:934 j 0:426 0:480 + j 2:403 0:453 + j 1:891 0:0
0:480 + j 0:240 1:069 j 4:727 0:588 + j 2:353 0 :0
0:453 + j 1:891 0:588 + j 2:352 1:042 + j 8:243 j 3:666
0 :0 0 :0 j 3:666 j 3:333
#
The execution of the system using both methods achieved the expected solution and convergence state. The result of the application of both methods are shown in tables 1 and 2. These tables, however, reveal that the CJG method takes two times more iteration steps than the NR method to converge to the solution.
i
f e = 0
i
ij
+
i
= 0 13 = 0 18 10 = 1 05 + 0 0
:
:
T
T
n
= 0 55 = 0 30 40 = 0 50
P 20 P 30 P
General Expression
(8)
(7)
i
where Pi0 and Qi0 are the requested real and imaginary power respectively at the ith node, and gij and bij are the real and imaginary parts of the admittance constant.
3. Application of the Jacobian Matrix to Distributed Power Load Flow Calculation The NR method is the most powerful method to solve nonlinear equations. However, this method is not appropriate when dealing with large systems as it takes more time to converge to the solution. Further, the method is also not suitable for parallel processing in distributed systems as it requires the calculation of an inverse matrix at each iteration step. The CJG method seems to be more appropriate for distributed processing of nonlinear systems as it is shown in this section. The CJG method and its application for the distributed power load flow are detailed next.
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3.1. Modification of the Jacobian Matrix for Distributed Systems
Table 1. The converged state on Constant Jacobian Gaussian method (initial value 6) x = 1:0, converged condition 10
time
k
1 2 3 4 5 6 7 8
Solu tion
k
dx2
0:02221 0 10808 0:04002 j 0:10801 0:04099 j 0:10838 0:04128 j 0:10838 0:04130 j 0:10839 0:04131 j 0:10839 0:04131 j 0:10839 0:04131 j 0:10839 0:95869 j 0:10839 j :
k
dx3
0:00370 0 00943 0:01494 j 0:00894 0:01510 j 0:00864 0:01535 j 0:00860 0:01536 j 0:00860 0:01536 j 0:00860 0:01536 j 0:00860 0:01536 j 0:00860 0:98464 j 0:00860 j :
k
dx4
1:10500 +j 0:12693 0:09143 +j 0:12691 0:09277 +j 0:12881 0:09240 +j 0:12890 0:09242 +j 0:12895 0:09241 +j 0:12895 0:09242 +j 0:12895 0:09242 +j 0:12895 1:09242 +j 0:12895
time
1 2 3 4
solu tion
k
x2
0:97779 j 0:10808 0:95904 j 0:10837 0:95869 j 0:10839 0:95869 j 0:10839 0:95869 j 0:10839
k
x3
0:99630 j 0:00943 0:98478 j 0:00859 0:98464 j 0:00860 0:98464 j 0:00860 0:98464 j 0:00860
k
0:01787 0:00323 4 :6 E
4
6 :8 E
5
1 :0 E
5
1 :4 E
6
2 :4 E
7
k
eps
0:23193 0:01902 0:00371 2 :4 E
7
=
X
0+ 0
k
Gi
n
0:23193
x4
1:10500 +j 0:12693 1:09249 +j 0:12893 1:09242 +j 0:12895 1:09242 +j 0:12895 1:09242 +j 0:12895
J x0 ii dxi
eps
Table 2. The converged state on Newton Raphson method (initial value x = 1:0, converged condition 10 6 ) k
In order to apply the CJG method to parallel processing in distributed systems using general nonlinear equations, the Eq. (4) is rewritten for the ith subsystem as the following equation:
j
=1j 6=j
J x
k
=
k
zi
=1j 6=j
= 0 1 0 + 0 1 0 .
where zi
=
ij
J x
J x
ii
ii
J x
Gi x
z
1
k i
dx
k j
1
n
+ j
zij
dx
X
then dxi
x
k ij
k i
dx
1
1
(9)
k j
dx
1
(10)
and
ij
The term zi is a function of the ith subsystem with x0 and dxik 1 variables and zij as a constant matrix. dx and k+1 dx show all variables in a given system and all varii ables in the ith subsystem respectively, whereas J x0 i and k k Gi x0 dxi i equations are related specificaly to the ith subsystem, and Gki x0 dxkj is composed of variables related to the ith subsystem and other jth subsystems. Note that Eqs. (9) and (10) contain all variables and relations required by the entire system. In addition, dxkj in Eq.(9) is the variable which connects directly the ith subsystem with other jth subsystems. Thus, when the ith subsystem receives a renewal state from a direct connected subsystem on the k-1th iteration step, the solution on Eq.(10) of the ith subsystem enables the calculation to more converged state on the kth iteration step. If this process is carried out on each subsystem simultaneously, all variables in a given system can reach one solution. Moreover, as the process can be divided into subsystems, parallel and distributed processing can be executed with several machines.
3.2. Application of the Power Load Flow Calculation The modified Jacobian matrix for distributed systems can be applied to the power load flow calculation using Eqs. (8), (9) and (10) on the ith subsystem. The example system shown in the previous section will be used for expressing this approach. It is supposed that one processor can support any of the four nodes of the example system. However, three processors are required for nodes 2, 3 and 4 in order to
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enable the parallel calculation of this system. The equation for one processor to calculate node 2 is shown below:
1:044 4:577
4:877 1:093
+F1 dx + k
1
dx1 =
0:588 2:353
k
2:353 0:588
0:526 0:019
dx2
k
1
where the Jacobian matrix of the entire system is:
2 6 6 4
1:044 4:577 0:588 2:352 0 0
4:877 1:093 2:353 0:588 0 0
0:588 2:353 1:019 8:480 0 0
2:353 0:588 8:005 1:064 3:666 0
0 0 0 3:666 0 2 :0
0 0 3:666 0 3:666 0
(11)
3 7 7 5
In the algorithm for parallel processing, two more equations similar to Eq. (11) should be given to two more processors to calculate nodes 3 and 4. The parallel processing is then executed at each processor using their respective equations and whenever necessary exchanging data among themselves in the iterative process. The result after the execution of the example system on the three processors is given in table 3. Such a result shows that (1) this approach can converge to the approximate solution; (2) when compared with the sequential approach more iteration steps are required to obtain the solution; and (3) less calculation time is required so that it makes it very fast.
4.3. Parallel Processing Let us consider n0 as the subsystem order of a distributed system and m0 the number of iteration steps to converge to the solution in such a subsystem. Thus,
n3+mnnm nn when n is small and n is large so that n 3 n n. n 3 is 0
0
0
0
0
0
0
0
0
the number of multiplications of the inverse matrix of order n0 and m0 n0 n is the order of the calculating process of the G function and operations of the Jacobian matrix. Consequently, each subsystem can be processed more efficiently as the parallel processing in this way has been shown as very effective and fast.
4.4. Ill Condition As far as ill condition is concerned, because the CJG method uses the G function it converges to the solution whereas the NR method does not[11]. Thus, in the CJG method even if the Jacobian matrix presents a singular point between the initial point and the solution, it still converges to the solution. In the case of the NR method, only with a convex function it will converge to the solution.
5. Conclusion 4. Complexity of the Power Load Flow Calculation Method 4.1. Convergency The error ratio of the NR method is "k 2 in the (k+1)th iteration step while in the CJG method the error ratio is "k . Consequently, the application of the NR method converges to the solution faster than the CJG method. Moreover, if the converged condition has m iterations, the CJG method takes two times more (2m iterations) than the NR method to converge to the solution.
4.2. Speed to Solve an Equation
This paper presented a new approach suitable for distributed power load flow calculation as well as an example of central and distributed power load flow calculation. The Constant Jacobian Gaussian (CJG) method has been applied in the approach for distributed power load flow calculation. Some of the characteristics of this approach are:
The speed to solve an equation depends on the number of the multiplication of the operators involved. Thus, if a Jacobian matrix is of order n, the inverse matrix takes n3 times order operations. Consequently, the total number of multiplication in the NR method is mn3 + n2 and in the CJG method is n3 + 2n2 m, where 2n2 depends on the calculating process of the G function and is an arbitrary number. Therefore, the larger the n, the NR method takes m times more operations than the CJG method, which makes the CJG method converges to the solution faster.
the CJG method takes about twice more iteration steps than the Newton Raphson (NR) method, to converge to the solution; the CJG method converges faster to the solution than the NR method when dealing with large systems; the CJG method by applying the G function makes parallel processing possible and easy in the power load flow calculation method.
Distributed power load flow calculation is more suitable to operate and control large power systems, which makes the proposed approach and algorithm very effective and useful.
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References [1] W. G. Tinny and C. E. Hart, “Power Flow Solutions by Newton’s Method”, IEEE Trans. on Power Apparatus and Systems, Vol.PAS-86, 1967, pp. 14491457. [2] B. Stott and O. Alsac, “Fast Decoupled Load Flow”, IEEE Trans. on Power Apparatus and Systems, Vol. PAS-93, no. 3, May/June 1974, pp. 859-869. Table 3. The converged state with CJG method on parallel processing (initial value x = 1:0, converged condition 10 5 ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
I teration k
solution x2
x3 x4
eps
2:4136E 1 6:7578E 2 2:7896E 2 1:9115E 2 1:6950E 2 1:2504E 2 1:0821E 2 7:8597E 3 6:7721E 3 4:9138E 3 4:2272E 3 3:0703E 3 2:6370E 3 1:9179E 3 1:6444E 3 1:1978E 3 1:0253E 3 7:4794E 4 6:3925E 4 4:6699E 4 3:9851E 4 2:9156E 4 2:4844E 4 1:8202E 4 1:5489E 4 1:1364E 4 9:6566E 5 7:0944E 5 6:0208E 5 4:4293E 5 3:7536E 5 2:7664E 5 2:3424E 5 1:7277E 5 1:4610E 5 1:0788E 5 9:0971E 6 0:95869 j 0:10839 0:98464 j 0:00860 1:09242 + j 0:12895
[3] K. Behnam Guilani, “Fast Decoupled Load Flow”, IEEE Trans. on Power Systems, Vol.3, no. 2, May 1988, pp. 734-742. [4] H. W. Dommel, W. F. Tinney and W. L. Powell, “Further Developments in Newton’s Method for Power Systems Applications”, Paper 70 CP 161-PWR, presented at IEEE PES Winter Meeting, New York, January 1970 . [5] W. F. Tinney and W. S. Meyer, “Solution of Large Space Systems by Orderd Triangular Factorization”, IEEE Trans. on Automatic Control, Vol. AC-18, no.4, 1973, p. 333. [6] G. X. Lue and A. Semlyen, “Efficient Load for Large Weakly Meshed Networks”, IEEE Trans. on Power Systems, Vol.5, no.4, November 1990, pp. 1309-1316. [7] B. A. Carre, “Solution of Load Flow Problems by Partitioning System into Trees”, IEEE Trans. on Power Apparatus and Systems , Vol. PAS-87, no.11, 1968 p. 1931. [8] A. M. Sasson, “Decomposition Techniques Applied to the Nonlinear Programming Load Flow Method”, IEEE Trans. on Power Apparatus and Systems, Vol. PAS-89, no.1, 1970, p. 78. [9] H. K. Keasvan, M. A. Pai and M. V. Bhat, “Piecewise Solution of the Load Flow Problem”, IEEE Trans. on Power Apparatus and Systems , Vol. PAS-70, no.11, 1971, p. 1382. [10] R Kasturi and M. S. N. Potti, “Piecewise Newton Raphson Load Flow - An Exact Method Using Ordering Elimination”, IEEE Trans. on Power Apparatus and Systems, Vol. PAS-95, no.4, 1976, p.1224. [11] S. Iwamoto and Y. Tamura, “A Load Flow Calculation Method for Ill-Conditioned Power Systems”, IEEE Trans. on Power Apparatus and Systems, Vol.PAS-100, no. 4, April 1981, pp. 1736-1743.
Proceedings of The Thirtieth Annual Hawwaii International Conference on System Sciences ISBN 0-8186-7862-3/97 $17.00 © 1997 IEEE
1060-3425/97 $10.00 (c) 1997 IEEE
