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Approximate linear decoupled solution as the initial value of power system load flow
ELEOTIqlO IOUAn ELSEVIER
Electric Power Systems Research 32 (1995) 161 163
,,
Approximate linear decoupled solution as the initial value of power system load flow e Georgios Leonidopoulos 11 Kilkis Street, Kalamata, 241 00, Greece
Received 10 October 1994
Abstract In this paper an approximate solution of the load flow equations is developed using the same principles employed for developing the fast decoupled method. The approximate decoupled solution takes less time to calculate than one iteration of the fast decoupled method and gives a good approximate picture of the V,D profile of a power system. In the paper, it is used as the initial value of the fast decoupled method, which is applied on standard IEEE test systems, and a solution is achieved to within 0.01 p.u. (1 p.u. = 100 MW/MVAr) of maximum busbar mismatches in less than 50% of the number of iterations that the method takes to converge when a flat initial value (V= 1.0 p.u., D = 0.0 rad) is considered. Keywords': Load flow
I. Introduction In power systems load flow calculation is basic and necessary in most cases [1-4]. The non-linearity of the load flow equations makes them difficult to solve exactly mathematically and they can only be solved numerically to within a specified accuracy (usually 0.01 p.u.). Many numerical methods exist for their solution such as the Gauss-Seidel method, Newton's method, and the fast decoupled method. Despite many numerical load flow methods having been developed, what is missing from the power system discipline is a good approximate mathematical solution of the load flow. The present paper tries to fill this gap, developing an approximate mathematical solution of the load flow problem using the assumptions that were employed to deduce the fast decoupled method from Newton's method. The approximate load flow solution is a decoupled solution, i.e. the voltage magnitude V is a function of the reactive power Q and the voltage angle D is a function of the active power P only. From it, the basic equations of the fast decoupled method are deduced very easily. The approximate decoupled solution can be used as the initial value of the fast decoupled method, considerably improving its convergence characteristic.
2. Approximate linear decoupled solution of load flow The load flow equations are as follows. (a) Active power Pk = Vk ~, V~ [Gk/CoS(Dk -- D~ ) + Bk/sin(D/, - Di )] ]~1
Taking into consideration the assumptions of the fast decoupled method (FDM), Vk = V~ = 1.0 p.u., and then expanding the functions s i n ( D ~ - D / ) and cos(D~.- D / ) around zero, we have sin(D k - D~ ) = (Dk -- D / ) cos(Dk-- D/ ) = 1 -
(D k - D/)3 3! + """
( D k - D~ )2 2! +" " "
Now, taking the first term of each expansion, Pk becomes P~ = ~
[Gk/+ B k / ( D k -- D~ )]
i= 1
/--J
= ~
/
I
Bk/(Dk-D/)
i=l
= Bkl (Dk -- DI) + " " " + Bk~.(Dk-- Dk) + " " " ' : T h e content of the paper is covered by Greek Patent No. 1001140. 0378-7796/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved S S D I 0378-7796(94)00911 -M
+ B~,,, (Dk -- D,, )
162
G. LeonMopoulos, Electric Power Systems Research 32 (1995) 161 163
Table 1 Performance of the F D M using the FIS and the ADS as the initial value
Iteration 1
1 2 3 4 5 6
Iteration 2
Max lAP[
Max IA Q I
Max IA Vl
Max IADI
Max IAPI
Max IAQI
Max IA Vl
Max IADt
0.91 0.12 0.92 0.08 2.85 0.27
0.60 0.34 0.83 0.24 1.94 0.22
0.068 0.021 0.054 0.039 0.170 0.089
0.31 0.07 0.35 0.08 0.43 0.07
0.510 0.095 0.150 0.082 0.270 0.180
0.085 0.025 0.082 0.024 0.230 0.110
0.018 0.003 0.022 0.007 0.050 0.020
0.048 0.006 0.072 0.007 0.099 0.023
Max
Max
Max
Max
of
IAPI
[zxal
Ivf - v,I
IDf-D~[
iterations
0.0075 0.0050 0.0077 0.0030 0.0099 0.0099
0.009 0.007 0.009 0.005 0.010 0.010
0.061 0.019 0.056 0.034 0.140 0.079
0.280 0.073 0.310 0.080 0.360 0.090
6 3 5 3 18 8
Final solution No.
1 2 3 4 5 6
All the values are in p.u. FIS = flat initial solution: V = 1.0 p.u., D = 0.0 rad. A D S = approximate decoupled solution. Vf and Df = final voltage magnitude and angle. Vi and D~ = initial voltage magnitude and angle. 1: IEEE 14-node system, F D M and FIS were used. 2: IEEE 14-node system, F D M and A D S were used. 3: IEEE 30-node system, F D M and FIS were used. 4: IEEE 30-node system, F D M and A D S were used. 5: IEEE 57-node system, F D M and FIS were used. 6: IEEE 57-node system, F D M and A D S were used. = -BklDI
-Bk2D2
+ Bk(k + I) + "
....
+(Bkl
• + Bk, , ) D k . . . . .
+'''-[-Bk(k
Taking
1)
into consideration
the F D M
assumptions,
i.e.
Bk, , D,,
V k = 1.0 p.u.
Since Bkl + • " " + Bk(k-~
c o s ( D k - D s ) = 1.0
+ Bk~k + l~ + " " " + Bk,, = - - B k k
and
then
Gkj sin(Ok -- Dj ) << Bk/ P k = - - B k l D1 . . . . .
BkkDk .....
In m a t r i x f o r m , for k = 1 . . . . . a n d D , = 0.0, we h a v e
{i }=I m+I
L
--
B(,,~ + ,~1
.
.
.
.
--
Bk,,D,,
m + l (P, V b u s b a r s )
Qk b e c o m e s Qk=
j= 1
~Bks
In m a t r i x f o r m , for k = 1 . . . . .
][i]
B ( , . + ~)(,.~+ D
-- ~
m (P,Q busbars),
,,~ + l
or
[P., +, ] = [PB][D,., +, ]
(1)
....
(b ) R e a c t i v e p o w e r
]
+
Qk = Vk ~ j=l
Vj [Gkj sin(Dk - - D i ) -- Bkj CoS(Dk -- Dj )] _--Bin(re+l)
--
mn
163
G. Leonidopoulos / Electric Power Systems Research 32 (1995) 161 163 Table 2 CPU times (in seconds) for different program stages System
1
2
3
4
5
6
G, B, Y matrix formation [PB] formation and inversion [QB~] formation and inversion ADS calculation AP calculation convergence test AD and new D calculation AQ calculation convergence test A V and new V calculation
0.12 0.07 0.02
0.13 0.08 0.03 0.01 0.01 0.00 0.01 0.00
0.37 0.38 0.18
0.29 0.32 0.19 0.00 0.03 0.01 0.04 0.01
1.00 1.85 1.14
0.98 1.79 1.06 0.04 0.13 0.02 0.12 0.02
0.01 0.00 0.01 0.01
0.02 0.01 0.03 0.0l
or
[Q,, ] = [QB, ][ V,, ] + [QB b ][ V,..... ]
(2)
Then, from Eq. (1), we get [D,,,+,] = [PB] '[P,,,+,]
(3)
On the other hand, from Eq. (2), we have [V,,] = [QBa] ' ( [ O , , ] - [QBb][V, _,, ])
(4)
Eqs. (3) and (4) constitute the approximate decoupled solution of the power system load flow equations. In order to find the corrections [A V,, ] and [AD m+/], we take the differences of Eqs. (3) and (4): [AVm] = [QB~] I[AQm ]
(5)
[AD,, +, ] = [PB] - ' lAP,, +, ]
(6)
Eqs. (5) and (6) are the basic equations of the fast decoupled method [1]. 3. Performance of the method
The performance of the fast decoupled method using the approximate decoupled solution as the initial value is shown in Table 1. The standard IEEE testing systems of 14, 30 and 57 nodes were used for the test. A VAX computer was used to obtain the results and the CPU times needed to complete the different stages of the program are shown in Table 2.
0.12 0.01 0.13 0.01
The approximate decoupled solution was also used as the initial value of the fast decoupled method, reducing the number of iterations needed for the method to converge to less than 50% of those required when a flat initial value (V = 1.0 p.u., D = 0.0 rad) is considered. A shorter time is needed to calculate the approximate decoupled solution than for one iteration of the fast decoupled method and this makes its use as an initial value advantageous. The approximate decoupled solution can also be used as the initial value of any other numerical method for load flow, considerably improving its convergence characteristic since it is very close to the actual load flow solution.
5. Nomenclature
Dk AD 1 m n Pk AP k Qk AQk
Vk AV
Yk~
voltage angle of busbar k voltage angle correction number of P, V busbars number of P,Q busbars total number of busbars net active power of busbar k active power mismatch of busbar k net reactive power of busbar k reactive power mismatch of busbar k voltage magnitude of busbar k voltage magnitude correction admittance of line k-j, Yk~= Gkj +jBk~
4. Discussion and conclusions References
An approximate mathematical solution of the load flow problem has been developed in this paper. The solution is a decoupled one. In other words, V depends on Q and D on P only. This is expected since the assumptions of the fast decoupled method were employed to deduce the approximate decoupled solution. This makes the solution carry all the well-known advantages and disadvantages inherent in the fast decoupled method.
[1] B. Stott and O. Alsac, Fast decoupled load flow, IEEE Trans. Power Appar. Syst., PAS-93 (1974) 859-867. [2] F.F. Wu, Theoretical study of the convergence of the fast decoupled load flow, 1EEE Trans. Power Appar. Syst., PAS-96 (1977) 189 197. [3] B. Stott, Review of load flow calculation methods, Proc. 1EEE, 62 (1974) 916-929. [4] J. Arrillaga, C.P. Arnold and B.J. Harker, Computer Modelling of Electrical Power Systems, Wiley, Chichester, UK, 1983.
Electric Power Systems Research 32 (1995) 161 163
,,
Approximate linear decoupled solution as the initial value of power system load flow e Georgios Leonidopoulos 11 Kilkis Street, Kalamata, 241 00, Greece
Received 10 October 1994
Abstract In this paper an approximate solution of the load flow equations is developed using the same principles employed for developing the fast decoupled method. The approximate decoupled solution takes less time to calculate than one iteration of the fast decoupled method and gives a good approximate picture of the V,D profile of a power system. In the paper, it is used as the initial value of the fast decoupled method, which is applied on standard IEEE test systems, and a solution is achieved to within 0.01 p.u. (1 p.u. = 100 MW/MVAr) of maximum busbar mismatches in less than 50% of the number of iterations that the method takes to converge when a flat initial value (V= 1.0 p.u., D = 0.0 rad) is considered. Keywords': Load flow
I. Introduction In power systems load flow calculation is basic and necessary in most cases [1-4]. The non-linearity of the load flow equations makes them difficult to solve exactly mathematically and they can only be solved numerically to within a specified accuracy (usually 0.01 p.u.). Many numerical methods exist for their solution such as the Gauss-Seidel method, Newton's method, and the fast decoupled method. Despite many numerical load flow methods having been developed, what is missing from the power system discipline is a good approximate mathematical solution of the load flow. The present paper tries to fill this gap, developing an approximate mathematical solution of the load flow problem using the assumptions that were employed to deduce the fast decoupled method from Newton's method. The approximate load flow solution is a decoupled solution, i.e. the voltage magnitude V is a function of the reactive power Q and the voltage angle D is a function of the active power P only. From it, the basic equations of the fast decoupled method are deduced very easily. The approximate decoupled solution can be used as the initial value of the fast decoupled method, considerably improving its convergence characteristic.
2. Approximate linear decoupled solution of load flow The load flow equations are as follows. (a) Active power Pk = Vk ~, V~ [Gk/CoS(Dk -- D~ ) + Bk/sin(D/, - Di )] ]~1
Taking into consideration the assumptions of the fast decoupled method (FDM), Vk = V~ = 1.0 p.u., and then expanding the functions s i n ( D ~ - D / ) and cos(D~.- D / ) around zero, we have sin(D k - D~ ) = (Dk -- D / ) cos(Dk-- D/ ) = 1 -
(D k - D/)3 3! + """
( D k - D~ )2 2! +" " "
Now, taking the first term of each expansion, Pk becomes P~ = ~
[Gk/+ B k / ( D k -- D~ )]
i= 1
/--J
= ~
/
I
Bk/(Dk-D/)
i=l
= Bkl (Dk -- DI) + " " " + Bk~.(Dk-- Dk) + " " " ' : T h e content of the paper is covered by Greek Patent No. 1001140. 0378-7796/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved S S D I 0378-7796(94)00911 -M
+ B~,,, (Dk -- D,, )
162
G. LeonMopoulos, Electric Power Systems Research 32 (1995) 161 163
Table 1 Performance of the F D M using the FIS and the ADS as the initial value
Iteration 1
1 2 3 4 5 6
Iteration 2
Max lAP[
Max IA Q I
Max IA Vl
Max IADI
Max IAPI
Max IAQI
Max IA Vl
Max IADt
0.91 0.12 0.92 0.08 2.85 0.27
0.60 0.34 0.83 0.24 1.94 0.22
0.068 0.021 0.054 0.039 0.170 0.089
0.31 0.07 0.35 0.08 0.43 0.07
0.510 0.095 0.150 0.082 0.270 0.180
0.085 0.025 0.082 0.024 0.230 0.110
0.018 0.003 0.022 0.007 0.050 0.020
0.048 0.006 0.072 0.007 0.099 0.023
Max
Max
Max
Max
of
IAPI
[zxal
Ivf - v,I
IDf-D~[
iterations
0.0075 0.0050 0.0077 0.0030 0.0099 0.0099
0.009 0.007 0.009 0.005 0.010 0.010
0.061 0.019 0.056 0.034 0.140 0.079
0.280 0.073 0.310 0.080 0.360 0.090
6 3 5 3 18 8
Final solution No.
1 2 3 4 5 6
All the values are in p.u. FIS = flat initial solution: V = 1.0 p.u., D = 0.0 rad. A D S = approximate decoupled solution. Vf and Df = final voltage magnitude and angle. Vi and D~ = initial voltage magnitude and angle. 1: IEEE 14-node system, F D M and FIS were used. 2: IEEE 14-node system, F D M and A D S were used. 3: IEEE 30-node system, F D M and FIS were used. 4: IEEE 30-node system, F D M and A D S were used. 5: IEEE 57-node system, F D M and FIS were used. 6: IEEE 57-node system, F D M and A D S were used. = -BklDI
-Bk2D2
+ Bk(k + I) + "
....
+(Bkl
• + Bk, , ) D k . . . . .
+'''-[-Bk(k
Taking
1)
into consideration
the F D M
assumptions,
i.e.
Bk, , D,,
V k = 1.0 p.u.
Since Bkl + • " " + Bk(k-~
c o s ( D k - D s ) = 1.0
+ Bk~k + l~ + " " " + Bk,, = - - B k k
and
then
Gkj sin(Ok -- Dj ) << Bk/ P k = - - B k l D1 . . . . .
BkkDk .....
In m a t r i x f o r m , for k = 1 . . . . . a n d D , = 0.0, we h a v e
{i }=I m+I
L
--
B(,,~ + ,~1
.
.
.
.
--
Bk,,D,,
m + l (P, V b u s b a r s )
Qk b e c o m e s Qk=
j= 1
~Bks
In m a t r i x f o r m , for k = 1 . . . . .
][i]
B ( , . + ~)(,.~+ D
-- ~
m (P,Q busbars),
,,~ + l
or
[P., +, ] = [PB][D,., +, ]
(1)
....
(b ) R e a c t i v e p o w e r
]
+
Qk = Vk ~ j=l
Vj [Gkj sin(Dk - - D i ) -- Bkj CoS(Dk -- Dj )] _--Bin(re+l)
--
mn
163
G. Leonidopoulos / Electric Power Systems Research 32 (1995) 161 163 Table 2 CPU times (in seconds) for different program stages System
1
2
3
4
5
6
G, B, Y matrix formation [PB] formation and inversion [QB~] formation and inversion ADS calculation AP calculation convergence test AD and new D calculation AQ calculation convergence test A V and new V calculation
0.12 0.07 0.02
0.13 0.08 0.03 0.01 0.01 0.00 0.01 0.00
0.37 0.38 0.18
0.29 0.32 0.19 0.00 0.03 0.01 0.04 0.01
1.00 1.85 1.14
0.98 1.79 1.06 0.04 0.13 0.02 0.12 0.02
0.01 0.00 0.01 0.01
0.02 0.01 0.03 0.0l
or
[Q,, ] = [QB, ][ V,, ] + [QB b ][ V,..... ]
(2)
Then, from Eq. (1), we get [D,,,+,] = [PB] '[P,,,+,]
(3)
On the other hand, from Eq. (2), we have [V,,] = [QBa] ' ( [ O , , ] - [QBb][V, _,, ])
(4)
Eqs. (3) and (4) constitute the approximate decoupled solution of the power system load flow equations. In order to find the corrections [A V,, ] and [AD m+/], we take the differences of Eqs. (3) and (4): [AVm] = [QB~] I[AQm ]
(5)
[AD,, +, ] = [PB] - ' lAP,, +, ]
(6)
Eqs. (5) and (6) are the basic equations of the fast decoupled method [1]. 3. Performance of the method
The performance of the fast decoupled method using the approximate decoupled solution as the initial value is shown in Table 1. The standard IEEE testing systems of 14, 30 and 57 nodes were used for the test. A VAX computer was used to obtain the results and the CPU times needed to complete the different stages of the program are shown in Table 2.
0.12 0.01 0.13 0.01
The approximate decoupled solution was also used as the initial value of the fast decoupled method, reducing the number of iterations needed for the method to converge to less than 50% of those required when a flat initial value (V = 1.0 p.u., D = 0.0 rad) is considered. A shorter time is needed to calculate the approximate decoupled solution than for one iteration of the fast decoupled method and this makes its use as an initial value advantageous. The approximate decoupled solution can also be used as the initial value of any other numerical method for load flow, considerably improving its convergence characteristic since it is very close to the actual load flow solution.
5. Nomenclature
Dk AD 1 m n Pk AP k Qk AQk
Vk AV
Yk~
voltage angle of busbar k voltage angle correction number of P, V busbars number of P,Q busbars total number of busbars net active power of busbar k active power mismatch of busbar k net reactive power of busbar k reactive power mismatch of busbar k voltage magnitude of busbar k voltage magnitude correction admittance of line k-j, Yk~= Gkj +jBk~
4. Discussion and conclusions References
An approximate mathematical solution of the load flow problem has been developed in this paper. The solution is a decoupled one. In other words, V depends on Q and D on P only. This is expected since the assumptions of the fast decoupled method were employed to deduce the approximate decoupled solution. This makes the solution carry all the well-known advantages and disadvantages inherent in the fast decoupled method.
[1] B. Stott and O. Alsac, Fast decoupled load flow, IEEE Trans. Power Appar. Syst., PAS-93 (1974) 859-867. [2] F.F. Wu, Theoretical study of the convergence of the fast decoupled load flow, 1EEE Trans. Power Appar. Syst., PAS-96 (1977) 189 197. [3] B. Stott, Review of load flow calculation methods, Proc. 1EEE, 62 (1974) 916-929. [4] J. Arrillaga, C.P. Arnold and B.J. Harker, Computer Modelling of Electrical Power Systems, Wiley, Chichester, UK, 1983.