method based on Beale's method for solving the total. OPF problem, i.e. both economic load dispatch and optimal reactive-power dispatch subproblems.
New optimal power-dispatch algorithm using Fletcher's quadratic programming method J. Nanda, PhD DP. Kothari, PhD S.C. Srivastava, PhD
Indexing terms: Optimisation, Algorithms
Abstract: A fresh attempt has been made to develop a new algorithm for optimal power flow (OPF) using Fletcher's quadratic-programming method. The new algorithm considers two decoupled subproblems needing minimum cost of generation and minimum system-transmission losses. These have been solved sequentially to achieve optimal allocation of real and reactive power generation and transformer tap settings with consideration of system-operating constraints on generation, busbar voltage and line-flow limits. The potential of the new algorithm for OPF has been demonstrated through system studies for two IEEE test systems and an Indian system. Results reveal that the proposed new algorithm has potential for online solving of OPF problems.
List of symbols i = e{+ jfi YtJ = Gjj — jBu
= complex voltage of busbar i = i/th element of busbar admittance matrix Zij = r(j + jxu = i/th element of busbar impedance matrix Ysert = Gsert + jBserj = series admittance of ith line S, = P{ + jQi = complex injected power at busbar i PLDi and QLDt = active power load and reactive power load at busbar i PL and QL = total real power loss and total reactive power loss in the system PD and QD = total real power and total reactive power demands in the system [J] and [J/] = Jacobian and inverse Jacobian Au A2, A3 and AA = sensitivity submatrices denning relation between change in reactive power and voltage magnitude at, bt and c, = cost coefficients of ith generating unit VGt and VLj = voltage magnitude at source busbar i, at load busbar j /; = magnitude of current flow in line i Paper 6599C (P9), received 9th November 1988 Prof. J. Nanda is at the Department of Electrical Engineering and Centre for Energy Studies, and Prof. D.P. Kothari is with the Centre for Energy Studies at the Indian Institute of Technology, Hauz Khas, New Delhi, 110016, India Dr. S.C. Srivastava is at the Department of Electrical Engineering at the Indian Institute of Technology, Kanpui, India IEE PROCEEDINGS, Vol. 136, Pt. C, No. 3, MAY 1989
DFtj = distribution factor relating real generation of ith generator to current flow in yth line
1
Introduction
In an optimal power flow problems, total fuel cost, total transmission loss or some other appropriate objective functions are minimised subject to the system constraints. A lot of research work has been carried out in the past in OPF area [1-5, 17] using several optimisation techniques including classical, linear, quadratic and nonlinearprogramming methods. Among them, linear and quadratic-programming methods have gained more attention in recent times because of their inherent simplicity, efficient handling of constraints and speed of solution, compared with nonlinear-programming methods. Since the two objective functions often used in OPF problems, generator-fuel costs, and total system losses are usually expressed as quadratic functions of real and reactive power generation, it will be worthwhile to use quadratic programming (QP) methods instead of linear programming (LP), which involve additional efforts and approximations in linearising the objective functions. Further LP methods in many situations result in zigzagging of solutions, causing convergence problems. In the literature various LP-based QP methods, as developed by Wolfe [8] and Beale [9], are available. Many researchers have applied both Wolfe's and Beale's methods for solving only economic load dispatch (ELD) problems. It has been established [10] that Beale's method is superior to Wolfe's method. Contaxis et al. [6] were possibly the first to use a QP method based on Beale's method for solving the total OPF problem, i.e. both economic load dispatch and optimal reactive-power dispatch subproblems. They have clearly demonstrated that Beale's QP method provides better results and faster solutions than the LP based on the Revised simplex method for OPF solution. A QP method developed by Fletcher [10] is found to be superior to the methods developed by Wolfe and Beale. Literature shows that Fletcher's QP method has yet not been applied to power-system OPF problems. The motivation for the present work is to apply Fletcher's QP method as a fresh attempt to the solution of OPF problems and to explore its potential as compared with the LP and LP-based QP methods applied to power-system OPF problems. The OPF problem considers the twin subproblems of minimising total fuel cost and total system active power loss, for optimal allocation of active and reactive-power 153
generation and transformer tap settings. The algorithm uses a new approach for linearising various system constraints. New formulations are used to handle the limits on busbar voltages, transformer taps and line flows. The two subproblems have been solved sequentially until both cost of generation and system loss converge to a prespecified tolerance. The potential of the new algorithm for OPF has been demonstrated through system studies on two IEEE test systems consisting of 14 and 30 busbar [13] and on an 89-busbar Indian system. Results clearly demonstrate the superiority of Fletcher's QP method over LP and LP-based QP methods, as applied to OPF problems until now.
equation as an equality constraint, and limits on reactivepower generation, voltage magnitudes of the generator and load buses and on transformer taps as inequalities. The control variables considered for optimisation are active-power generation (PG) for the P-optimisation subproblem, and reactive-power generation {QG) for the Q-optimisation subproblem. Consider a system having N busbars and NL lines. Let the first Ng be the generator busbar, the first Nq be the reactive-source busbars (including the Ng generator busbars) and the first Na lines contain transformers with OLTC control. The two subproblems can be formulated as in the following Sections.
2
3.1 P-Optimisation subproblem Minimise total cost of generation (F t ) expressed as
Fletcher's QP method
Fletcher's QP method [10] does not depend on any LP subroutine and can be used to minimise a quadratic function /(3c) of n variables, subject to m linear equality and inequality constraints. The formulation of the problem is to minimise T
T
/(3c) = \x Ax - b x
subject to equality constraint
(1)
subject to U2J3C 3S I
(4)
(5)
and inequality constraints (2)
and
PGimin^PGt^PGimax,
(6)
and (3)
where eqn. 2 restricts the values of the variables 3c (3c > 0) to remain within their upper and lower limits u and 1, and eqn. 3 represents a set of linear equality and inequality constraints (total m — 2ri) in terms of variables 3c. The algorithm is developed from classical formulation of the minimisation of the quadratic function, subject to linear equality constraints. The method is of exchange type and is solved while keeping a basis of inequality constraints (active constraints) which are also treated as equality constraints. The problem which is generated by active constraints is said to be an equality problem (EP). Each iteration attempts to reduce/(3c) by moving towards the minimum of EP corresponding to the current basis. The constraints which are violated on doing this are added to the basis until a feasible solution of the equality problem is reached. At such a point the Lagrange multipliers associated with the EP are examined to see whether the solution is a solution of the inequality problem. If not then a constraint is removed from the basis and further reductions in/(3c) are sought as above. Constraints which are equalities in the original problem are dealt with trivially by including them in the initial basis and by not considering them as candidates for removal. The details of the algorithm is described in Reference 10.
//) owing to a change in transformer tap settings. The optimal voltage magni-
and \_{BAJ)i}QG?+ !>] > VG\min,
i=l,...,Nq
(29)
where
OLTC
[
KG
— VGk~\
'7G?
i t h transformer
w
«
'] + ll(BAJ)ijQGJ]
and
fictitious reactive power source
busbarj
^ Q /' Q G k a
V k /6 k —j—busbar k
For load busbars: Fig. 1 Representation of an OLTC transformer
i = N.+ l,...,N
l,...,N
(30)
(31)
where
[
VI
— VIk~\ yrk
+
"« L
and
a One-line diagram b Reactance diagram
tudes Vj and Vk at busbars ; and k, respectively, can be found from eqns. 26 and 27 as the changes in reactivepower generation are known. Presuming no change in complex-voltage angles, the expressions for old and new reactive-power flow entering busbar k from busbar ; can be written as ri cos (dj - dk) - VlBsert and similarly QW = aj")VjVkBsert cos ( - VG';m
(47)
.7=1
l,...,N
(48) 157
NQ
increment iteration count by 1 and repeat Steps 3 to 8 until the convergence is achieved. (49)
where VG"min
VGimln-VGi VGk +
f[_(BAJ)ij(QGkJ-QGjmian
VGimax-VGk VGk
VLk
VLimax-VLk VLk
The optimal reactive-power generation at source busbars and OLTC controlled busbars is found from QGi = Qgt + QGimin,
i = 1 , . . . , NQ
(50)
4.3 Computational steps Step 1: Read system data; from busbar admittance and impedance matrices. Step 2: Run a base load flow and compute total cost of generation and system losses. Compute factors DF and initialise optimisation iteration count to 1. Step 3: Compute loss coefficients, using eqn. 15 and compute sensitivity matrices [BAJ] and \_BAB~\ as defined in eqns. 24 and 25. Step 4: Solve the real-power dispatch subproblem using eqns. (39-42), applying Fletcher's QP method. Find the optimal values of PG, considering QG as constant. Step 5: Compute reactive-power generation limits of additional Q-sources (due to transformer taps) from eqns. 34 and 35. Step 6: Considering PG found in Step 4 as constant, solve the optimal reactive-power dispatch subproblem using eqns. (43-50) and Fletcher's QP method. Obtain optimal QG at all the source busbars (including fictitious sources due to transfer taps). Step 7: Compute optimal voltages at the transformer end and start busbars using eqns. 26 and 27, and determine optimal values of transformer taps using eqn. 33. Modify the corresponding elements of [Ybusbar] and compute [Zbusbar]. Find modified slack busbar voltage using eqn. 26. Step 8: Keeping constant optimum PG and QG of sources obtained in Steps 4 and 6 except for slack busbar and using the modified slack busbar voltage and [Y busbar] obtained in Step 7, run an optimal load flow. Compute the cost of generations and system losses. Step 9: If both the cost of generation and total system active power loss have converged to a prespecified tolerance, optimal power-flow solution is reached. Otherwise, 158
5
System studies
System studies have been carried out on IEEE 14 busbar, IEEE 30 busbar [13] and on an 89 busbar Indian system. The constraints imposed on the busbar voltage at the generator busbars are 1.0 p.u. (minimum) and 1.1 p.u. (maximum), and at load busbars they are 0.9 p.u. (minimum) and 1.1 p.u. (maximum). The 89 busbar Indian system has 27 generating units with seven hydro units and twenty thermal units. Only thermal units are assumed to participate in P-optimisation subproblem. Thus active power generation of hydro units are kept constant at the values obtained from the base load-flow results during optimisation. The optimal power-flow results have been obtained on an ICL 2960 computer with convergence criteria: 0.0001 p.u., 0.1 $/h, 1 Re/h and 0.001 p.u., for power mismatches in load flow, cost of generation (IEEE 14 busbar and 30 busbar systems), cost of generation (89 busbar Indian system) and total system loss, respectively, on a 100 MVA base. Two case studies have been conducted on IEEE 14 busbar and 30 busbar systems. In Case Study 1, the optimal power-flow results have been obtained by considering the limits on real and reactive-power generation and busbar voltages. Case Study 2 considers transformer OLTC settings and line-flow limits as additional constraints in the model. In the absence of any data for lineflow limits, maximal line-current limits have been presumed, considering the margin over their values for the base-case load flow. Current limits of certain lines are purposely kept below their base values to study the potential of the new algorithm. NGGDFs were obtained at the end of base load flow using a perturbation of 1 % in PG and kept constant for the optimisation iterations. A literature survey shows that probably no researcher has considered discrete OLTC settings in the OPF studies. Hence a more realistic situation has been simulated in Case Study 2 by considering five discrete OLTC tap settings for transformers in both IEEE 14 and 30 busbar systems at 0.95, 0.975, 1.0, 1.025 and 1.05 p.u. After each optimisation iteration, the optimal tap settings found are approximated to their nearest discrete tap values. The optimal load flow results for both case studies have been obtained for IEEE 14 and 30 busbar systems using Fletcher's QP method, employing Fortran subroutines [11, 12], and compared with those obtained by a QP method based on Beale's approach. Results for Case Study 1 for these two IEEE systems have also been obtained by a LP method based on the 'Revised simplex' technique and compared with Fletcher's QP method. The application of Beale's QP technique for solving optimal power-flow problems in the present work mainly follows Contaxis' approach [6] but differs in several respects. Whereas in real-power subproblems Contaxis et al. have used generalised generation distribution factors (GGDFs) to consider line-flow limits, new generalised generation distribution factors (NGGDFs) have been used in the present work. In the reactive-power subproblem, whereas Contaxis et al. have taken the minimisation of cost of slack power generation as the objective function, in the proposed algorithm the objective function considered is the minimisation of total system real-power loss. Contaxis et al. have not considered transformer tap settings in their model, whereas IEE PROCEEDINGS, Vol. 136, Pt. C, No. 3, MAY 1989
in the present work tap settings have been duly considered in the model. The optimal power-flow algorithm using LP based on the 'Revised Simplex' technique has been developed in line with the method presented by Contaxis et al. [6] except that the line-flow limits have not been considered in the model. The model considers minimisation of total cost of generation for real-power dispatch subproblems, which have been linearised using a piecewise approach, and takes into account the limits on real-power generation. To linearise the quadratic cost of generation, the cost curve has been split into three segments, involving three generating units of smaller capacity in place of each unit in the original problem. The reactive-power dispatch subproblem has been solved to minimise real-power gen-
eration at slack busbars and, considering limits on reactive-power generation, slack busbar real-power generation and voltages at all busbars. The control variables in solving the reactive power-subproblem are real-power generation at slack busbars and busbar voltages at generator busbars. The two subproblems have been solved alternatively, after converging on each of them separately involving in-between load flows. The convergence on both the subproblems is achieved when real-power generation at the slack busbar converges to within a prespecified tolerance of 0.001 p.u. at 100 MVA base. The OPF results for both Case study 1 and Case study 2 using Fletcher's QP, Beale's QP and LP [6] are presented in Tables 1 and 2 for an IEEE 14 busbar system and in Tables 3 and 4 for an IEEE 30 busbar
Table 1: IEEE 14 busbar OPF results — cost, loss and CPU time figures Case study
Method used Cost of Total system loss generation, ~JT~ L L' $/h MW MVAR r
Base — case Case 1 Fletcher's QP Beale's QP LP based on 'Revised simplex' Case 2 Fletcher's QP Beale's QP
1156.394
11.70
1134.858
8.92
-15.75
1135.959 1133.785
9.22 8.91
1134.277 1135.948
No. of optimisation iterations
-3.22 —
No. of LF runs
CPU time, s Each P-optimisation
Each Q-optimisation
Total, including load flows but excluding base LF
—
—
3
3
0.45
1.1
21.15
-14.67 -19.56
5 P-optimisation 3 Q-optimisation 4
5 7
0.64 0.74
1.6 3.73
38.70 40.48
8.79
-16.29
5
5
0.52
1.5
38.30
9.18
-14.99
9
9
1.40
2.2
92.34
Table 2: IEEE 14 busbar OPF results — optimum generation and transformer taps Control variables
Base case
PG,, MW PG 2 , MW PG 3 , MW QG,, MVAR QG 2 ,MVAR QG 3 , MVAR QG 4 , MVAR QG 5 ,MVAR
210.70 160.58 68.77 40.00 20.00 38.57 18.31 -7.34 22.06 26.43 24.00 6.51 17.17 17.34 20.99 19.18 0.962 0.962 0.978 0.978 0.969 0.969
t,,p.u. t 2 ,p.u. t 3 ,p.u.
Case study 1 Fletcher's QP
Case study 2
Beale's QP
LP based on 'Refised simplex'
Fletcher's QP
Beale's QP
161.43 68.97 38.32 -10.81 26.42 4.64 18.59 20.00 0.962 0.978 0.969
167.89 66.68 33.34
160.74 69.73 38.28 -6.68 22.16 1.53 16.68 23.52 0.95 1.00 0.95
160.53 68.92 38.73
-20.58 36.52 -3.30 36.03 5.27 0.962 0.978 0.969
-11.13 26.42 4.64 18.58 20.00 0.95 1.05 1.05
Table 3: IEEE 30 busbar OPF results —cost, loss and CPU time figures Case study
Method used
Base — case Case 1 Fletcher's QP Beale's QP LP based on 'Revised simplex' Case 2 Fletcher's QP Beale's QP
Cost of Total system loss generation, p L $ h / MW MVAR
No. of optimisation iterations
1288.281
15.38
1244.197
10.25
-15.95
3
1245.310 1252.270
10.51 11.39
-14.10 -16.31
6 P-optimisation 4 Q-optimisation 6
1244.426
10.32
-7.17
1246.435
10.81
-2.47
6.44 —
IEE PROCEEDINGS, Vol. 136, Pt. C, No. 3, MAY 1989
No. of LF runs
—
Each P-optimisation
CPU time, s Each Total, Q-optimisation including load flows but excluding base LF
—
3
0.45
2.2
45.45
6 10
0.64 0.74
3.4 12.15
108.24 200.86
6
6
0.54
3.1
123.60
9
9
3.10
4.6
215.64 159
Table 4: IEEE 30 busbar OPF results — optimum generation and transformer taps Control variables
Base case
PG1; MW PG 2 ,MW PG 3 , MW QG,,MVAR QG 2 , MVAR QG 3 , MVAR QG 4 , MVAR QG 5 , MVAR QG 6 , MVAR
238.78 166.48 40.00 75.60 20.00 51.22 -13.05 -7.86 50.00 26.02 22.08 32.99 24.00 23.49 37.08 23.34 12.52 12.26 0.978 0.978 0.969 0.969 0.962 0.962 0.968 0.968
t,,p.u. t 2 ,p.u. t 3 ,p.u. t 4 ,p.u.
Beale's QP
Discussion
The choice of an optimal power-flow algorithm for online implementation depends mainly on it's superiority over other methods in terms of faster and more reliable solution, lesser memory requirement and a better value of optimality. A comparison between Fletcher's QP method for solving OPF problems and Beale's QP method, and the LP method based on the 'Revised simplex' technique, using these criteria is stated below. 6.1 CPU time The CPU time required by Fletcher's QP, Beale's QP and the LP method are presented in Tables 1 and 3 for various case studies conducted on IEEE 14 and 30busbar systems. Average CPU time for each Poptimisation and Q-optimisation blocks have been provided. The total CPU time quoted includes the time taken for solving the complete optimal power-flow problem, excluding the time taken for the base load flow. In Case Study 2, the total CPU time does not include the time for computation of NGGDFs, which are computed only once from base load-flow results and kept constant for subsequent optimisation iterations. Results clearly 160
LP based on Revised simplex'
166.95 200.05 75.62 50.00 51.34 44.77 1.31 -8.82 45.92 44.11 -6.91 32.81 23.74 10.82 32.27 36.20 15.76 -5.21 0.978 0.978 0.969 0.969 0.962 0.962 0.968 0.968
system. Results (Tables 1 and 3) reveal that significant reductions in cost and loss figures are obtained using any of the three methods for OPF solution. In both the case studies, Fletcher's QP method provides better cost and loss figures and takes significantly fewer iterations and less CPU time for solution than Beale's QP method. Consideration of OLTC and line-flow limits in Case Study 2, however, increases the number of iterations for convergence. The LP based method takes appreciably more iterations as compared with both Beale's and Fletcher's QP methods and requires more CPU time. Cost and loss figures however are found to be slightly better with the LP method in the case of IEEE 14 busbar system but are much higher in case of the IEEE 30 busbar system, as compared with the figures obtained by both Fletcher's and Beale's methods. In the absence of details of transformer tappings and line-flow limits for the 89-busbar Indian system, results for only Case Study 1 were obtained using Fletcher's QP method. These results show a considerable reduction in the total cost of generation and system loss from the base-case (the total cost of generation was reduced from Rs.204, 606/h to Rs.199, 172/h, and the total real-power loss was reduced from 74.6 to 63.5 MW). 6
Case study 2
Case study 1 Fletcher's QP
Fletcher's QP
Beale's QP
167.01 75.78 50.93 14.24 33.32 28.55 -6.00 24.92 24.00 0.950 1.025 1.050 1.050
167.80 75.29 51.13 24.14 22.27 35.25 -6.00 24.08 24.00 0.950 1.050 1.025 1.050
reveal (Tables 1 and 3) that Fletcher's QP method provides much faster solution than Beale's QP method, whereas Beale's method is faster than the LP based on the 'Revised simplex' technique. As an example in Case Study 1 for the IEEE 30 busbar system, Fletcher's QP method required about 42% of the CPU time taken by Beale's QP method, and only about 23% of the CPU time taken by the LP method for the OPF solution. 62 Memory requirements With the advancement in computer technology, the memory requirement is no longer a major constraint in solving a medium-to-big size system. However, because of the rapid growth in power-system size caused by the increase in grid interconnections, memory requirement may assume significant importance. Consider an optimisation problem involving N independent (control) variables and K constraints. Let Fletcher's QP method form a basis of K' constraints {K' ^ K). The LP method needs N' (N' > N) independent variables to linearise the objective functions and correspondingly involves K" constraints (K" ^ K). The memory and computational requirements can be visualised from the fact that whereas Fletcher's QP method will involve (N + K') variables and K' constrained equations for solution of the optimisation problem, Beale's QP method involves (2N + K) variables and (N + K) constrained equations, and the LP method will involve (2N' + K") variables and (AT + K") constrained equations. Thus the memory required by Fletcher's QP method will be minimal, followed by Beale's QP and the LP method in sequence. 6.3 Value of optimality and reliability of solution Comparison of optimal cost and loss figures obtained by the three methods in Tables 1 and 3 for IEEE 14 busbars and 30 busbar systems clearly reveal that Fletcher's QP method provides better cost and loss figures, in both case studies, than Beale's QP method. The LP method has provided slightly lower cost and loss figures in Case Study 1 for the IEEE 14 busbar system; however, it has provided much higher figures than both Fletcher's and Beale's QP methods for the IEEE 30 busbar system. A literature survey reveals that LP-based methods often result in zigzagging of the solution mainly because of the linearisation of objective functions [15 and 16], thereby providing a less reliable solution for OPF problems involving quadratic objective functions. The QP methods have, however, been found to be much more reliable in IEE PROCEEDINGS, Vol. 136, Pt. C, No. 3, MAY 1989
solving OPF problems, considering various practical constraints. Considering all the desirable features discussed above, Fletcher's QP method proved to be a much better choice for OPF solution than both the LP and Beale's QP methods. 7
Conclusions
Optimal power-dispatch problems, considering practical constraints, have been solved for the first time using Fletcher's QP method. The models developed for applying QP methods to OPF problems use several new formulations to take into account transformer taps, voltage constraints and the linearisation of equality constraints. A more exact set of 'new generalised generation distribution factors' to account for line-flow constraints is suggested. OLTC in discrete steps has been considered for the first time in the model by using the new concept of a fictitious reactive-power source. An algorithm based on Fletcher's QP method for OPF provides cost and loss figures better than the ones obtained by Beale's QP and LP methods. Moreover, studies reveal that Fletcher's QP method provides much faster solution than Beale's QP method, although the latter is much faster than the LP based algorithms. It is envisaged that Fletcher's QP method should appear to the utilities to have a distinct edge over all simplex-based LP and QP techniques for online implementation in optimal power-dispatch problems. 8
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