Optimal weight assessment based on a range of objectives in a multiobjective optimal load flow study

U. Nangia N.K. Jain C.L. Wadhwa

Indexing terms: Progrummirzg and planning, Multiobjectives, Weighting methods, Optimal loud jlow study

Abstract: In a multiobjective optimal load flow study (MOLF) two objectives (cost of generation and transmission losses) have been considered for minimisation. A weighting method has been used to formulate the problem. An attempt is made to obtain the relation between the range of each objective, i.e. cos1 of generation and transmission losses, and the weight associated with each of them. If the range of an objective function is large, a small weight has to be associated with it. If the decision makers follow this relation, then appropriate weights will be elicited. The ‘optimal weight’ is derived on the basis of the maximum range of each objective function. If the ‘optimal weight’ is attached to the multiobjective function consisting of cost of generation and transmission losses, the target point is achieved in a single step, thereby saving milch computational effort.

1 Introduction

Multiple objective programming and planning repre- sents a very useful generalisation of more traditional single objective appt oaches to planning problems [ 1-41. The consideration of many objectives in the planning process [5-91 accomplishes three major improvements in the problem solving. 1. Multiobjective programming and planning promotes more appropriate roles for the participants in the plan- ning and decision making process. 2. A wider range of alternatives is usually identified when a multiobjective methodology is employed. 3. Models and the analyst’s perception of a problem will be more realistic: if many objectives are considered.

In this paper, multiple objectives for the optimal load flow (OLF) solution are considered as cost of genera- tion (Fc) and system transmission losses (FL). It is well known that the OLF problem refers to the twin sub- problems of optimum active power generation (eco- nomic load dispatch) and optimum reactive power generation. In economic load dispatch, cost of genera- tion is considered a:; the objective function to be mini-

0 IEE, 1998 IEE Proceedings online no 19981655 Paper first received 2nd Dceember 1996 and in revised form 22nd August 1997 The authors are with the Electrical Engineering Department, Delhi College of Engineering, Ddhi University, Delhi, India

mised. In optimum reactive power generation, transmission losses (FL) can be considered as the objec- tive function to be minimised. This is because in a decoupled sense, transmission losses are mainly dependent on the voltage magnitudes which are dependent on the reactive powers. Therefore, optimisa- tion of reactive power dispatch can be considered as the minimisation of system active power losses.

It is therefore obvious that, for the OLF solution, the mathematical model can be based on the objective functions cost of generation and svstem transmission losses. If the OLF solution is achieved by considering only the cost of generation as the objective function then we achieve minimum cost of generation, but not minimum transmission losses. Similarly, if the OLF is solved considering only system transmission losses (FL) as the objective function then we achieve minimum sys- tem transmission losses but not minimum cost of gen- eration (Fc). When simultaneous minimisation of both the objectives is considered, it is not possible to attain their absolute minimum values obtained when the objectives are considered individually [8].

The ideal situation where one would like to operate the power system is where both objectives, i.e. cost of generation (FC) and transmission losses (FL) are mini- mum. Such a point (Fcmin, FLmin) in 2-D space is termed the ‘ideal point’, but such a point is not feasi- ble. Therefore, an attempt is made to obtain an operat- ing point as close as possible to the ideal point, this point being called the ‘target point’.

2 Formulation of multiobjective optimal load flow (MOLF) problem

The two aspects of the OLF problem considered here are (i) to minimise the cost of generation; (ii) to minimise the system transmission losses. The objective function to minimise the cost of genera- tion is given as

NC:

i=l

where Pgi is the active power generation at the ith gen- erator, Ci is the cost of generation for the ith generator and NG is the total number of generators in the system.

The objective function to minimise the system trans- mission losses is given as

FL = Pp n

p= 1

IEE Proc.-Gener. Transm. Distrib., Vol. 145, No. 1, January 1998 65

where P is the active power at the pth node and n is the totafnumber of nodes in the system.

Under normal operating conditions the system trans- mission losses depend upon the voltage profile, and in turn govern the reactive power requirements of the sys- tem. System transmission losses have been considered here for reactive power dispatch and for improving the voltage profile [4, 101. Moreover, this would result in a reduction in the cost of installing extra equipment for VAR generation and voltage adjustment. The solution of the OLF problem is subjected to inequality con- straints on active and reactive power generation, load voltages, and equality constraints [ l l ] of the load flow equations. The objective function (q is formulated using the weighting method [6, 121 as

and w1 and w2 are the weights attached to the cost of generation function and system transmission losses, respectively. These weights represent the tradeoff between the two objectives, cost of generation and sys- tem transmission losses. In other words these represent the preferences stated by the decision makers, so that it is only relative values of weights which are of impor- tance. Therefore, one of the weights can be arbitrarily set to unity or any positive number. If the specified weight is equal to the optimal weight, then the target point is achieved in a single step.

el, e2, ..., en are the real parts of the voltages at the n busbars andfi, f2, ..., f n are the imaginary parts of volt- ages at the n busbars. The nodal voltages are taken as the independent variables of the problem.

3 Computational procedure

Zangwill's transformation [ 131 has been used to trans- form the constrained MOLF problem into a sequence of unconstrained problems, and a standard algorithm [I41 using a quasi-Newton variable metric method has been used for unconstrained minimisation.

3. I Zangwill transformation Zangwill [13] developed a formulation for solving the general nonlinear constrained minimisation problem by transforming it into a sequence of unconstrained mini- misation problems. This technique applies a penalty to any solution X which is outside the feasible region. So the initial estimated solution need not be a feasible solution. It is basically an outside in technique, which starts from any point outside the feasible region, and works towards the feasible region. Zangwill proposed the following transformation:

m P

a= 1 z=m+l

where h,(X) is the deviation of the ith equality con- straint from its specified value, g,(X) is the deviation of the ith violated inequality constraint, m is the number of equality constraints, p is the total number of con- straints and r is a positive decreasing penalty parame- ter.

The penalty parameter is changed according to rk = t.r"l in the kth Zangwill iteration, where t is a positive

66

constant less than unity. The minimisation process moves such that the minimum for the unconstrained function F(X,rk) and the constrained function AX) are reached simultaneously. In practice when F(X,rk) has been minimised and its value is close enough to f ( X ) the process can be stopped. Alternatively, every time F(X,rk) is minimised, the magnitude of maximum of constraint violations is checked and when this falls below a certain specified value, the minimum is assumed to have been reached. The value in the present work has been taken as 0.0015. The algorithm for sequential unconstrained minimisation using the Zang- will technique is explained below through a sequence of steps and is also depicted in Fig. 1 in the form of a flow chart.

I ossume a solution x I

set COUNT K=O & /compute f i x K ) and g i X K I I

in and

1 estimate voiue o:a 1 a - m i n I Z i E S T - f i / X g Hg 101

aetermtne a*(K) by minimising f IXK.aHK91 I usina unidimensional search

update inverse Hessmn H HK.lIHK-HKYKIYKITHK.

I b K l s ~ 7

lYK)THKYK gTY i 0

YTHY:O 16KlTYK

yes

Fig. 1 Flow chart for standard algorithm

i. Assume a suitable initial estimate 2). Set the iteration count K = 0. ii. Calculate go (first-order partial derivatives). iii. Take @ (inverse of the second-order partial deriva- tive matrix) as the identity matrix. iv. Advance the iteration count by 1, if lg kl 5 E; stop, else go to step (v). E is a prespecified small positive value. A value of 0.00001 for E has been considered here. v. Determine Ak such that f (xk - Ak Hk 9) is mini- mised in the direction -Hk gk. Let & be the value of Ak where fixk - ;Ik Hk 2) is minimised.

IEE Proc -Gener Trunsm D u m b , Vol 145, No 1, Junuary 1998

vi. Calculate

vii. If Nxk) - f (Xk+' ) + E] 2 0 continue; otherwise, rein- itialise the Hk+l to tfe identity matrix. viii. If (81 < E stop; otherwise continue. ix. If B Y = 0 or Y i Y Y = 0; reinitialise the Hk+l to the identity matrix, othe *wise continue. x. Update the matrix Hk according to the following equation:

H ~ Y ~ ( Y ~ ) ~ H ~ bk (bk)T Hk+fl = Hk - ( Y k ) T H k Y + ( 6 k ) T Y k

xi. Go to step (iv) Some important points in respect of the above algo-

rithm are listed below. (a) If the function v.ilue has not decreased in the final iteration, the search for the minimum is terminated provided the gradient is already sufficiently small. Oth- erwise, the next step is in the direction of the steepest descent, i.e. reinitialising H to the identity matrix. (b) During the process of minimisation, if the direc- tional derivative -Hk $" becomes positive and greater than E, H should be reinitialised to the identity matrix. (c) When the approximation to H is initially chosen to be positive definite. during successive iterations this property is retained. (d) If the function to be minimised is quadratic positive definite then the minimisation process terminates in at most n iterations, where n is the number of variables. This also guarantees that the H matrix tends to G-' (G being the Hessian matrix). So the ultimate convergence of a quadratic function is justified.

4 Assessment of optimal weight

The multiobjective Junction F is formulated using the weighting [6, 121 method as defined by eqn. 1.

F = W ~ F C + W ~ F L where wI and w2 are the weights attached to the cost of generation and transmission losses function, respec- tively. These weights represent tradeoff between the two objectives, cost of generation Fc and system trans- mission losses FL. The tradeoff depends on the range of the objective function that each weight is implicitly rep- resenting. In most decision situations, there is no unique definition of range. The range can be defined as the possible (or available, relevant, etc.) range of the objective function. If the range Ri of objective Zi is var- ied over a larger (smaller) range it has to lead to a smaller (larger) weight [15].

Weights can be used to determine the target point which will depend ,311 the decision makers willingness to specify the weights. Economists such as Marglin and Major [16, 171 ac'.vocated the prior assessment of weights for public decision-making problems. They suggested a planning process in which decision makers ponder the relative importance of objectives before spe- cific design or po1ic:y issues are raised. Prior specifica- tion of weights can be used for any level of decision making; it is not ctsnfined to public decision making.

IEE Proc-Gener. Transm. Distrib., Vol. 145, No. 1, January 1998

This could be a favourable approach for decision mak- ing because of its simplicity but extensive sensitivity analysis should be pursued, i.e. weights should be var- ied systematically and the weighted problem should be solved for each new set of weights. From the solutions so obtained, some preference criterion should be used by the decision makers to determine the target point.

There are various weight elicitation methods which are different with respect to the intensity with which they bring the range of an objective function to the decision maker's attention. In ranking, rating and ratio methods [ 181, the decision maker directly determines the importance of each objective without explicitly con- sidering the range of each objective.

A second group of methods explicitly considers the full range of objectives so that the range is apparent to the decision maker, e.g. regression approach.

A third group of weight elicitation methods consists of tradeoff approaches [19] in which the decision mak- ers give preference statements representing tradeoff between two objectives.

In this paper, we have tried to explore the relation between the range of each objective, i.e. cost of genera- tion and transmission losses and its weight. The objec- tive function F as defined by eqn. 1 is rewritten as

F = Fc + FL In order to scale the function, we multiply Fc by the

range R, of transmission losses FL, and multiply FL by the range Rc of the cost of generation F,, i.e.

F = Rr, Fc + RcF, (2) In order to normalise this function, we divide

throughout by RL.

F/RL = F c + (Rc/RL) * FL (3) Comparing the above equation with eqn. 1, it is observed that w , = 1.0 and w2 = RCJRL, i.e.

where Fc and FL are the cost of generation and system transmission losses values obtained at any randomly selected weight, FCm,, is the minimum value of the cost of generation obtained by setting w1 = 1.0 and w2 = 0.0, FLm, is the minimum value of transmission losses obtained by setting w1 = 0.0 and w2 = 1.0.

If the decision maker is made aware of the above relation between range of objective functions and the weights associated with them, this leads to reliable methods of assessing preferences.

In order to achieve the target point in a single step, the weight w2 should be equal to the optimal weight. The optimal weight is derived by the decision maker based on the full range of objective functions. In other words,

where Fc at FLmjn is the value of cost of generation obtained at FLmin and F L at Fcmin is the value of trans- mission losses obtained at F,,,,.

At each solution USC (unit saving in cost of genera- tion ) and USL (unit saving in transmission losses) are calculated which are defined as below:

61

where (Fe at FLmjn - Fcmin) = MPSC (maximum possi- ble saving in cost of generation).

Similarly,

(FL at Fcmin - FLNzin) = MPSL (maximum possible sav- ing in transmission losses)

At w ~ ~ ~ ~ ~ ~ ~ ~ , USC and USL are equal. The algorithm to determine optimal weight is as follows: 1(a) Individually minimise each of the objectives to obtain the ideal point, i.e. (Fcmi,, FLmin). (b) Compute the value of each objective at each of the ideal solutions. From this, obtain F, at FLnzin and FL at

2 Compute the maximum range of each of the objec- tives as follows:

FCmin.

Max. range of Fc; Rc = Fc at FLmin ~ FCm, Max. range of FL; RL, = FL at FCm, - FLmin

3 Compute wZoptimal = RclR, = (Fc at F L ~ ~ ~ - Fcmin)l(FL

4 Optimise the function F defined by eqn. 1 by keeping wI = 1.0 and w2 = w~~~~~~~~ and obtain the target point which gives equal satisfaction in both the objectives.

4.1 Sample calculation The optimal weight for a 14-bus system is calculated as below.

The results of individual optimisation of cost of gen- eration and system transmission losses are shown in the first and second rows of Table 2. From these

at Fcmin - FLmin)

F c at F~~~~ = 1062.49 $/h

FL at F~~~~ = 8.83 MW Fcmin = 998.75 $/h

FLmin = 6.69 MW range Rc of cost of generation

= 1.062.49 - 998.75 = 63.74 $/h range RL of transmission losses

Scaling the objective function we get

Normalising the above equation,

F12.14 = Fc + 2 9 . 8 F ~ Comparing this equation with eqn. 1, we get w1 = 1.0 and w2 = 29.8. Solving the above weighted problem, we get the target point in a single step as Fc = 1016.09$/h and FL = 1.21MW. USC and USL at this target point are 0.73 and 0.73, which are equal. This is shown in the third row of Table 2. Thus, the above w2, determined on the basis of maximum range of each objective func- tion, is the optimal weight.

5 Computational results

Three standard test systems of 5 , 14 and 30 bus systems have been considered [20], and the computational results are shown in Tables 1, 2 and 3, respectively. The second and third column of these Tables show the weights w1 and w2, respectively. The fourth column shows the cost of generation Fe This cost includes the cost of generation of supplying electric power to all the loads of the system including the local loads at the gen- erating station. The fifth column shows the cost of gen- eration of supplying electric power to all the loads excluding the local loads. This is represented by FXwI

8.83 - 6.69 = 2.14 MW

F = 2.14Fc + 63.74F~

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L). In multiobjective optimal load flow studies, FdwlL) has been considered for optimisation. The seventh and eighth columns show the USC (unit saving in cost of generation) and USL (unit saving in transmission losses). In other words, the first and second rows of these Tables show the results of individual optimisation of the cost of generation and transmission losses, respectively. The third row shows the results at optimal weight. It is observed that USC and USL are equal at optimal weight, i.e. this is the target point which gives equal satisfaction in both objectives. The target points are T1 (wl = 1.0, w2 = 20.81) for the 5 bus system, T2 (wl = 1.0, w2 = 29.8) for the 14-bus system and T3 (wl = 1.0, w2 = 17.0) for the 30-bus system. For 5 and 14- bus systems w~~~~~~~~~ is exactly the same as calculated by eqn. 5. For the 30-bus system w20ptimal as calculated by eqn. 5 is 14.21, shown in the fourth row of Table 3, whereas the target point is obtained at w2 = 17.0. This shows that the formula for wZoptlmal can be used for the 5 and 14-bus systems but this needs slight modification for a 30-bus system. Research is under progress and the results will be communicated as soon as they are avail- able.

Table 1 5-bus system

1 1.0 0.0 760.95 760.95 5.18 1.0 0.0 2 0.0 1.0 764.43 764.43 5.01 0.0 1.0 3 1.0 20.81 761.82 761.82 5.05 0.75 0.75

Table 2 14-bus system

S.No. W, ~2 fc FC(W/L) FL USL

1 1.0 0.0 1137.11 998.75 8.83 1.0 0.0

2 0.0 1.0 1212.09 1062.49 6.69 0.0 1.0

3 1.0 29.80 1160.83 1016.09 7.27 0.73 0.73

Table 3 30-bus system

S.NO. W, wz Fc Fc(w,~, FL USC USL ~~

1 1.0 0.0 1245.57 1023.59 9.77 1.0 0.0

2 0.0 1.0 1292.73 1055.02 7.56 0.0 1.0

3 1.0 17.0 1264.17 1033.31 8.24 0.69 0.69

4 1.0 14.21 1258.93 1029.33 8.39 0.82 0.62

6 Conclusions

An attempt has been made to establish the relation between the ranges of two objective functions (cost of generation and transmission losses) and their weights. This relation is established for 5, 14 and 30-bus sys- tems. Methods based on importance judgements such as simple ranking, rating or ratio methods should only be used with great care, perhaps not at all. Prior assess- ment of weights can be used in decision making, but requires extensive sensitivity analysis.

Optimal weight is determined for three standard sys- tems based on the maximum range of each objective function. Using optimal weights gives the target point in a single step which results in considerable saving of computational effort.

7

1

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