Generation and load uncertainties incorporated in load flow studies

J. Tom6 Saraiva Vladimiro Miranda Manuel A. C. C. Matos

INESC - lnstituto de Engenharia de Sistemas e Computadores FEUP, DEEC - Faculdade de Engenharia da Universidade do Porto

Depart. de Eng. Electrotecnica e Computadores Rua dos Bragas, 4099 Porto Codex, Portugal

telef. 351 -02-321 006, telex 23023 INESC P, fax 351 -02-31 8692

Abstract- Power system modelling incorporat ing Uncertainties having a non probabilistic nature or qualitative knowledge Is becoming more important. This paper presents a Fuzzy AC Load Flow model where fuzzy data is used to obta in poss ib i l i ty d is t r ibut ions o f voltages, active and reactive flows and losses, currents and generated powers. These distributions are compared with the ones obtained through a Monte Carlo based simulation in order t o evaluate the errors inherent to the Fuzzy AC Load Flow.

1 Introduction

Robust planning methodologies in Power System Engineering require some kind of formal approach to uncertainties. In the planning period the load flow studies usually use data having deterministic or probabilistic nature. However, it is easily accepted that, in several situations, some data can not be classified this way. In a planning environment the engineer often produces propositions, based in his knowledge and experience, having a fuzzy nature such as:

- "The active load at bus k will be,

- "The active power flow at line 5 will be, aoDroximatelv. 10 MW";

m ~ ~ e or Igss, 12 MW";

This kind of uncertainty is due, most of the cases, lo incomplete human knowledge. It is not

argued that fuzzy models will replace probabilistic ones as there are cases where these ones are appropriate. However, it is important to use concepts and develop tools which are able to capture the fuzzy nature commonly found in most of the human activities. These techniques can be important decision aid tools as they provide a more global vision of the system behavior.

2 Fuzzy AC load flow model

The Fuzzy AC Load Flow model [6] reflects the bus data uncertainty on the voltages, angles, active and reactive flows and losses, generated active and reactive powers and currents. In order to obtain their possibility distributions the following steps should be considered:

a) The possibility distributions are built using an incremental technique departing from a previous deterministic load flow using load values related to the medium point of the associated possibility distributions. Through this study deterministic values of voltages (Vd), angles (ed), active and reactive flows (Pdik and a&), active and reactive generated power (pdg and Qdgh bsses (pdloss and Qdloss) and currents (Id) are evaluated. The final possibility distributions are obtained by superimposing these deterministic values to their respective fuzzy increments.

b) Once an operating point is obtained, the fuzzy deviations of the injected active (on PV and PQ buses) and reactive powers (on PQ

CH2964-5/91/0000-1339$01.00 01991 IEEE 1339

buses) both referring to their deterministic values (1) are evalu-ated. In this expression the elements of [Z] are the possibility distributions of the active and reactive injected powers while [z] are the increment possibility distributions referring to the deterministic values [Zd].

= El - [zdl ( 1 1

c) The increments of angles (EV and PQ buses) and voltages (PQ buses), [AX], are evaluated using the Jacobian matrix [J] built in the last iteration of the deterministic study as defined in the Newton-Raphson method (2). The p2ssibility distribution of voltages and angles, [XI, are the superimposing of the deterministic values, [Xd],,and the distributions of their increments, [AX].

d) The increments of active and reactive line flows are evaluated considering they are non linear functions of both V and 8 in the extreme buses. If the active power flow between buses i and k is considered (4) the deviation APik can be, approximately, evaluated by (5).

( 4 )

Considering, in t h s esres_sion, t&e possibility distributions AVi, AVk, Aei and A8k it is possible_to obtain ATik. The possibility distribution Pik is thus:

For line currents and losses a similar technique can be used. An extensive study, however, showed this was not a satisfactory method for lightly loaded lines or lines where reversing of power flow may occur. In these cases the corrective method described in [6] shouM be considered.

e) The possibility distributions of the generated active power in the slack and the generated reactive power in the slack and PV buses can be evaluated considering they are non linear functions of V and 8 in all buses. These functions can also be linearized, so that their possibility distributions are obtained by superimposing the deterministic values to the respective increment.

3 Monte Carlo based simulation

The fuzzy load flow model can produce a linearization error which is function of the deterministic load flow initially performed and of the data uncertainty. To evaluate an upper bound of this error a Monte Carlo based simulation was performed by running and processing, in a possibilistic way, a huge number of load flow studies associated to different load and generation specified values so that the possibility distributions of voltages, angles, currents, power generations, flows and losses could be built. This simulation was performed considering the following steps:

a) The data possibility distributions were discretized by selecting 7 a-cuts: from 0.0 to 1.0 in steps of 0.2, plus the a-cut 0.5;

b) For each of these cuts the extreme values of any output variable, Z. were obtained. According to the Extension Principle (31, given n possibility distributions @Ai( X i ) i=l ... n) and a composition law, f(x1 ,..., Xn), the output possibility distribution pA(z) is obtained by:

pA(z)= SUP min (pAi(Xi), ..., pAn(xn)) (7) Z=f(Xt ,..., Xn)

If the extreme values associated to a cut of an output value Z are to be obtained, this indicates that the input values x i , ..., xn must have a possibility not less than a and the possibility of, at least, one of them must be a.

c) Considering the previous discussion, seven Monte Carlo simulations were run, one for each a-cut. In each of them the value of one active or reactive specified generated power or load was randomly selected fixing its possibility at a.

1340

I I I , I , , I

I

The other input values were randomly selected with uniform probability so that their possibility was not less than a. This way for each Monte Carlo simulation, 12500 load flows were performed.

d) For each output variable the extreme values obtained via the seven Monte Carlo simulations were used to build its possibility distribution.

4 An application study

The 60 kV distribution system of Oporto, Portugal, was used to test the Fuzzy Load Flow model by comparing it to the results obtained via the Monte Carlo simulation. The simplified network is presented in fig. 1.

220kV lSOkV 4

6 7

Figure 1: Oporto simplified 60 kV distribution network.

For this network, the specified active and reactive generation and loads are printed in ref. [6] (a 500 MVA power base was considered). Their distributions were assumed as trapezoidal fuzzy numbers represented by Pi, P2, P3 and P4 of f i i . 2. Other system elements can also be found in [6].

p3 p4

Figure 2: Trapezoidal fuzzy number.

In fig. 3 to 5 one may find the possibility distributions obtained for the generated active power at the slack bus and for the active and reactive flows at line 10-13, respectively. The distributions obtained using the Fuzzy AC Load Flow and the results of the Monte Carlo simulations are drawn in thin and strong line, respectively. In these figures the values obtained through two deterministic load flows (using Newton-Raphson) considering maxima and minima load values and generated powers of each possibility distribution are also represented.

eXtr8mecasesMluss n

Figure 3 - Possibility distribution of the active generated power at bus I.

a%tmmc8smwlm

-45 -3.5 'f 1.7. 3 5 6.9 8.1 10.1 14.K 15.1 P(W) 0.4

Figure 4 - Possibility distribution of the active power flow at line 10-13.

oxtremcasesvaluss

Figure 5 - Possibility distribution of the reactive power flow at line 10-13.

From the analysis of these results and the results obtained by the authors for other case studies, one may conclude that:

- the "exact" possibility distributions cover a wider range of values than the interval delimited by the values obtained with the two

1341

I ~~

I li I I 1 I ' I I

deterministic load flows using the maxima and the minima load and generated powers;

- the fuzzy load flow model reveals that some power flows may occur in either direction at some lines (due to different combinations of loads and generated powers). This could remain undetected if only some deterministic studies were performed;

- given the specified power uncertainties. the extreme values of some output variables cannot be found using the extreme (maxima or minima) values of the specified possibility distributions nor with the medium ones:

- the possibility distributions built via the Monte Carlo simulations have not a trapezoidal shape. Their left branches are concave while the right ones are convex. This was expected as the product, the sine or the cosine of trapezoidal fuzzy numbers, using (7 ) , are fuzzy numbers also affected by this kind of distortion;

- to compare two possibility distributions ref. [4] indicates three criteria to be applied in hierarchical order: the Removal, the Central Value and the Amplitude. In table 1 the relative errors corresponding to each of the three pairs of distributions of figs. 3, 4 and 5 are displayed considering trapezoidal approximations of the distributions built with the Monte Carlo simulations. In these three cases, both the relative errors of the Removal and the Central Value are satisfactory. The Amplitude relative errors seem bigger, namely, in the first case; however, these values must be interpreted considering that the Removal and the Central Value relative errors take into account the relative position in the real axis of the distributions to be compared. On the contrary, the Amplitude relative error doesn't consider this aspect so that small differences between, for instance the two possibility distributions represented in fig. 3, produce a large relative

I

ECV (Yo)

Table 1: Relative errors resulting from comparing the distributions represented in fig. 3, 4 and 5.

error. These values indicate that trapezoidal approximations of slightly distorted distributions are sufficient to capture the associated information and that the approximate distributions obtained with the Fuzzy Load Flow model are very satisfactory.

4 Conclusions

As it is recognized, fuzzy modelling can give a qualitative insight to system behavior. It also has the flexibility needed to be used as a man- machine interface as it condenses and translates large amounts of data and results into information understandable and manageable by decision makers.

The advantages of the Fuzzy AC Load Flow model were stressed by comparing its results with the ones obtained via a Monte Carlo simulation. It was possible to conclude that results having the same level of quality and information can be achieved by performing only one study (the Fuzzy AC Load Flow) instead of thousands of deterministic runs, achieving therefore a remarkable gain in efficiency.

References

[l] Zadeh, L.A., "Fuzzy Sets", Central. no. 8, pp. 338 - 353, August1965. [2] Zadeh, L.A, "Fuzzy sets as a basis for a theory of possibility", ~ ~ K Z Y Sets andSvstems, vol. 1 , no. 1 . pp 3 - 28, January 1978. [3] Zimmermann, H.-J., Fuzzy Set Theory and Its Applications, Kluwer Nijhoff Publishing, Boston, 1985. [4] Kaufmann, A., Gupta, M.M., Fuzzy Mathematical Models in Engineering and Management Science, North Holland,. Amsterdam, 1988. [5] Miranda, V., Matos, M.. "Distribution system planning with fuzzy models and techniques", Proceedings of ClRED 89, Brighton, August 1989. [SI Miranda. V.. Matos, M., Saraiva, J. T.. "Fuzzy Load Flow - New Algorithms Incorporating Uncertain Generation and Load Representation", 10 th PSCC, Graz, August 1990; in Proceedings of the 10 th PSCC, Butterwords, London, 1990.

1342

r