Optimal load flow using sequential unconstrained minimization technique (SUMT) method under power transmission losses minimization

Electric Power Systems Research 52 (1999) 61 – 64 www.elsevier.com/locate/epsr

Optimal load flow using sequential unconstrained minimization technique (SUMT) method under power transmission losses minimization M. Rahli a,*, P. Pirotte b b

a Electrical Institute, USTO, B.P 1505 Oran El M’naouer, Oran, Algeria Electrical Institute of Monte´fiore, Sart-Tilman, B-28 LiegeB 4000, Belgium

Received 15 July 1998; accepted 30 November 1998

Abstract The paper presents a new approach for solving the optimal power flow problem. The problem is solved by a sequential unconstrained minimization technique (SUMT) algorithm which was developed by Fiacco and McCormick. Firstly, we compute real power transmission losses and maintain a constant optimal power flow calculation procedure. Secondly, we propose a new approach for total real power transmission losses, as a linear function of generated real power; its effect and influence on the minimum cost of fuel and the result of real power transmission losses. In the second case, the real power transmission losses are 28% lower then the first case. The proposed approach has been applied for the first time to the 14 bus test system and the results are judged satisfactory. © 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Power transmission losses; Optimal load flow; Sequential unconstrained minimization technique algorithm

1. Introduction In an electrical power system, a continuous balance must be maintained between electrical generation and varying load demand, while system frequency, voltage levels and security also have to be maintained. Furthermore, it is desirable that the cost of such generation be minimal [1]. Moreover, the division of load in the generating plant becomes an important operation, as well as an economic aspect, which must be solved every changing load (1%) or every 30 min. Operations research techniques have been successfully used for solving optimal load flow problems by using linear or nonlinear programming. In our case, we retain the nonlinear programming problem. We use for the first time a SUMT algorithm [2,3] for solving the optimal load flow under some equality and inequality constraints. The equality constraints reflect a real and reactive power balance and the inequality constraints reflect the limits of real and reactive generation. We * Corresponding author. Fax: +213-642-1581.

suppose that voltage levels and security are maintained in the two cases. The proposed approach has been applied for the first time to a 14 bus test system [4] and the results are judged satisfactory.

2. Objective The problem of the economic dispatch [3,5] which exist to minimize the cost of production of the real power can generally be stated as follows: min



n

ng

% Fi (Pgi )

i=1

(1)

Under the following constraints: M Pm gi 5 Pgi 5 P gi

(2)

Q 5 Q gi 5 Q

(3)

m gi

ng

n

i=1

j=1

ng

n

i=1

i=1

M gi

% Pgi = % Pchj + PL % Qgi = % Qchj + QL

0378-7796/99/$ - see front matter © 1999 Published by Elsevier Science S.A. All rights reserved. PII: S 0 3 7 8 - 7 7 9 6 ( 9 9 ) 0 0 0 0 8 - 5

(4) (5)

M. Rahli, P. Pirotte / Electric Power Systems Research 52 (1999) 61–64

62

where generally Fi (Pgi ) is quadratic curve: Fi (Pgi )=ai + bi Pgi + ci P 2gi

(6)

and ai, bi and ci are the known coefficients and ng, is number of generator; Pgi, is real power generation; Qgi, is reactive power generation; Pch, is real power load; Qch, is reactive power load; PL and QL are respectively real and reactive losses.

3. Mathematical formulation

r 1 \ …\ 0}. Note that Fiacco and McCormick originally chose to make the function of the inequality constraints in the form of an added ‘barrier’: p

1 (k) g (x ) i=1 i

G(g(x (k)))= %

for as one or more gi (x (k))“ 0 from the feasible region, G(g(x (k)))“ 8; hence the concept of a barrier. As r (k) is reduced, the effect of the barrier is reduced, and x may move closer to an inequality constraint boundary. As mentioned before, other possible choices exist for G(g(x (k))), such as:

The nonlinear programming problem can be formally stated as:

G(g(x (k)))= % min{0, gi (x (k))}2

Minimize: f(x)

or

(7)

Subject to constraints

m

hi (x) = 0

i=1, …, m

and (p–m) constraints gi (x) ] 0

linear

linear

and/or

nonlinear

equality (8)

and/or

nonlinear

inequality

p

i=1

p

p

i=1

i=1

G(g(x (k)))= − % ln(gi (x (k)))= % ln

and in the 1970 coded version of SUMT the penalty function used was P(x (k), r (k))= f(x (k))+

i=m + 1, …, p

(9)

The nonlinear programming algorithm SUMT [2], developed at the Research Analysis Corp., McLean, VA, is an extension of the ‘created response surface technique’ proposed by Caroll, Fiaco and McCormick subsequently developed and validated the method and extended it to accommodate equality constraints. Computer codes are available from the Research Analysis Corp. and the Ballistics Research Laboratories, Aberdeen Proving Ground, MD. The SUMT algorithm has been developed to solve the nonlinear programming problem stated by Eqs. (7) – (9), in which the objective functions f(x) and inequality constraints gi (x), can be nonlinear functions of the independent variables, but the equality constraints hi (x) must be a linear function of the independent variables if convergence to the solution of the nonlinear programming problem is to be guaranteed. The basic idea underlying SUMT is to solve repetitively a sequence of the unconstrained problems whose solutions at the limit approach the minimum of the nonlinear programming problem. In the 1967 coded version of SUMT the nonlinear programming problem is converted into a sequence of unconstrained problems by defining the P function as follows: P(x (k), r (k))=f(x (k))+

1 r

(k)

1 gi (x (k))

m

1 r

(k)

% h 2i (x (k)) i=1

p

%

−r (k)

ln gi (x (k))

(11)

i=m+1

In both versions of the code the form of H(h(x (k))) chosen was simply the sum of the squares of the respective equality constraints, so that as r (k) “0, the equality constraints are more and more closely satisfied. Although in principle each equality constraint might be split up into two inequalities and treated as such, in practice this type of approach is quite unsatisfactory, it slows the search excessively and tends to cause premature termination. The minimization of Eq. (10) or Eq. (11) is initiated at an interior point (or boundary point), that is, a point x (0) at which all the inequality constraints are satisfied. After r (0) is computed, x (1) is determined by minimizing P(x, r (0)). Then r (1) is computed and x (2) determined by minimizing P(x, r (1)), and so forth. The procedure described suffers from several handicaps. First, the hessian matrix of the P function becomes progressively more ill-conditioned as the extreme is approached; hence the search directions may become mix-leading. Second, the rate of convergence depends on the initial choice of r (0) and the method of reducing r (k).

m

% h 2i (x (k))

4. Summary of the computational procedure

i=1

p

+r (k)

1 (k) ) i = m + 1 gi (x %

(10)

Where the weighting factors r are positive and form a monotonically decreasing sequence of values {r/r 0 \

Step l: k= 0, x (0), h (0) Step 2: compute S (k) = − h (0)G (k) h

(0)

2

(k)

= [9 f(x )]

−1

and

G

(k)

where = 9P(x (k))

M. Rahli, P. Pirotte / Electric Power Systems Research 52 (1999) 61–64

Step 3: x (k + 1) =x (k) +l (k) +S (k)

63

Table 1 Real constraints only

Step 4: Dx (k) = x (k + 1) −x (k) Step 5: DG (k) =G (k + 1) −G (k)

PG1 (optimal value) (MW) PG2 (optimal value) (MW) QG1 (optimal value) (MVAR) QG2 (optimal value) (MVAR) PL (MW) Fuel cost ($/h) Computing time (s) Iteration number

Step 6: h (k + 1) =h (k) +Dh (k) Step 7: if P(x (k + 1))− P(x (k)) 5o1 P(x (k)) and D(x (k + 1)) 5 o2 (x (k))

1st variant

2nd variant

187.233 90 617 – – – 985.385 0.00 20

154.446 118.155 – – 13.601 978.561 0.00 20

The initial points are:

the problem becomes optimal and terminate the procedure. Otherwise, go to step 8. Step 8: k= k +1 and go to step 2.

5. Test system and results The proposed method is applied to a 14 bus test system [4] to assess the suitability of the algorithm. The fuel cost (in US$/h) equations for the two generators are: F1(PG 1)= 100+1.5PG 1 +0 006PG 2 1

F2(PG 2)= 130+2.1PG 2 +0 009PG 2 2

and the constraints are: 1355PG 1 B195 (MW)

PG (0) = 150 (MW) 1

and

PG (0) = 120 (MW) 2

The results of the real generated optimal power, minimum fuel cost and computing time are given in Table 1.

5.1.2. Second 6ariant The transmission line losses are considered as a linear function of real generated power. The coefficients were calculated by the Gauss–Seidel’s method [3,5]. PL = 0.08996PG 1 + 0.03828PG 2 The power balance equation will become therefore: 0.91004PG 1 + 0.96172PG 2 = 259 We take the same initial points as the first variant. The results of the real generated optimal power, minimum fuel cost transmission line losses and computing time are given in Table 1.

705 PG 2 5145 (MW)

5.2. Case 2

−305QG 5 100 (MVAR) − 205 QG 5 100 (MVAR)

We will take into account real and reactive constraints with the same considerations in the first case. In the two variants, we will take the same initial points:

SPk = 259 (MW)

PG (0) = 150 (MW)

1

2

1

and

PG (0) = 120 (MW) 2

SQk =98 (MVAR)

QG (0) = 30 (MVAR)

The total load is 259 MW and the transmission line losses are respectively 18.85 MW and 75 MVAR after calculation by Newton – Raphson’s method [3,5]. We considered two cases with for every case two variants Tables 3 and 4.

The results of the real and reactive generated optimal power, total real transmission losses, minimum fuel cost and computing time are given in Table 2.

1

and

5.1.1. First 6ariant The transmission line losses are calculated and maintained constant (PL = 18.85 MW). The power balance equation will become: PG 1 +PG 2 =277.85 (MW)

2

Table 2 Real and reactive constraints

5.1. Case 1 We will take into account only real constraints.

QG (0) = 20 (MVAR)

PG1 (optimal value) (MW) PG2 (optimal value) (MW) QG1 (optimal value) (MVAR) QG2 (optimal value) (MVAR) PL (MW) Fuel cost ($/h) Computing time (s) Iteration number

1st variant

2nd variant

155.866 120.520 17.56 10.00 – 993.381 0.00 1

158.892 113.17 14.039 10.00 13.868 972.795 0.00 10

M. Rahli, P. Pirotte / Electric Power Systems Research 52 (1999) 61–64

64 Table 3 Transmission line data in p.u: P–q

Impedance

1–2 1–5 2–3 2–4 2–5 3–4 4–5 4–7 4–9 5–6 6–11 6–12 6–13 7–8 7–9 9–10 9–14 10–11 12–13 13–14

6. Conclusion Line charging

0.01938+j0.05917 0.05403+j0.22304 0.04699+j0.19797 0.05811+j0.17632 0.05695+j0.17388 0.06701+j0.17103 0.01335+j0.04211 0.00000+j0.02091 0.00000+j0.55618 0.00000+j0.25202 0.09498+j0.19890 0.12291+j0.25581 0.06615+j0.13027 0.00000+j0.17615 0.00000+j0.11001 0.03181+j0.08450 012711+j0.27038 0.08205+j0.19207 0.22092+j0.19988 0.17093+j0.34802

j0.0264 j0.0246 j0.0219 j0.0187 j0.0170 j0.0170 j0.0006 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000 j0.0000

The complete determination of the optimum power system steady-state operating condition is a nonlinear programming problem. A new approach to the solution has been developed in this paper, based on the Newton’s method and called SUMT method. The numerical results in the two cases indicate that the proposed method can be used to determine the optimum control for generation power with minimum fuel cost and lower transmission line losses with the expected accuracy in a time which is very short enough to be compatible with on-line applications.

References

Table 4 Bus data in p.u: Number

Bus type

Real power

Reactive power

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Reference Generator Load Load Load Load Load Load Load Load Load Load Load Load

0 0.6825 −0.9500 −0.4900 −0.0780 −0.1199 −0.0200 −0.1099 −0.3100 −0.1100 −0.0450 −0.0700 −0.1400 −0.1600

0 0.2489 −0.1079 −0.0370 −0.0200 −0.1535 −0.0100 −0.2162 −0.1800 −0 0700 −0.0200 −0.0180 −0.0600 −0.0600

.

[1] M.A Sheirah, M.B. Eteiba, M.A. El Sayed, M.I. Auf, On-line adaptive optimal load flow, Electr. Power Syst. Res. 10 (1986) 205 – 214. [2] D.M. Himmelblau, Applied Non Linear Programming, McGraw – Hill, New York, 1972. [3] M. Rahli, Applied Linear and Nonlinear Programming to Economic Dispatch, Ph.D. Thesis, Electrical Institute, USTO, Oran, Algeria, 1996. [4] L.L. Freris, A.M. Sasson, Investigation of the load-flow problem, I.E.E. 115 (10, October) (1968) 1459 – 1470. [5] G.W. Stagg, A.H. E1 Abiadh, Computer Methods in Power System Analysis, McGraw – Hill Book Company, New York, 1968.