On-line economic dispatch using a new loss coefficient formula

Electric Power Systems Research, 25 (1992) 131-136

131

On-line economic dispatch using a new loss coefficient formula Yung-Chung Chang, Wei-Tzen Yang and Chun-Chang Liu Department of Electrical Engineering, National Taiwan University, Taipei (Taiwan)

(Received May 23, 1992)

Abstract This paper presents an on-line economic power dispatch method by c a l c u l a t i n g the B-coefficients from the n e t w o r k sensitivity factors. The sensitivity factors are established from a new line flow solution based on the DC load flow model. The d e r i v a t i o n of a new loss coefficient formula is performed and the solution a l g o r i t h m is described. M a n y assumptions in the use of the c o n v e n t i o n a l loss coefficients are eliminated. Example systems are tested to demonstrate the performance of the proposed method. Compared with the exact solution of the optimal load flow method and the c o m p u t a t i o n a l r a p i d i t y of the base-case d a t a b a s e method, the proposed method has good a c c u r a c y and a fast execution time. This method is very suitable for on-line a p p l i c a t i o n in the economic o p e r a t i o n of power systems.

1. I n t r o d u c t i o n

The economic dispatch problem is to distribute load demand among the units available while minimizing the generation costs and satisfying the power balance equations and unit operating limits. The Langrangian multiplier method is still widely used to solve the problem and the penalty factors or the incremental losses are always the key points in the solution technique. The incremental losses can be calculated rapidly by the conventional B-coefficient method which maintains a database of many sets of coefficients calculated from average load demands. However, the B-coefficients are not constant when the system operating points change. This leads to low accuracy. The method based on load flow to evaluate incremental losses has good accuracy but is not practical in on-line application because of the long computation time. To overcome these shortcomings, Lin et al. [1] presented a method which involved calculating the penalty factors from a base-case database. This method has good accuracy and is rapid, but it is assumed that the load change in each bus conforms. This assumption can hardly exist in practical power systems. In particular, it will be invalid if the on-line units alter when the system loading conditions in the load pick-up and dropdown periods are the same. An on-line B-coefficient method was proposed by Li [2]. This method finds the solution rapidly, but it must

satisfy the assumptions that the voltage magnitude and the voltage angle in each bus are known in advance, and the voltage and system loss are fixed during the iteration procedure. We present a new method for calculating the B-coefficients to overcome the defects stated above. The algorithm starts with an initial dispatch without considering power loss. The DC load flow method [3] is then used to calculate the line flows and the power loss can be found immediately. The B-coefficients can be expressed in terms of the generalized generation shift distribution factor (GGDF) [4] which is evaluated by taking these line flows as a base condition. Finally, the generations are solved by using an iterative technique. 2. P r o b l e m f o r m u l a t i o n

2.1. Description of the economic dispatch problem The conventional economic dispatch problem is to find a set of unit generations which do not violate the operating limits while minimizing the objective function: NG

J = ~ fi(PG~)

(1)

i=1

Simultaneously, the power balance must be satisfied:

equation

NG

PGi ----PD + P L

(2)

i=l

Elsevier Sequoia

132

The cost function of the ith g e n e r a t i o n unit is u s u a l l y r e p r e s e n t e d as a second-order polynomial of P(~, :

approximation, we can calculate the G S D F by ,

Fm

A ( m , i) - ~P(~i

(3)

fi (PGi) = a, + biP(~ i + c,P(~, 2

-

m

xm

=

1

.....

NL

(10)

The L a g r a n g i a n multiplier m e t h o d is a d o p t e d to find the optimal solution of the convex function. By combining the objective function (1) with the p o w e r b a l a n c e e q u a t i o n (2) multiplied by a Lag r a n g i a n multiplier 2, the L a g r a n g i a n function is expressed as

w h e r e l and k are the initial bus and terminal bus, respectively, of line m. The new line flows after shifting can be expressed in an incremental form as

L = ~. f~ (PGi) + )~ PD

Fm = F ° , + ~ A ( m , i) AP(;~

Nc~

Pa~

i=l

(4)

The o p t i m u m g e n e r a t i o n dispatch is achieved by t a k i n g the derivative of L with respect to Pa~ and the c o o r d i n a t i o n e q u a t i o n is obtained: ~fi(Pai)

-

-

P F i = ).

(5)

w h e r e PF~ is the p e n a l t y factor which can be expressed as PF~ -

1 1 - OPL/~PGi

(6)

with the i n c r e m e n t a l loss term OPL/~P~. From (3) and (5), the g e n e r a t i o n of the ith unit can be expressed as Pc~i =

)~/PF i

-- b i

(7)

2c~

The system g e n e r a t i o n is expressed by the summ a t i o n over all g e n e r a t i o n units as NG

1

Arc"

2PF~c~

--~1 ~=

NG

=~1 i=1 PGi = ~" t"=

= ci

NG

NG

Fm

=

2

D(m, i) PGi

m = 1. . . . .

NL

(12)

i=I

w h e r e D(m, i) is the G G D F which can be calculated by D(m, i) = D(m, r) + A ( m , i)

(13)

and NG

NG b i

2(PD + P L ) + ,~--1 - -

2.2.2. Generalized generation shift distribution factor ( G G D F ) Since the solution a c c u r a c y of the G S D F is g u a r a n t e e d only when the total system generation remains unchanged, Ng [4] proposed the G G D F from the concept of the G S D F to overcome this limit. The G G D F can completely replace the G S D F and calculates line flows in an integral form when the total system generation changes. It can be defined by the equation

F°,, - ~ A ( m , i ) P ~ ,

(8)

By s u b s t i t u t i n g (2) into (8) and rearranging, the L a g r a n g i a n multiplier can be r e w r i t t e n as )~=

NL

(11)

bi

2c~

m = l .....

i=1

l

(9)

i=l

D(m, r) =

i ~i N(~

(14)

i=1

where D(m, r) denotes the G G D F for line m due to the generation of the reference bus.

1

~1 PF~c~ 2.2. S e n s i t i v i t y factors 2.21. Generation shift distribution factor (asDg) B e c a u s e it possesses the feature of reflecting the shift of bus g e n e r a t i o n to each transmission line, the G S D F [3] can be used to c a l c u l a t e line flows after g e n e r a t i o n rescheduling. Using the definition of the r e a c t a n c e matrix and the DC

2.3. Expression for p o w e r loss A p o w e r system delivers unit generations to c u s t o m e r s via transmission lines and produces p o w e r loss. There exists a relation b e t w e e n power loss, transmission lines and unit generations, and the B-coefficients (or incremental losses) may be c a l c u l a t e d immediately if we can find this relation. As derived by Lee et al. [5], the p o w e r loss can be a p p r o x i m a t e d by PL = R , . F , . 2

(15)

133

Substituting (12) into (15), we obtain the relation

PL =

Rm =1

D(m, i)

(16)

i

2.4. Determination of the loss coefficients The power loss expressed by using the B-coefficients and bus generations in a quadratic form was first developed by George [6]: NG

NG

PL = ~ ~ B~jPG~PGj

(17)

i=lj=l

The incremental loss can be derived as ~PL NG ~PGi

- 2 ~ B~jPaj

(18)

j =1

Taking the derivative of ~PL/0PGi with respect to

PGj, we acquire the B-coefficients: 1

~2PL

B~j - 2 5PG~ OPGy

(19)

A new loss coefficient formula is obtained by substituting (16) into (19): NL

Bij = ~, R,nD(m, i) D(m,j)

(20)

rn=l

The B-coefficients are constant owing to the GGDF remaining constant during the iterative procedure. It is noted that the summation of incremental loss in eqn. (18) is only over the generation buses, while that in ref. 2 is over all system buses; this leads to a shorter execution time for the proposed method throughout the iteration loop.

3. M e t h o d

Based on the previous derivation, a base-case flow calculation is necessary to establish the GGDF. In order to avoid using the AC load flow method which is time consuming, we use the DC load flow formula to find the line flows [3] as follows: F m = 0i - 0j

xm

m --- 1,. . . , N L

(21)

and 0 = XP

(22)

where 0~ and 0i are the voltage angles of the initial bus and terminal bus, respectively, on line m. The power loss is then obtained by substituting (21) into (15). The network configuration is unchanged, so the reactance matrix X can be

built in advance. Since the DC load flow model is involved in the process of establishing the GSDF, it does not increase assumptions using the model to evaluate base-case flows. The convergence speed can be accelerated by applying a smoothing factor which smooths the penalty factor change during iterations [ 7]: PF'i(new) = PFi(old) + a(PFi(,ow) - PFi(old))

(23)

The starting value of all penalty factors is unity. Because the generation of the reference bus is dependent on other bus generations, the penalty factor is assigned to be unity in the conventional method. But here it is considered as an independent variable and no reference bus is necessary in the proposed method. The solution steps are as follows. Step 1. Read in system information. Step 2. Execute initial dispatch without considering power loss. Step 3. Calculate the base-case flows by (21) and (22). Step 4. Calculate the power loss by (15) and absorb it with a generation unit (here, we use

Pox). Step 5. Calculate the GGDF by (10), (13), and (14).

Step 6. Calculate the B-coefficients by (20). Step 7. Calculate the incremental losses by (18) and the penalty factors by (6). Step 8. Modify the penalty factors by (23) and calculate the power loss by (17). Step 9. Calculate ~ by (9) and the generations by (7). Step 10. If max/JGi ,~(k) - - P(~i- 1) I ~<~, go to step 11; else, go to step 7. Step 11. If the generations exceed their limits, they are fixed on the limits and go to step 7; else, go to step 12. Step 12. Output results.

4. N u m e r i c a l e x a m p l e

In order to demonstrate the performance of the proposed method, the IEEE 30-bus system [8] is used to compare its solution accuracy with that of the optimal load flow (OLF) method and the IEEE 14-bus system [9] is used to compare its execution time with that of the real-time basecase database method. All tests were performed on a PC-AT 386 computer. The results using the IEEE 30-bus system are shown in Table 1. The elements in the last column are deviations of the proposed method

134 TABLE 1. Results for the I E E E 30-bus system

P(~ Pc~2 P(~,~ Pos P(~I~ P~ ~:~ Loss Cost

OLF method (W)

Proposed m e t h o d (V)

l

V / W (%)

1.7624 0.4884 0.2151 0.2215 0.1214 0.1200 0.0948 802.40

1.7462 0.4895 0.2166 0.2305 0.1255 0.1200 0.0944 802.73

0.9192 0.2252 - 0.6974 4.0632 3.3773 0.0000 0.4219 - 0.0411

TABLE 2. Load d e m a n d s for the I E E E 14-bus system

D e m a n d (p.u.) D e m a n d dev. (%)

A 2.590 100

B 2.409 93

C 2.460 95

D 2.668 103

E 2.720 105

TABLE 3. Results for the I E E E 14-bus system by the approximate m e t h o d

PG1 P62 P(~ Loss Cost CPU time (s)

A

B

C

D

E

1.6033 0.6869 0.3930 0.0932 1136.647 7.223

1.5420 0.6177 0.3316 0.0830 1057.064 7.231

1,5589 0,6368 0.3505 0.0858 1079.752 7.207

1.6263 0.7131 0.4258 0.0976 1171.171 7.250

1.6433 0.7323 0.4447 O.1008 1194.344 7.215

from the OLF method. It is obvious that the proposed method has good accuracy; the deviations in system loss and operating cost are only 0.4219% and -0.0411%, respectively. Table 2 shows the load demands and their percentage deviations for the I E E E 14-bus system. A is a base-case point and B, C, D, and E are four forecasting operating points of which the demands are 93%, 95%, 103%, and 105%, respectively, with reference to A. The system demands for each point are 2.590p.u. (A), 2.409 p.u. (B), 2.460p.u. (C), 2.668p.u. (D), and 2.720p.u. (E). The results are shown in Tables 3 5. The approximate method [10] is used to calculate the base-case economic dispatch for comparison in ref. 1. Table 3 shows the economic dispatch solutions for each operating point utilizing this method. The solutions for B, C, D, and E by the d a t a b a s e method are shown in Table 4. It reveals that the method has a rapid computation time and good accuracy, b u t the more the demand changes from the base case, the more the a c c u r a c y is reduced. The relative deviations in system loss and operating cost from the approximate method are given in the last two rows. Table 5 shows the results by the proposed method. It is obvious that the solution method has a rapid c o m p u t a t i o n time, only 0.11 second, and the deviations are i n d e p e n d e n t of the location of the points. The relative deviations from

TABLE 4. Results for the I E E E 14-bus system by the d a t a b a s e method

Pcu PG2 P¢~6 Loss Cost CPU time (s) Loss dev. (%) Cost dev. (%)

A

B

1.6033 0.6869 0.3930 0.0932 1136.647

1.5434 0.6211 0.3307 0.0866 1058.65I 0.198 4.3373 - 0.1501

0.0000 0.0000

C 1.5599 0.6392 0.3498 0.0884 1080.817 0.203 3.0303 0.0986

1)

E

1.6255 0.7] 16 0.4265 0.0959 1170.446 0.200 1.7418 0.0619

1.6418 0.7297 0.4457 0.0977 1193.498 0.202 3.0754 0.0708

TABLE 5. Results for the I E E E 14-bus system by the proposed method

p¢~ PG~ Pc6 Loss Cost CPU time (s) Loss dev. (%) Cost dev. (%)

A

B

C

1.5905 0.6977 0.3968 0.0949 1137.599 0.11 - 1.8240 -0.0838

1.5315 0.6305 0.3318 0.0848 1058.246 0.ll - 2.1687 -0.1118

1.5481 0.6494 0.3501 0.0876 1080.422 0.11 - 2.0979 -0.0639

D

E 1.6159 0.7268 0.4248 0.0995 l 172.236 0.11 1.9467 0.0909

1.6329 0.7462 0.4436 0.1026 1195.507 0.11 1.7857 0.0974

135

(ii) the voltage m a g n i t u d e at every source bus remains constant; (iii) the p o w e r factor at each source remains constant; (iv) the voltage angle at every source bus remains constant; (v) the ratio Rm/Xm is the same for all transmission lines; (vi) the phase angles of all load currents are the same. These assumptions will be i n c o n v e n i e n t in practical application and the results o b t a i n e d are unreliable. This paper has presented a fast economic dispatch m e t h o d by e v a l u a t i n g the loss coefficients from the sensitivity factors and only assumes: (i) the ratio RmXm is m u c h smaller t h a n u n i t y for all transmission lines; (ii) the voltage m a g n i t u d e at every bus is unity; (iii) the ratio of reactive to real flow is much smaller t h a n u n i t y for all transmission lines. Assumptions (i) and (ii) are a c c e p t a b l e in practical operation, (iii) comes from the derivation of the approximate expression (15) for the power loss and can be achieved by supplying the reactive p o w e r requirements from a local source [5]. Obviously, m a n y assumptions are eliminated. M a i n t a i n i n g many sets of loss coefficients in the d a t a b a s e and executing an AC load flow are not necessary, so the storage requirements are small. The n o n c o n f o r m i t y condition of demand changing which the d a t a b a s e method c a n n o t handle and which does not meet the assumptions of the c o n v e n t i o n a l B-coefficient m e t h o d can be t r e a t e d easily. Two s t a n d a r d systems are tested to d e m o n s t r a t e the effectiveness of the proposed method and the results show that it m a i n t a i n s the a d v a n t a g e s of a fast c a l c u l a t i o n speed and good accuracy. This method is very suitable for on-line application in power systems.

.ii,,,,,,,,,,,,,,,,,,

#.

,

[

,

l

,

,,

$315

,

I

,,

,,

I gO

~*

[

, , ,

10~

0

% PD

,,

Fig. 1. P e r c e n t a g e d e v i a t i o n in s y s t e m loss for t h e p r o p o s e d m e t h o d a n d t h e d a t a b a s e m e t h o d r e l a t i v e to t h e a p p r o x i m a t e method: - - , p r o p o s e d m e t h o d ; -, d a t a b a s e method.

0 . 2

' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' '

0.1

0 ° 0

~J

,

,

,

~-

,

95

. . . .

,. 1 gO

% PD

,

,

,

,

,

,

I 05

,

,

1

O

a-

Fig, 2. P e r c e n t a g e d e v i a t i o n in o p e r a t i n g cost for t h e p r o p o s e d m e t h o d a n d t h e d a t a b a s e m e t h o d r e l a t i v e to t h e a p p r o x i m a t e m ethod: - - - , p r o p o s e d m e t h o d ; - - , d a t a b a s e method.

the approximate m e t h o d are shown in Figs. 1 and 2. The smoothing factor and the convergence t o l e r a n c e for the example systems are set to 0.85 and 0.001 p.u., respectively.

5. C o n c l u s i o n s The classical dispatch a p p r o a c h with the use of B-coefficients involves a large n u m b e r of assumptions [11]: (i) all load c u r r e n t s m a i n t a i n a c o n s t a n t ratio to the total current;

Nomenclature

A(m, i) G S D F for line m, due to g e n e r a t i o n shift of ith unit ai, bi, Ci cost e q u a t i o n coefficients loss coefficients Bij D(m, i) G G D F for line m, due to generation of ith unit Fm active p o w e r flow on line m F° base-case p o w e r flow on line m fi (Pal) cost function of ith unit L L a g r a n g i a n function NB total n u m b e r of buses

136

total number of generation units total number of lines P bus injection power vector total system demand PD generation of ith unit P(li kth iterative value of Pc~ P(~! generation shift of ith unit P~ power loss penalty factor of ith unit PF~ PFi(.ew) penalty factor of ith unit before smoothing PF~(~w) penalty factor of ith unit after smoothing penalty factor of ith unit in previous PFi(old) iteration resistance of line m Rm reactance matrix X Xli, Xki elements of X reactance of line m x., 3( G

0

Oi 2

smoothing factor convergence tolerance bus voltage angle vector ith element of 0 Lagrangian multiplier

References 1 C. E. Lin, Y. Y. Hong and C. C. Chuko, Real-time fast economic dispatch, IEEE Trans., P W R S - 2 (1987) 968 972. 2 W. Li, An on-line economic power dispatch method with security, Electr. Power Syst. Res., 9 (1985) 173 181. 3 A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control, Wiley, New York, 1984. 4 W. Y. Ng, Generalized generation distribution factors fbr power system security evaluation, IEEE Trans., PAS-IO0 (1981) 1001 1005. 5 T. H. I, ee, D. H. Thorne and E. F. Hill, A transportation method for economic dispatching, IEEE Trans., PAS-99 (1980) 2373 2385. 6 E. E. George, lntrasystem transmission losses. Trans. AIEE, 62 (1943) 153 158. 7 H. H. Happ, Optimal power dispatch, I E E E Trans., PAS-93 (1974) 820 830. 8 0 . Alsac and B. Stott, Optimal load flow with steady-state security, I E E E Trans., PAS-93 (1974) 745 751. 9 G. F. Reid and L. Hasdorff, Economic dispatch using quadratic programming, IEEE Trans., PAS-92 (1973) 2015 2023. 10 C. E. Lin and Y. Y. Hong, Approximate penalty factor eco nomie dispatch, IEEE Midwest Power Syrup., Houghton, MI, USA, 1985, pp. V-E-1 5. 11 W. D. Stevenson, Jr., Elements o[ Power System Analysis, McGraw-Hill, Kogakusha, Tokyo, 1962.