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distribution systems
A new decomposition method for optimal operation of transmission/generation and subtransm ission/d istri bution systems M i c h e l e Brucoli, M a s s i m o La Scala, Francesco Torelli and M i c h e l e Trovato Politecnico di Bari, Dipartimento di Elettrotecnica ed Elettronica, Via E. Orabona 4, 7 0 1 2 5 Bari, Italy
The level of automation in electric power systems and, particularly, in subtransmission/distribution ( SD) systems is becomin9 hi qher and higher. The continuous surprisin9 increase of the number of control actions should be properly taken into account in optimal power flow ( O PF) programs. This is important, in order to avoid deficiencies and inaccuracies which can degrade the usefulness of the OPF application. In the first part of the paper a 9eneralized formulation of the OPF problem is provided which takes into account the entire set of controls and involves specific goals for the SD systems. This leads to the definition of a non-linear multiobjective optimization problem which can be solved off-line only with a very heavy computational burden. To overcome this difficult)', the second part of the paper deals with the development of an optimization procedure which exploits the naturally decomposable structure of the power system model. Namely, on the basis of a Decomposition-Coordination technique, the proposed method turns the 9lobal optimization problem to the solution of several decoupled subproblems. The validity of the su99ested procedure and the accuracy of the results are illustrated by a numerical example. Keywords." model/in9 oJ" energy systems, optimal power flow, mathematical programmin 9 I. I n t r o d u c t i o n
Over the past years, three main activities have been developed by the electric utilities dealing with planning, Received 15 June 1992; revised 7 December 1992; accepted 25 February 1993
Volume 1 5 Number 5 1 993
design and operation of subtransmission/distribution (SD) systems: (i) development of demand-side management (DSM) strategies for achieving savings in fuel or capacity at the generation level 1 11; (ii) utilization of dispersed and storage generation devices (DSGs), as a managed part of total utility supply system12'13; and (iii) implementation of new powerful control and monitoring functions for improving the operation of the SD system 14 21. Referring to the third activity, it should be pointed out that intensive investigations have been carried out to assess the technical and economic feasibility of automated distribution systems and to integrate these new control actions into the energy control organization of power systems14 iv. Moreover, the impact of these new control actions on the real-time control problem of these systems has also been studied 18 21. As is well known, the recent advancements in Optimal Power Flow (OPF) methods 22 24 have made more realistic the implementation of OPF programs on the Energy Management System for on-line applications. However, there are some aspects which have not been completely exploited 25'26. A critical question concerns the accuracy to be adopted for modelling SD systems and how to take into account the control structures used for these systems. On the other hand, a formulation of the OPF problem which globally includes accurate representations of the SD systems with the entire set of their control actions is clearly unrealistic, owing to the great dimension of the problem and the consequent computational burden.
0142-061 5 / 9 3 / 0 5 2 7 3 - 1 0 •
1993 Butterworth-Heinemann Ltd
273
To overcome this serious drawback, several techniques have been proposed very recently which utilize decomposition coordination concepts 27'2s for solving the generation planning and OPF problems for large-scale systems29 32. These procedures decompose the overall optimization problem into several subproblems which are separately solved. The various solutions are then coordinated according to a centralized policy and the process is repeated until it converges to a global optimum operating point, i.e. the pool objective. These procedures allow a considerable reduction of the computational time, but it should be pointed out that they refer to a unique objective function which often takes into account only specified goals of the TG system. Moreover, the presence of control strategies at the level of SD systems is scarcely considered. On the basis of these observations, in this paper a new procedure for solving the OPF problem is presented which takes into account not only the optimal operation of the TG system but also explicitly includes specific objectives, control actions, technical and economic requirements of the SD systems. Namely, starting with the observation that a large power system consists of several SD networks (HV or MV levels) connected to a TG network (EHV level), a mathematical representation which reflects this naturally decomposable form has been derived for the overall system. In particular, the system is described as an interconnection of subsystems, each one modelled by its steady-state equations. In these equations control variables, inputs and couplings with other subsystems are clearly displayed. Then, a multiobjective non-linear optimization problem for the overall system is formulated by assigning a specified goal, characterized by an appropriate objective function, to each subsystem. Finally, on the basis of an optimization approach which utilizes the Decomposition Coordination method 2~, a new computational procedure is suggested which avoids the time-consuming coordination phase of the general method. Namely, by using realistic assumptions at the decomposition phase, the global optimization problem is solved by using a hierarchical computational scheme characterized by two levels. At the first level, the solution of several decoupled subproblems, associated with each SD system, is determined. At the second level, the optimal operation conditions of the TG system are obtained taking into account the above determined optimality requirements of all SD systems. In this way, the reduction of the computational burden is considerable and, at the same time, the accuracy of the results is satisfactory. Furthermore, the proposed procedure proves suitable to be implemented in Energy Management Systems with a hierarchical and distributed control organization. The capability and usefulness of the method are illustrated using a 275/132 kV 40-busbar test system.
I
transmission/generation system
1
| m,,)~" O(i) m~,)-~b ,~(,)mO,-(~.,,)
Q(J)[ l ' ~
~(J) ......
1
(J)
w':I ' k ' ~
......
Q~I N I j ~
--
subtransmtssion/distribution system
Figure 1. Scheme of interconnection between the TG system and the jth SD system
outputs, the transformer tap setting and the outputs of compensating devices, such as synchronous compensators, capacitors and reactor banks and static var systems. Now, let us consider an interconnected power system composed of N SD networks connected to a TG system. The TG system includes N T nodes: N~ load nodes and N G generating nodes (NT = ND + N~). Thejth SD network includes Nj nodes: Noi load nodes and N ~ generating nodes, which take into account the presence of dispersed storage and generation. The link between the TG network and the jth SD network is established by NIj interconnection buses (j= 1.... N) and N~ transformers or autotransformers, each one represented by an ideal autotransformer with real transformation ratio ~,u) Hqk (k = 1.... Nu) or an ideal phase shifter with a pure angular shift ,~u) V'1k, as shown in Figure 1. In addition, in this figure pu) nu) (k = 1, ' • • Nlj), indicate the real and reactive sk and "z~sk, net power flows, respectively, delivered to the jth SD system through the kth interconnection bus. Finally, it is assumed that a voltage-var control shunt device is available at each interconnection bus and .~x~TktO(J)~,~/"---- l , . . . Nlj ) represents the corresponding reactive power injection. 11.2 System equations and constraints Two general classes of constraints are involved in an OPF problem: the network constraints and the constraints due to the operating limits. The network constraints refer to the voltage, current and power-flow relationships and are usually expressed in terms of real and reactive power supply and demand balance equations. Under the above assumptions, the equations representing the overall interconnected power system can be synthetically written in the following form:
PG-P(O, V,m,¢~,Oi, V , , m , , O , ) = 0
(1)
II. Formulation of the global OPF problem
QG - Q(0, V, m, ~, 01, Vl, mT,~i) = 0
(2)
I1.1 Assumptions
- P D - P ( 0 , V,m, dp,Ol, Vl, ml,~l):0
(3)
- QD -- Q(0, V, m, ~, 01, V~, mr, ~bl) = 0
(4)
For an electric TG system, the optimal power flow problem consists in minimizing the fuel costs, system losses, or any other appropriate objective function by adjusting certain controllable variables and satisfying physical and operating constraints on various control parameters and dependent variables. Usually, the control variables are the generator real and reactive power
274
-- eli{O, V, O,j, VU, m , j ,
~lj, O j,
Vj) = 0
j = l ... N
QTj - - Qb{ O, V, O,j, Vu, mu, Oij, Oj, Vj) = 0
(5)
j = 1... N (6)
PGj-
Pj(OIj, Vii, m~, 0j,
V~) = 0
j = 1... N
(7)
Electrical Power & Energy Systems
QGj -- Qj(OIj, Vu, m~, Oj, Vj) = 0 - Po
-
Vlj, m j, o j,
j = 1... N
= o
Qcj - QDj - Q_.J(Ou,Vii, m j, Oj, Vj) = 0
j = 1... N
j=I...N
(8) (9) (10)
with 01 =
[0tI ,' 0tl ......
v,=
m, = [ m',,,
0 1 t, . . . . .
.....
0~
tions among the control actions. To this purpose, equations (5) and (6) can be restated as
-Psj-P'j(O, r, OU,Vu, mlj,~lj)=O
j=I...N
(11)
QTj -- Q , j - ~(0, V, Oij , rlj , mlj , ~lj) = 0 j = 1... N (12) P,j-- ~'(0,i, V,j, Oj, Vj)=0 j = 1... N (13)
It
.....
..... ml, ..... roll'
where PG, Q6
Nc-dimensional vectors of real and reactive powers at the generating nodes of the TG network, including the slack bus Po, QD No-dimensional vectors of real and reactive consumptions at load nodes of the TG network 0, V Nv-dimensional vectors of phase angles and voltage magnitudes at the nodes of the TG system m, ~ vectors of proper dimension of the tap settings and angular shifts of the transformers of the TG system 0u, Vu Nu-dimensional vectors of the phase angles and voltage magnitudes at the interconnection buses between the TG network and the jth SD network mlj, ~lj Nifdimensional vectors of the tap settings and angular shifts of the transformers linking the jth SD network to the TG system QTj Nu-dimensional vector of the reactive powers supplied by voltage-control shunt devices at the interconnection buses of the jth SD network with the TG system 0i, Vj Nfdimensional vectors of the phase angles and voltage magnitudes at nodes of the jth SD network P~j, Q~j N~fdimensional vectors of real and reactive powers of the generating resources of thejth SD system, including large or intermediate sized dispersed storage and generation devices PDj, QDj NDfdimensional vectors of real and reactive loads of the jth SD network m~ vector of appropriate dimension of the transformer tap settings present in the jth SD network Qc~ ND;dimensional vector of the reactive powers supplied from the compensation devices at each node ofthejth SD network The equations (1~(10) represent the load flow equations of the interconnected power system. Particularly, the vectors P~, Q~, m, gp, QTj, m~ and S u ( j = I . . . N ) summarize the main control variables that affect the operation of the transmission system and, consequently, of the SD systems too. Analogously, the variables P~, Q~, Qc~ and m~ (j= 1... N) are mainly concerned with the operation of each SD system, but their effects have repercussions on the EHV level. Owing to the presence of so many different control actions on the system, it is important that all these control actions be coordinated to gain the maximum degree of efficiency. To reach this objective it is first necessary to individuate the correla-
Volume 15 Number 5 1993
Qs~-Q](0u, Vu, Oj, V~)=0
j= I...N
(14)
with
QTj--r/-l(J) -- L~TI,
/')(J) ~,T2~
/')(J) [)(j) ]t • • • , Y-~Tk~ • • • , - ~ T N I j - I
Psj = [P(J) P(j) P(J) i_ si , s2 ~ ' " " , ~ S k '
Q s.J = [oo) o(J) o (j) I - r . , S l , -Y-,S2 ~ " " " ' I ¢ . , S k , " ' 01j :
p(j) ]t
" " " ' ~SNIjA
o (j) i t
" , If--SNIjJ
[-~(J) ~(j) ~(J) ~(J) -It LVlt ~ vie ' " " " ' 'JIk ~ " ' ' ' VlNlj/
0j=e0?,0?
.....
.....
where ~ ' ) and Q'j<') are Nu-dimensional vectors, the kth element of which is a function representing the net real and reactive power leaving from the kth interconnection bus and determined only by the network flows on the TG-side. Analogously, ~'(.) and ~'(.) represent the same powers as determined by network flows on the SD-side. Finally, since line flows depend only on the difference between the terminal-voltage phase angles, the vectors 01j and 0i appearing in equations (1 3) and (14) are referred to one of the interconnection buses of thejth SD network. Thus, equations (11}-(14) allow P,j and Qsj to be considered, simultaneously, as 'loads', at the EHV level, and 'generated powers', i.e. control variables, at the HV level. This means that the interconnection buses can be included into the set of the buses of the TG system as load buses with a reactive power source for the voltage control. On the other hand, they have to be considered as also present in thejth SD network as generating buses, whose real and reactive powers can be optimized accordingly to a specified goal. In addition to the network constraints, engineering constraints have to be considered, which take into account specified operating and security requirements, such as voltage levels, generator loading, reactive source loading, transformer-tap limits, transmission-line loading. These constraints are usually given in the form of inequalities and can be summarized as h(PG, QG, QT, m, ~b,O, V, ml, ~bl,el, 111)~<0 (1 5)
h~P,¢, Qsj, Pc~j, Qcq, Q~, mr, oj, Vj, ¢~,~,ru) <~0 j= I...N
(16)
where h(.) and hk) are vectors, of appropriate dimensions, of functions representing all the inequality constraints on the TG network and on thejth SD network, respectively.
11.3 Global OPF problem Usually, when solving the classic OPF problem, the objective function represents specific goals for the TG system, such as the cost of real power generation. The aim of this paper is to assess the global OPF problem for a large power system including several SD systems, which are explicitly taken into account. Thus, the following general form can be assumed as objective function V = c(P~, Q~, QT, m, qS,0, V, mT,~bl,01, VI) N
+ ~ Q(P,j, Qsj, P~j, Qcj, Q¢j, m~, Oj, vj, 0 u, vu) (17) j=l
275
where c(.) is the cost criterion for the TG system and cxl.) the objective function of thejth SD system. Now, under the above assumptions, the O P F problem for the overall power system can be formulated as follows: N
minimizec(w,y,z~ ..... z; ..... ZN)+ ~ Cj4%,Yj, X i) (18) j=l
b y 3 3 , 34
given
8L 8w
Ow
8L
-
\Ow)
\Owl
\ Oy)
\~Oy)
+
oy
OL _ c< -]- (Og]t
subject to
O(w,y,p, zl ..... zj ..... zN) = 0
(19)
h(w,y, z~ ..... zj ..... zN) <~0
(20)
Os(wj, xj, yj, pj)=O
(21)
hj(wi, xj, yj)<~O xj = Dszj
j = 1 ... N j = 1 ... N
j = 1... N
(22) (23)
(29)
DZ;
Ozs
(30)
2f_(Oh~ t
\OZsl
p-D}pj=O
(31)
OL 82
OL 8p
- g ( w , y , p , zl ..... Zj..... ZN)=0
(32)
- h ( w , y , zl ..... zj ..... ZN)=O
(33)
with
w = EetG, QtG, QtT, mt, f~t, mtl, ~tl] t P = [P~D, Q ~ I t
0%
y = [0 t, V t, 0',]'
z i = [e'~i' Qtsi, ~j]' Yj = E01u,~, ~ ] '
-
oL
pj = EPbj, Qi)j] t
8xj
L= c(w,y, gl . . . . .
Owj o<
\Ow/
+
Zj . . . . .
ZN)
+ ~,to(w,y,P, Zl ..... Zj..... ZN)
+ pth(w,Y, Zl ..... Zj..... ZN) N + E [Cx(Wj'YJ' X j) + ).~gj~wj, xj, yj, pj)
-
%
(og?I'
,~j+
8xj
/agfi'_
+ I--I
20Xfl
j=I...N
(34)
j=I...N
(35)
\Owfi
wj = [ptj, QtGj' Qtj, m}]t
where D r is a (3NIj x 3NIj) identity matrix and xj is a 3Nu-dimensional vector. The objective functions c(') and cj{.) ( j = l , 2 .... N) are identical to those defined in equation (17), in addition the inequality constraints (20) and (22) correspond to equations (15) and (16), respectively. Moreover equation (19) includes equations (1)-(4) and (11)-(12), thus equation (19) is the steady state equation of the TG network. Equation (21) includes equations (7)-(10) and (13)-(14), thus it synthetically represents the load flow equations of thejth SD network. Finally, equation (23) represents the interactions between the TG system and the SD systems. A general method for solving the multiobjective optimization problem (18)-(23), consists in applying the conditions established by the theorem of Kuhn and Tucker 33'3.. To this purpose, by assuming that the functions c('), O('), h(') and cj(.), O~(') and hi(.) are convex, the following Lagrangian can be associated to the optimization problem (18)-(23):
j=l ...N
\DZj}
•j
/Ohj '
+/--/
20Xfi
[l~j -t- pj
=
0
j=l ...N (36)
8L =gj~wj, xj, yj, pj)=O 8,~j 8L -h(wj, xj, y ) = O 8pj 8L - x s - Diz ~= 0 0pj
j = 1... N
(37)
j = 1... N
(38)
j = 1... N
(39)
It should be observed that, in addition to the traditional problem of determining the optimal operating condition of the TG system, the above considered formulation of the O P F problem involves further conditions to be satisfied with the aim of achieving, at the same time, analogous goals for each SD system. The suggested statement of the O P F problem allows the structure of each involved subnetwork to be preserved. Moreover, the interactions among subsystems are explicitly taken into account. This approach proves useful, as will be shown in the next section, when a DecompositionCoordination procedure is used for solving the problem (18)-(23), because it reduces the mathematical complexity of the problem and the computational effort needed.
j=l
+ p)hj(w~, x~, y j) + p}(x~ - Dsz~)]
(24)
where L 2j and ps (j = 1 ... N) are vectors of appropriate dimensions of Lagrange multipliers which take into account all equality constraints; p and pj (j = 1 ... N) are vectors of appropriate dimensions of Kuhn-Tucker multipliers, for which the following conditions hold (25)
p~>O /zj >/0
j = 1... N
(26)
pth(w,y, z~ ..... zj..... zN) = 0
(27)
~}hs(wj, xj, yj)=O
(28)
j = 1... N
Then, the set of necessary conditions for a minimum are
276
III.
Decomposition-Coordination
procedure
The conditions (29)-(39) require the solution of a system of non-linear equations. This problem is very difficult to tackle in a global way, because its solution requires too much computational time making on-line application impossible. In attempting to give an efficient and practical solution to this problem, in this section an approach is developed on the basis of a Decomposition-Coordination procedure 27'28. In order to apply these methods, two important conditions have to be satisfied. First, it is necessary to have a mathematical representation of the complex system under study in a decomposable form. Namely, the overall system has to be modelled as an interconnection of subsystems, each one described by its steady state equations, coupling with other subsystems
Electrical Power & Energy Systems
~
subject to
Y ~
subtransmission system
W
g(w,y,p, Zl ..... zi .... , ZN)= 0
(44)
h(w,y, Z l , . . . , z j . . . . . ZN)<~ O
(45)
and
!
minimize cj(w~,y~, xj) + p)xj
el Zj
Xj
(46)
subject to subtransmission
gj(wj, xj, yj, pj) = 0
system
~0
h~(%, x j, y~) <~0
P ~
j= 1...N
subtransmission system
Figure 2. Schematic representation of the structure of the considered large-scale power system
and an objective function which takes into account the specific goal to be achieved for this subsystem. Secondly, the objective function of the global optimization problem must have an additively separable form. Now it should be noted that the mathematical description of the optimization problem (18)-(23) allows us to identify S= N + 1 subsystems: the TG system and N SD systems. The couplings among the subsystems are expressed by the equation (23) and are schematically shown in Figure 2. In particular, the vectors w, p, y and wi, pj, y j (j = 1... N) represent the control variables, inputs and outputs for the transmission system and the jth SD system, respectively. The vectors zj (j= 1... N) are the outputs of the 'transmission' subsystem which act as inputs, i.e. as intermediate-control variables x j, to the jth SD subsystem. Under these assumptions, the Goal Coordination Method 27'28 can be applied for solving the global optimization problem (18)-(23). This method involves a two level computation scheme and assumes the Lagrangian multipliers pj (j--1... N) as coordination variables. To this purpose, we can associate, with simple mathematical manipulations in the Lagrangian (24), the term p~rj to the jth SD system, so that equation (24) can be rewritten as N L = Lo + Y Lj (40) j=l
j = 1...N j = 1... U
(47) (48)
The solution of these optimization subproblems constitutes the decomposition level of the two level calculation structure which is employed for solving the global problem (18)-(23). Practically, this is equivalent to solving independently the system of equations (29)-(33) and (34)-(38), with fixed p j = ( j = 1... N). Then, the main task of the coordination process is to update the interaction variables. To this purpose, let w*, y*, z* (j= 1 ... N) and w*, y*, x* ( j = I . . . N ) be the solution of the above subproblems and ~j = x* - DjZ*
(49)
a 3N,idimensional difference-vector. If ej=0, by comparing the objective function (18) and the objective functions (43) and (46), the solution of the global optimization problem (18)-(23) is also achieved. If not, a second level of calculation, i.e. the coordination level, is necessary in order to ensure that constraint (39) is satisfied. At this level, a new set of parameters pj ( j = I . . . N ) can be derived by using the following gradient-type coordinator algorithm (see Appendix) =
+ Kj x - Djz9
j = 1... N
(50)
with Kj>0
(41)
where Kj is a (3N~j× 3N~j) constant diagonal matrix and zj and xj derive from the solution, at the step v, of the decoupled optimization subproblems (43)-(45) and (46)-(48), respectively. The developed Decomposition-Coordination procedure presents some advantages. First, the solution of the overall optimization problem (18)-(23) is reduced to the solution of N + 1 independent optimization subproblems. Secondly, the procedure can be carried out by using parallel or distributed computing architectures. In this case the computational task corresponding to the Decomposition phase can be carried out by a unique parallel computer located in the power system control centre, or by using a suitable distributed computing scheme, with several processors located at the lower levels in the control hierarchy, for example into the regional control centres.
(42)
IV. Solution of the optimization problem by a non-iterative decomposition procedure
Now, if the parameters pj (j= 1..... N) are fixed, the function (40) assumes a 'separable' form. Thus, we can associate the following N + l decoupled optimization subproblems to the previous Lagrangians (41)-(42) N minimize c(w,y, z~ . . . . . z i . . . . . ZN)-- ~. p)Djzj (43) j=l
It should be pointed out that the general solution scheme illustrated in Section III requires, owing to the coordination phase, an iterative process for obtaining the optimal operating condition of the overall interconnected power system. For large power systems this constitutes a heavy task, from the CPU time-consuming viewpoint, even if the above mentioned computing architectures are adopted.
with L o = c(w,y, Zl . . . . . ZN) + ~tg(w,y,P, Zl,... ,Zj . . . . . ZN) N
+ pth(w,Y, zl . . . . . zN)- ~ p)Djzj j=l
+ I~hy(w~, x j, y j) + p~x~
Volume 15 Number 5 1993
277
In order to overcome this drawback, a suboptimal optimization procedure characterized by a two-level computational hierarchy can be suggested. At the first level, the optimal operating condition of each SD system is determined by solving independently the N optimization subproblems defined by equations (46)448) with p~=0, j = 1.... N. At this level, according to the assumptions made in Section II, the interconnection buses, for each SD network, have to be considered as generating buses with a fixed voltage, one of which is assumed as a slack bus. At the second level, the O P F problem (43)-(45) is solved, for the TG system, by assuming p j = 0,j = 1.... N, and considering the interconnection buses as load buses with fixed voltage magnitudes. At this level, the loads of the interconnection buses are the power flows which optimize the operation of the SD systems, as determined at the first level. Note that the suggested computational scheme can be deduced from the general method developed in Section III by using realistic assumptions. Namely, let wj,1 yj1 and x) (] = 1 ... N) be the solution of subproblems (46)-(48) for v = 1 and p) = 0
j= I...N
(51)
Figure 3. One-line diagram of the test system
The O P F problem (43)-(45) can be solved, for the TG system, by using equations (29)-(33) and assuming z) =xJ
j= I...N
(52)
Under the assumptions (51) and (52), equation (50) gives for v=2 Or2 - 0
j=l ...N
(53)
Thus the coordination phase proves unnecessary and the iterative solution scheme of the Decomposition Coordination method can be avoided. It should be noted that the suggested suboptimal procedure gives priority to the optimization of the SD systems. This is important because the variables Psi and Qsj can be properly taken into account. Namely, they have been explicitly considered as control variables. Moreover, it should be noted that, at present, these variables may represent the only available control actions in each SD system. On the other hand, since most of the control actions are concentrated in the TG system, it should not be difficult for the TG system to meet the operating conditions at the interconnection buses resulting from the solution of the decoupled optimization subproblems for the SD systems (see equation (52)). Finally, it is interesting to note that with the suggested suboptimal procedure the previously discussed distributed computing architecture can be still used.
Table 1. Operating conditions of SD-system generators, on 100 MVA base Generator
Real power (p.u.)
Reactive power (p.u.)
GS1 GS 2 GS 3 GS4
0.40 0.50 0.70 0.15
0.10 0.13 0.18 0.06
the generator nodes of the TG system; (ii) the operating conditions shown in Table 1 for the generators of the SD systems; (iii) the real nominal transformation ratio for all the interconnection transformers; (iv) a fixed voltage magnitude of 1.02 p.u. at the interconnection buses 8, 9, 25, 28, 32 and 36. In addition, only the network constraints have been considered for the dependent and control variables. Taking into account the above mentioned assumptions, the following cost function for the overall system is proposed
F=c+cl
V. Numerical example To demonstrate the effectiveness of the suggested approach, a 40-busbar modified version of the sample power system reported in Reference 34 has been used (Figure 3). This power system consists of three 132kV subtransmission networks SD1, S O 2 and SD 3 connected to a 275 kV transmission system through the interconnection buses 8, 9, 25, 28, 32 and 36. In this case there are three generators in the TG system and four generators in the SD systems. The network data are available from the authors. All simulations have been carried out by assuming: (i) a fixed voltage magnitude of 1.05p.u. for
278
with 3 C=Cp-~CQ-~- 2 Coi i-1 3 3
C1:
2 i=1
L'Ps,+ 2 CQ~, i=1
and
ep = ~ (ah + flhPch + 7.P~h)
oo: E
hG.CfG
2
E (eoh-eo,)2/2
hE,~a kE,CC°G
Electrical Power & Energy Systems
Z Q~+ ~_, ~ (QTj-QT,)E/2
cQ=
j~Jl i
i r~li
j~I
+ y' ~, (Qc.-QT)2/2 h~.SPG jeSF li
Cps/= ~
~
BjrPsjPs,
jES'ali rE~li
cos,--
J~,~b
Y
Je'~allr~.~f li
IQs,-Qs)2/2
where AeG is the set of the generating buses of the TG system; 6~, is the set of the interconnection buses between the ith SD system (i= 1,2,3) and the TG system; Cth, flh and 7h, hE'G, are the heat-cost coefficients of the generators of the TG system, the values of which are given in Table 2. The generating sources on the subtransmission systems are considered as dispersed generation, for which no cost curves are assumed. The terms B~ are the loss coefficients of the interconnection buses, considered as generating buses for each SD system 35. Thus, these coefficients are calculated by using only the line data of the SD systems (see Figure 3). The cost function F depends upon both real and reactive system variables and consists of two objectives. The function c represents the two goals to be achieved for the TG system: the minimization of the total operating cost and the optimization of the reactive power delivered by
all the reactive-power sources in the TG system3~. The function cx takes into account the specific goal for each SD system, which consists, in this case of the optimal management of real and reactive flows delivered to the SD systems. In particular, Cps' represents the power losses of the ith SD system aS. Table 3 presents the most significant results when the suggested decomposition procedure is applied for solving the problem of minimizing the function F subject to the network constraints of the overall power system. For purposes of comparison, Table 3 also shows the corresponding results when the same problem is solved by using the Decomposition-Coordination method. The comparison clearly reveals the validity of the suggested procedure. To illustrate better the importance of the presence of a specific goal dealing with the optimal operation of the SD systems, different cost functions have been considered and compared with the cost function F. To this purpose, the previously considered load condition has been assumed as daytime peak load and a typical daily load curve has been assigned to each bus of the test system. Figure 4 shows, for example, the load curves of buses 10
Table 2. Heat-cost coetticients for the generators of the TG system on 100 MVA base (heat cost f = ~ + flPc + ~t~c) Generator
~
fl
?
G1 G2 G3
0 0 0
170 170 85
2 2 2
OoT, 12 A.M.
' 4
8
12 N
4
8
12 P.M.
Figure 4. Daily load curves of buses 10 and 15
Table 3. Comparison of results of the Decomposition-Coordination method and the suggested non-iterative decomposition procedure
Total generation cost cp (£/h) CQ Jr- ZCQi ~,Cpsi ~CQs i CQ "q- ~,CQi -~ ~-dCpsi "~ ~ C Q s
i
Real-generation of TG system Reactive-generation of TG system Real-interchange between the TG System and the SD systems Reactive-interchange between the TG system and the SD systems Total real-generation Total active load Total real line-losses Real line-losses of TG system Real line-losses of SD systems Total reactive-generation Total reactive load Total reactive line-losses
Volume 15 Number 5 1993
DecompositionCoordination method
Suggested decomposition procedure
3611.421 28.832 1.057 4.222 34.111 26.336 4.142 16.220
3618.215 28.737 1.219 4.238 34.194 26.353 4.151 16.223
4.008
4.151
28.086 26.43 1.656 0.701 0.955 7.531 6.600 2.181
28.103 26.43 1.673 0.689 0.984 7.6111 6.600 2.265
279
and 15. Then, the following different cost functions have been defined F 1
AA
P(i IOI (p.u.)
= Cp~- CO
F 2 = Cp -~ C(~
24-
F 3 = Cp
22-
Note that these functions do not include any specific goal for the SD systems and deal with three different classical optimization problems for the overall test system. The cost functions F 1 and F 2 include optimum reactive power allocation. In particular, the function F 2 differs from F~ because the term c~ does not include the optimal management of the reactive sources at the interconnection buses. The cost function F 3 takes into account only the
2
201816 12 A.M.
4
8
12 N
4
8
12 t'.M.
Figure 8. T o t a l r e a l - g e n e r a t i o n r e q u i r e m e n t for the selected d a i l y l o a d pattern and for d i f f e r e n t cost functions: F , F~ • F2 ..... ; F3 ....
Cp (£/h) 3600-
s/~ ' i ,,/
260(]"
...',,
..,
\ i
',,,',,,
.;:. .f;"
"1 ....... 1600 12 A.M.
~I.
'8
1'2N
~1
I 2 P).l.
'8
l 2 A.M.
Figure 5. T o t a l p r o d u c t i o n c o s t of the o v e r a l l system for t h e selected d a i l y l o a d pattern a n d for d i f f e r e n t cost functions: F ; F~ ; F2 ..... ; F3 ....
4
8
l2 N
4
8
12 I'.M.
Figure 9. T o t a l r e a c t i v e - g e n e r a t i o n r e q u i r e m e n t for the selected d a i l y l o a d pattern and for d i f f e r e n t cost functions: F F~ • F 2 ..... ; F3 ....
pL/ (p.u.)
tat
2.o1
,
t
1.0 12 A.M.
iI
J,
'8
,
,,
%
*~V I
12~
%
/ .t
4
%%
12 I'.M.
8
Figure 6. Real losses of t h e overall system for the selected d a i l y l o a d pattern and for d i f f e r e n t cost functions: F ' F~ • F 2 ..... ; F3 ....
PI.
SD (p.u.) I'" ~ ~.~. ~,; / \ "%,
1.2"
/' /
,:al / /
0.8.
f / / /f
l, /
~,It
4X
"~ ,%
"x~/\~_'",,, \v /
\ ~ \ ' t\ ~\ \
6'/ /
0.0, 12 A.M.
"~'1,\
/..j/
0.4.
4
v..'~,
8
]2 N
4
8
I 2 P.M.
Figure 7. T o t a l real losses of t h e SD systems for the selected d a i l y l o a d pattern and for d i f f e r e n t cost functions: F " F1 • F2 ..... ; F3 ....
280
economic allocation of generation. Under these assumptions, the above defined optimization problems have been solved for the overall test system and the results have been compared with the ones obtained when the cost function F is adopted. Figures 5 and 6 show the total production cost cp and the real losses PL of the overall system, respectively, for the selected daily load condition. These figures indicate that dispatching the real and reactive power flows delivered to the SD systems, produces a remarkable saving of the operation costs and yields a considerable reduction of the total system losses. These results are confirmed by Figure 7, where the total real loss PL~o of the SD system are shown. Analogous conclusions can be derived by Figures 8 and 9 which show the total real and reactive power generations for the overall test system. Finally, it can be of interest to illustrate how the presence in the objective function of specific goals assigned to each SD system can have a remarkable effect on the network voltage profile. To this purpose, the following performance index can be used Jv = 2
PDIAV/2
iE,~D
where J D is the set of load buses of the overall system, PD, the real-power demand and AV~ the voltage magnitude deviation from the nominal value of the ith load bus. Figure 10 shows the values of Jv when the above defined cost functions are used for assessing the optimal operation problem of the power system.
Electrical P o w e r & Energy Systems
12
Jv
13
0.03s m,% t
0.02-
•, o / ,, ; r-':-'-r-'--J , ' `
It~
t S-°
aa i t
~ ....
•
I •
I
14
ss S
.... ~lwl ':;
',;,--
.
""~
15
0.0 | 2 A.M.
16 F i g u r e 10. V o l t a g e p r o f i l e i n d e x f o r t h e s e l e c t e d d a i l y load pattern and for different cost functions: F ; F1 ; F2 . . . . . ; F3 . . . .
17 18
VI. C o n c l u s i o n s In this paper, a generalized formulation of the OPF problem has been proposed for a large power system, consisting of a TG system connected to several SD systems. In addition to the classical objectives of the TG system, such as the minimization of the production costs, the approach has utilized specific goals for the SD systems which have been explicitly modelled with their control actions, inputs and coupling variables with the TG system. In this way a non-linear multiobjective optimization problem has been first stated. Then, to make this large problem more tractable, a Decomposition-Coordination scheme has been developed which uses the Goal Coordination approach. Finally, a non-iterative suboptimal procedure has been suggested which avoids the coordination phase of the Decomposition-Coordination method, with a remarkable simplification of the problem from the computational viewpoint. The validity of the proposed procedure and the accuracy of the results have been illustrated by a numerical example.
VII. R e f e r e n c e s 1
2 3 4 5 6 7 8 9 10 11
Willis, L H, Powell, R W and Tram, H N 'Computerized methods for analysis of impact of demand side management on distribution systems" IEEE Trans. Vol PWRD-2 No 4 (1987) pp 1236-1243 Runnels, J E "Impacts of demand-side management on T and D - now and tomorrow', IEEE Trans, Vol PWRS-2 No 3 (1987) pp 724-729 Yau, T S, Huff, R G and Willis, H L 'Demand-side management impact on the transmission and distribution system" IEEE Trans. Vol PWRS-5 No 2 (1990) pp 506-512 Morgan, M G and Talukdar, S N "Electric power load management: some technical, economic, regulatory and social issues', IEEE Trans. Vol PAS-67 No 2 (1979) pp241-313 Nordell, D E 'Principles for effective load management', IEEE Trans. Vol PAS-104 No 6 (1985) pp 1450-1454 Salehfar, H and Patton, A D 'Modelling and evaluation of the system reliability effects of direct load control' IEEE Trans. Vol PWRS-4 No 3 (1989) pp1024-1030 Warnock, V J and Kirkpatrik, T L 'Impact of voltage reduction on energy and demand: phase I1' IEEE Trans. Vol PWRS-1 No 2 (1986) pp92-97 Civanlar, S, Grainger, J J, Yin, H and Lee, H S S 'Distribution feeder reconfiguration for loss reduction' IEEE Trans. Vol PWRD-3 No 3 (1988) pp 1217-1223 Adibi, i i and Thorne, D K 'Local load shedding' IEEE Trans. Vol PWRS-3 No 3 pp 1220-1229 Sanghvi, A P "Flexible strategies for load/demand management using dynamic pricing" IEEE Trans. Vol PWRS-4 No 1 (1989) pp83-93 Edvinsson, M J 'Load management in Europe - marketing aspects' IEEE Trans. Vol PWRS-1 No 4 (1986) pp83-89
Volume 15 Number 5 1993
19 20 21
22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
Kirkham, H, Nightingale, D and Koerner, T 'Energy management system design with dispersed storage and generation"/EEE Trans. Vol PAS-100 No 7 (1981) pp3432-3441 Kirkham, H and Klein, J 'Dispersed storage and generation impacts on energy management systems' IEEE Trans. Vol PAS-102 No 2 (1983) pp339-345 Fernandes, R A, Bunch, J B, Chestnut, H, Easley, J H, Fiedler, H J and Rushden, F A 'Evaluation of a conceptual distribution automation system' /EEE Trans. Vol PAS-101 No 7 (1982) pp2024-2031 'The distribution system of the year 2000', Report by I EEE Task Group on Long Range Distribution System Design,/EEE Trans. Vol PAS-101 No 8 (1982) pp2485-2490 Lee, R E and Brooks, C L 'A method and its application to evaluate automated distribution control' /EEE Trans. Vol PWRD-3 No 3 (1988) pp1232-1240 Bose, A and Anderson, P M "impact of new energy technologies on generation scheduling" IEEE Trans. Vol PAS-103 No 1 (1983) pp66-71 England. W A and Harrison, L W "Integrating the load management function into the energy control center" /EEE Trans. Vol PAS-104 No 6 (1985) pp1281-1285 Chan, M L and Albujeh, F "Integrating load management into energy management system's normal operations - primary factors' IEEE Trans. Vol PWRS-1 No 4 (1986) pp 152-157 Bhatnagar, R and Rahman, S "Dispatch of direct load control for fuel cost minimization' /EEE Trans. Vol PWRS-1 No 4 (1986) pp 96-102 Vachtsevanos, G J and Kalaitzakis, K C 'A methodology for dynamic utility interactive operation of dispersed storage and generation devices'/EEE Trans. Vol PWRS-2 No 1 (1987) pp 45-51 Sun, D I, Ashley, B, Brewer, B, Hughes, A and Tinney, W F 'Optimal power flow by Newton approach' IEEE Trans. Vol PAS-103 No 10 (1984) pp2864-2875 Lee, K Y, M o h t a d i , M A, Ortiz, J L and Park, Y M 'Optimal operation of large-scale power systems" IEEE Trans. Vol PWRS-3 No 2 (1988) pp413-420 Papalexopoulos, A D, Imparato, C F and Wu, F F 'Large-scale optimal power flow: effects of initialization, decoupling and discretization'/EEE Trans. Vol PWRS-4 No 2 (1989) pp 748-758 Tinney. W F, Bright, J M. Demaree, K D and Hughes, B A "Some deficiencies in optimal power flow'/EEE Trans. Vol PWRS-3 No 2 (1988) pp676~583 Vaahedi, E and Zein-EI-Din, H M 'Considerations in applying optimal power flow to power system operation' IEEE Trans. Vol PWRS-4 Vol No 2 (1989) pp694-703 Singh, M G and Titli, A System: decomposition, optimization and control Pergamon Press, Oxford, USA (1978) Jamshidi, M Large-scale systems, modelling and control Elsevier Science, New York (1983) Bloom, J A 'Long-range generation planning using decomposition and probabilistic simulation" IEEE Trans. Vol PAS-101 No 4 (1982) pp797-802 Mansour, M O and Abdel.Rahman, T M "Nonlinear var optimization using decomposition and coordination" IEEE Trans. Vol PAS-103 No 2 (1984) pp246-255 W a d h w a , C L and Jain, N K 'Multiple objective optimal load flow: a new perspective', IEEProc. C Vol 137 (1990) pp 13-18 Peschon, J, Piercy, D S, Tinney, W F, Tveit, O J and Cuenod, M 'Optimum control of reactive power flow', IEEE Trans. Vol PAS-87 Vol 1 (1968) pp 40-48 D o m m e l , H W and Tinney, W F 'Optimal power flow solutions" IEEE Trans. Vol PAS-87 No 10 (1968) pp18661876 Shen, C M and Laughton, M A 'Determination of optimum power-system operating conditions under constraints" lEE Proc. Vol 116 (1969) pp 225-239 Bergen, A R Power system analysis Prentice-Hall (1986) Elgerd, O J Electric energy systems theory." an introduction McGraw-Hill, New York (1971)
Appendix I AI.1 Derivation of equation (50)
The computational task at the first level is to solve the N + 1 decoupled subproblems (43)-(45) and (46)-(48), for
281
fixed p j ( j = l . . . N ) . This is equivalent to solving the system of equations (29)-(33) and (34)-(38), respectively. At the second level, it is necessary to update pj such that the equation (39) is satisfied. Since pj does not appear explicitly in equation (39), it is necessary to introduce an iterative coordination algorithm for updating it. To this purpose, by differentiating equation (24) the following relationship is obtained 8L t d,~,q-(~L~ t dp (~L~tdw.-~-(~L~tdy.-~-(~)
\Sw)
\~y/ N [-f~L~t
N f~LX~t
+ E/~z./dzj+ j=~\ z j /
\~I~1
L//~..!
j=~L\
+l--ld~j+l
\O.~jJ
//~L"~ t
//¢~LX~ t
dwj"b{~v ,} dyj+ |\ -S-x]j J j/ \ yj/ Id~j÷
\8#j]
\~pfl
dxj
8L dpj = Kj~p~
(56)
j = 1... N
(57)
where Kj is a constant diagonal matrix. Since a solution is required which gives a minimum w.r.t, the variables and a maximum w.r.t, the Lagrange multipliers 27, it is necesssary to assume that the elements of Kj matrix (j = 1 ... N) are positive. Discretizing equation (57) gives AL Apj
Now, when the computational task at the first level is concluded, equation (54) reduces to
282
8L
--=0 j = 1...N ?Pj Then, let us assume
Apj = K j - -
dpj =0 (54)
do=o
or
(58)
j = 1... N
Finally, substituting equation (39) into equation (58) yields Apj = Kj~xj- Djzj)
j = 1... N
(59)
Thus, equation (59) allows the following iterative formula to be introduced p~' +' = p~ + K1(x ~ - Djz~)
j --- 1... N
(60)
Electrical Power & Energy Systems
The level of automation in electric power systems and, particularly, in subtransmission/distribution ( SD) systems is becomin9 hi qher and higher. The continuous surprisin9 increase of the number of control actions should be properly taken into account in optimal power flow ( O PF) programs. This is important, in order to avoid deficiencies and inaccuracies which can degrade the usefulness of the OPF application. In the first part of the paper a 9eneralized formulation of the OPF problem is provided which takes into account the entire set of controls and involves specific goals for the SD systems. This leads to the definition of a non-linear multiobjective optimization problem which can be solved off-line only with a very heavy computational burden. To overcome this difficult)', the second part of the paper deals with the development of an optimization procedure which exploits the naturally decomposable structure of the power system model. Namely, on the basis of a Decomposition-Coordination technique, the proposed method turns the 9lobal optimization problem to the solution of several decoupled subproblems. The validity of the su99ested procedure and the accuracy of the results are illustrated by a numerical example. Keywords." model/in9 oJ" energy systems, optimal power flow, mathematical programmin 9 I. I n t r o d u c t i o n
Over the past years, three main activities have been developed by the electric utilities dealing with planning, Received 15 June 1992; revised 7 December 1992; accepted 25 February 1993
Volume 1 5 Number 5 1 993
design and operation of subtransmission/distribution (SD) systems: (i) development of demand-side management (DSM) strategies for achieving savings in fuel or capacity at the generation level 1 11; (ii) utilization of dispersed and storage generation devices (DSGs), as a managed part of total utility supply system12'13; and (iii) implementation of new powerful control and monitoring functions for improving the operation of the SD system 14 21. Referring to the third activity, it should be pointed out that intensive investigations have been carried out to assess the technical and economic feasibility of automated distribution systems and to integrate these new control actions into the energy control organization of power systems14 iv. Moreover, the impact of these new control actions on the real-time control problem of these systems has also been studied 18 21. As is well known, the recent advancements in Optimal Power Flow (OPF) methods 22 24 have made more realistic the implementation of OPF programs on the Energy Management System for on-line applications. However, there are some aspects which have not been completely exploited 25'26. A critical question concerns the accuracy to be adopted for modelling SD systems and how to take into account the control structures used for these systems. On the other hand, a formulation of the OPF problem which globally includes accurate representations of the SD systems with the entire set of their control actions is clearly unrealistic, owing to the great dimension of the problem and the consequent computational burden.
0142-061 5 / 9 3 / 0 5 2 7 3 - 1 0 •
1993 Butterworth-Heinemann Ltd
273
To overcome this serious drawback, several techniques have been proposed very recently which utilize decomposition coordination concepts 27'2s for solving the generation planning and OPF problems for large-scale systems29 32. These procedures decompose the overall optimization problem into several subproblems which are separately solved. The various solutions are then coordinated according to a centralized policy and the process is repeated until it converges to a global optimum operating point, i.e. the pool objective. These procedures allow a considerable reduction of the computational time, but it should be pointed out that they refer to a unique objective function which often takes into account only specified goals of the TG system. Moreover, the presence of control strategies at the level of SD systems is scarcely considered. On the basis of these observations, in this paper a new procedure for solving the OPF problem is presented which takes into account not only the optimal operation of the TG system but also explicitly includes specific objectives, control actions, technical and economic requirements of the SD systems. Namely, starting with the observation that a large power system consists of several SD networks (HV or MV levels) connected to a TG network (EHV level), a mathematical representation which reflects this naturally decomposable form has been derived for the overall system. In particular, the system is described as an interconnection of subsystems, each one modelled by its steady-state equations. In these equations control variables, inputs and couplings with other subsystems are clearly displayed. Then, a multiobjective non-linear optimization problem for the overall system is formulated by assigning a specified goal, characterized by an appropriate objective function, to each subsystem. Finally, on the basis of an optimization approach which utilizes the Decomposition Coordination method 2~, a new computational procedure is suggested which avoids the time-consuming coordination phase of the general method. Namely, by using realistic assumptions at the decomposition phase, the global optimization problem is solved by using a hierarchical computational scheme characterized by two levels. At the first level, the solution of several decoupled subproblems, associated with each SD system, is determined. At the second level, the optimal operation conditions of the TG system are obtained taking into account the above determined optimality requirements of all SD systems. In this way, the reduction of the computational burden is considerable and, at the same time, the accuracy of the results is satisfactory. Furthermore, the proposed procedure proves suitable to be implemented in Energy Management Systems with a hierarchical and distributed control organization. The capability and usefulness of the method are illustrated using a 275/132 kV 40-busbar test system.
I
transmission/generation system
1
| m,,)~" O(i) m~,)-~b ,~(,)mO,-(~.,,)
Q(J)[ l ' ~
~(J) ......
1
(J)
w':I ' k ' ~
......
Q~I N I j ~
--
subtransmtssion/distribution system
Figure 1. Scheme of interconnection between the TG system and the jth SD system
outputs, the transformer tap setting and the outputs of compensating devices, such as synchronous compensators, capacitors and reactor banks and static var systems. Now, let us consider an interconnected power system composed of N SD networks connected to a TG system. The TG system includes N T nodes: N~ load nodes and N G generating nodes (NT = ND + N~). Thejth SD network includes Nj nodes: Noi load nodes and N ~ generating nodes, which take into account the presence of dispersed storage and generation. The link between the TG network and the jth SD network is established by NIj interconnection buses (j= 1.... N) and N~ transformers or autotransformers, each one represented by an ideal autotransformer with real transformation ratio ~,u) Hqk (k = 1.... Nu) or an ideal phase shifter with a pure angular shift ,~u) V'1k, as shown in Figure 1. In addition, in this figure pu) nu) (k = 1, ' • • Nlj), indicate the real and reactive sk and "z~sk, net power flows, respectively, delivered to the jth SD system through the kth interconnection bus. Finally, it is assumed that a voltage-var control shunt device is available at each interconnection bus and .~x~TktO(J)~,~/"---- l , . . . Nlj ) represents the corresponding reactive power injection. 11.2 System equations and constraints Two general classes of constraints are involved in an OPF problem: the network constraints and the constraints due to the operating limits. The network constraints refer to the voltage, current and power-flow relationships and are usually expressed in terms of real and reactive power supply and demand balance equations. Under the above assumptions, the equations representing the overall interconnected power system can be synthetically written in the following form:
PG-P(O, V,m,¢~,Oi, V , , m , , O , ) = 0
(1)
II. Formulation of the global OPF problem
QG - Q(0, V, m, ~, 01, Vl, mT,~i) = 0
(2)
I1.1 Assumptions
- P D - P ( 0 , V,m, dp,Ol, Vl, ml,~l):0
(3)
- QD -- Q(0, V, m, ~, 01, V~, mr, ~bl) = 0
(4)
For an electric TG system, the optimal power flow problem consists in minimizing the fuel costs, system losses, or any other appropriate objective function by adjusting certain controllable variables and satisfying physical and operating constraints on various control parameters and dependent variables. Usually, the control variables are the generator real and reactive power
274
-- eli{O, V, O,j, VU, m , j ,
~lj, O j,
Vj) = 0
j = l ... N
QTj - - Qb{ O, V, O,j, Vu, mu, Oij, Oj, Vj) = 0
(5)
j = 1... N (6)
PGj-
Pj(OIj, Vii, m~, 0j,
V~) = 0
j = 1... N
(7)
Electrical Power & Energy Systems
QGj -- Qj(OIj, Vu, m~, Oj, Vj) = 0 - Po
-
Vlj, m j, o j,
j = 1... N
= o
Qcj - QDj - Q_.J(Ou,Vii, m j, Oj, Vj) = 0
j = 1... N
j=I...N
(8) (9) (10)
with 01 =
[0tI ,' 0tl ......
v,=
m, = [ m',,,
0 1 t, . . . . .
.....
0~
tions among the control actions. To this purpose, equations (5) and (6) can be restated as
-Psj-P'j(O, r, OU,Vu, mlj,~lj)=O
j=I...N
(11)
QTj -- Q , j - ~(0, V, Oij , rlj , mlj , ~lj) = 0 j = 1... N (12) P,j-- ~'(0,i, V,j, Oj, Vj)=0 j = 1... N (13)
It
.....
..... ml, ..... roll'
where PG, Q6
Nc-dimensional vectors of real and reactive powers at the generating nodes of the TG network, including the slack bus Po, QD No-dimensional vectors of real and reactive consumptions at load nodes of the TG network 0, V Nv-dimensional vectors of phase angles and voltage magnitudes at the nodes of the TG system m, ~ vectors of proper dimension of the tap settings and angular shifts of the transformers of the TG system 0u, Vu Nu-dimensional vectors of the phase angles and voltage magnitudes at the interconnection buses between the TG network and the jth SD network mlj, ~lj Nifdimensional vectors of the tap settings and angular shifts of the transformers linking the jth SD network to the TG system QTj Nu-dimensional vector of the reactive powers supplied by voltage-control shunt devices at the interconnection buses of the jth SD network with the TG system 0i, Vj Nfdimensional vectors of the phase angles and voltage magnitudes at nodes of the jth SD network P~j, Q~j N~fdimensional vectors of real and reactive powers of the generating resources of thejth SD system, including large or intermediate sized dispersed storage and generation devices PDj, QDj NDfdimensional vectors of real and reactive loads of the jth SD network m~ vector of appropriate dimension of the transformer tap settings present in the jth SD network Qc~ ND;dimensional vector of the reactive powers supplied from the compensation devices at each node ofthejth SD network The equations (1~(10) represent the load flow equations of the interconnected power system. Particularly, the vectors P~, Q~, m, gp, QTj, m~ and S u ( j = I . . . N ) summarize the main control variables that affect the operation of the transmission system and, consequently, of the SD systems too. Analogously, the variables P~, Q~, Qc~ and m~ (j= 1... N) are mainly concerned with the operation of each SD system, but their effects have repercussions on the EHV level. Owing to the presence of so many different control actions on the system, it is important that all these control actions be coordinated to gain the maximum degree of efficiency. To reach this objective it is first necessary to individuate the correla-
Volume 15 Number 5 1993
Qs~-Q](0u, Vu, Oj, V~)=0
j= I...N
(14)
with
QTj--r/-l(J) -- L~TI,
/')(J) ~,T2~
/')(J) [)(j) ]t • • • , Y-~Tk~ • • • , - ~ T N I j - I
Psj = [P(J) P(j) P(J) i_ si , s2 ~ ' " " , ~ S k '
Q s.J = [oo) o(J) o (j) I - r . , S l , -Y-,S2 ~ " " " ' I ¢ . , S k , " ' 01j :
p(j) ]t
" " " ' ~SNIjA
o (j) i t
" , If--SNIjJ
[-~(J) ~(j) ~(J) ~(J) -It LVlt ~ vie ' " " " ' 'JIk ~ " ' ' ' VlNlj/
0j=e0?,0?
.....
.....
where ~ ' ) and Q'j<') are Nu-dimensional vectors, the kth element of which is a function representing the net real and reactive power leaving from the kth interconnection bus and determined only by the network flows on the TG-side. Analogously, ~'(.) and ~'(.) represent the same powers as determined by network flows on the SD-side. Finally, since line flows depend only on the difference between the terminal-voltage phase angles, the vectors 01j and 0i appearing in equations (1 3) and (14) are referred to one of the interconnection buses of thejth SD network. Thus, equations (11}-(14) allow P,j and Qsj to be considered, simultaneously, as 'loads', at the EHV level, and 'generated powers', i.e. control variables, at the HV level. This means that the interconnection buses can be included into the set of the buses of the TG system as load buses with a reactive power source for the voltage control. On the other hand, they have to be considered as also present in thejth SD network as generating buses, whose real and reactive powers can be optimized accordingly to a specified goal. In addition to the network constraints, engineering constraints have to be considered, which take into account specified operating and security requirements, such as voltage levels, generator loading, reactive source loading, transformer-tap limits, transmission-line loading. These constraints are usually given in the form of inequalities and can be summarized as h(PG, QG, QT, m, ~b,O, V, ml, ~bl,el, 111)~<0 (1 5)
h~P,¢, Qsj, Pc~j, Qcq, Q~, mr, oj, Vj, ¢~,~,ru) <~0 j= I...N
(16)
where h(.) and hk) are vectors, of appropriate dimensions, of functions representing all the inequality constraints on the TG network and on thejth SD network, respectively.
11.3 Global OPF problem Usually, when solving the classic OPF problem, the objective function represents specific goals for the TG system, such as the cost of real power generation. The aim of this paper is to assess the global OPF problem for a large power system including several SD systems, which are explicitly taken into account. Thus, the following general form can be assumed as objective function V = c(P~, Q~, QT, m, qS,0, V, mT,~bl,01, VI) N
+ ~ Q(P,j, Qsj, P~j, Qcj, Q¢j, m~, Oj, vj, 0 u, vu) (17) j=l
275
where c(.) is the cost criterion for the TG system and cxl.) the objective function of thejth SD system. Now, under the above assumptions, the O P F problem for the overall power system can be formulated as follows: N
minimizec(w,y,z~ ..... z; ..... ZN)+ ~ Cj4%,Yj, X i) (18) j=l
b y 3 3 , 34
given
8L 8w
Ow
8L
-
\Ow)
\Owl
\ Oy)
\~Oy)
+
oy
OL _ c< -]- (Og]t
subject to
O(w,y,p, zl ..... zj ..... zN) = 0
(19)
h(w,y, z~ ..... zj ..... zN) <~0
(20)
Os(wj, xj, yj, pj)=O
(21)
hj(wi, xj, yj)<~O xj = Dszj
j = 1 ... N j = 1 ... N
j = 1... N
(22) (23)
(29)
DZ;
Ozs
(30)
2f_(Oh~ t
\OZsl
p-D}pj=O
(31)
OL 82
OL 8p
- g ( w , y , p , zl ..... Zj..... ZN)=0
(32)
- h ( w , y , zl ..... zj ..... ZN)=O
(33)
with
w = EetG, QtG, QtT, mt, f~t, mtl, ~tl] t P = [P~D, Q ~ I t
0%
y = [0 t, V t, 0',]'
z i = [e'~i' Qtsi, ~j]' Yj = E01u,~, ~ ] '
-
oL
pj = EPbj, Qi)j] t
8xj
L= c(w,y, gl . . . . .
Owj o<
\Ow/
+
Zj . . . . .
ZN)
+ ~,to(w,y,P, Zl ..... Zj..... ZN)
+ pth(w,Y, Zl ..... Zj..... ZN) N + E [Cx(Wj'YJ' X j) + ).~gj~wj, xj, yj, pj)
-
%
(og?I'
,~j+
8xj
/agfi'_
+ I--I
20Xfl
j=I...N
(34)
j=I...N
(35)
\Owfi
wj = [ptj, QtGj' Qtj, m}]t
where D r is a (3NIj x 3NIj) identity matrix and xj is a 3Nu-dimensional vector. The objective functions c(') and cj{.) ( j = l , 2 .... N) are identical to those defined in equation (17), in addition the inequality constraints (20) and (22) correspond to equations (15) and (16), respectively. Moreover equation (19) includes equations (1)-(4) and (11)-(12), thus equation (19) is the steady state equation of the TG network. Equation (21) includes equations (7)-(10) and (13)-(14), thus it synthetically represents the load flow equations of thejth SD network. Finally, equation (23) represents the interactions between the TG system and the SD systems. A general method for solving the multiobjective optimization problem (18)-(23), consists in applying the conditions established by the theorem of Kuhn and Tucker 33'3.. To this purpose, by assuming that the functions c('), O('), h(') and cj(.), O~(') and hi(.) are convex, the following Lagrangian can be associated to the optimization problem (18)-(23):
j=l ...N
\DZj}
•j
/Ohj '
+/--/
20Xfi
[l~j -t- pj
=
0
j=l ...N (36)
8L =gj~wj, xj, yj, pj)=O 8,~j 8L -h(wj, xj, y ) = O 8pj 8L - x s - Diz ~= 0 0pj
j = 1... N
(37)
j = 1... N
(38)
j = 1... N
(39)
It should be observed that, in addition to the traditional problem of determining the optimal operating condition of the TG system, the above considered formulation of the O P F problem involves further conditions to be satisfied with the aim of achieving, at the same time, analogous goals for each SD system. The suggested statement of the O P F problem allows the structure of each involved subnetwork to be preserved. Moreover, the interactions among subsystems are explicitly taken into account. This approach proves useful, as will be shown in the next section, when a DecompositionCoordination procedure is used for solving the problem (18)-(23), because it reduces the mathematical complexity of the problem and the computational effort needed.
j=l
+ p)hj(w~, x~, y j) + p}(x~ - Dsz~)]
(24)
where L 2j and ps (j = 1 ... N) are vectors of appropriate dimensions of Lagrange multipliers which take into account all equality constraints; p and pj (j = 1 ... N) are vectors of appropriate dimensions of Kuhn-Tucker multipliers, for which the following conditions hold (25)
p~>O /zj >/0
j = 1... N
(26)
pth(w,y, z~ ..... zj..... zN) = 0
(27)
~}hs(wj, xj, yj)=O
(28)
j = 1... N
Then, the set of necessary conditions for a minimum are
276
III.
Decomposition-Coordination
procedure
The conditions (29)-(39) require the solution of a system of non-linear equations. This problem is very difficult to tackle in a global way, because its solution requires too much computational time making on-line application impossible. In attempting to give an efficient and practical solution to this problem, in this section an approach is developed on the basis of a Decomposition-Coordination procedure 27'28. In order to apply these methods, two important conditions have to be satisfied. First, it is necessary to have a mathematical representation of the complex system under study in a decomposable form. Namely, the overall system has to be modelled as an interconnection of subsystems, each one described by its steady state equations, coupling with other subsystems
Electrical Power & Energy Systems
~
subject to
Y ~
subtransmission system
W
g(w,y,p, Zl ..... zi .... , ZN)= 0
(44)
h(w,y, Z l , . . . , z j . . . . . ZN)<~ O
(45)
and
!
minimize cj(w~,y~, xj) + p)xj
el Zj
Xj
(46)
subject to subtransmission
gj(wj, xj, yj, pj) = 0
system
~0
h~(%, x j, y~) <~0
P ~
j= 1...N
subtransmission system
Figure 2. Schematic representation of the structure of the considered large-scale power system
and an objective function which takes into account the specific goal to be achieved for this subsystem. Secondly, the objective function of the global optimization problem must have an additively separable form. Now it should be noted that the mathematical description of the optimization problem (18)-(23) allows us to identify S= N + 1 subsystems: the TG system and N SD systems. The couplings among the subsystems are expressed by the equation (23) and are schematically shown in Figure 2. In particular, the vectors w, p, y and wi, pj, y j (j = 1... N) represent the control variables, inputs and outputs for the transmission system and the jth SD system, respectively. The vectors zj (j= 1... N) are the outputs of the 'transmission' subsystem which act as inputs, i.e. as intermediate-control variables x j, to the jth SD subsystem. Under these assumptions, the Goal Coordination Method 27'28 can be applied for solving the global optimization problem (18)-(23). This method involves a two level computation scheme and assumes the Lagrangian multipliers pj (j--1... N) as coordination variables. To this purpose, we can associate, with simple mathematical manipulations in the Lagrangian (24), the term p~rj to the jth SD system, so that equation (24) can be rewritten as N L = Lo + Y Lj (40) j=l
j = 1...N j = 1... U
(47) (48)
The solution of these optimization subproblems constitutes the decomposition level of the two level calculation structure which is employed for solving the global problem (18)-(23). Practically, this is equivalent to solving independently the system of equations (29)-(33) and (34)-(38), with fixed p j = ( j = 1... N). Then, the main task of the coordination process is to update the interaction variables. To this purpose, let w*, y*, z* (j= 1 ... N) and w*, y*, x* ( j = I . . . N ) be the solution of the above subproblems and ~j = x* - DjZ*
(49)
a 3N,idimensional difference-vector. If ej=0, by comparing the objective function (18) and the objective functions (43) and (46), the solution of the global optimization problem (18)-(23) is also achieved. If not, a second level of calculation, i.e. the coordination level, is necessary in order to ensure that constraint (39) is satisfied. At this level, a new set of parameters pj ( j = I . . . N ) can be derived by using the following gradient-type coordinator algorithm (see Appendix) =
+ Kj x - Djz9
j = 1... N
(50)
with Kj>0
(41)
where Kj is a (3N~j× 3N~j) constant diagonal matrix and zj and xj derive from the solution, at the step v, of the decoupled optimization subproblems (43)-(45) and (46)-(48), respectively. The developed Decomposition-Coordination procedure presents some advantages. First, the solution of the overall optimization problem (18)-(23) is reduced to the solution of N + 1 independent optimization subproblems. Secondly, the procedure can be carried out by using parallel or distributed computing architectures. In this case the computational task corresponding to the Decomposition phase can be carried out by a unique parallel computer located in the power system control centre, or by using a suitable distributed computing scheme, with several processors located at the lower levels in the control hierarchy, for example into the regional control centres.
(42)
IV. Solution of the optimization problem by a non-iterative decomposition procedure
Now, if the parameters pj (j= 1..... N) are fixed, the function (40) assumes a 'separable' form. Thus, we can associate the following N + l decoupled optimization subproblems to the previous Lagrangians (41)-(42) N minimize c(w,y, z~ . . . . . z i . . . . . ZN)-- ~. p)Djzj (43) j=l
It should be pointed out that the general solution scheme illustrated in Section III requires, owing to the coordination phase, an iterative process for obtaining the optimal operating condition of the overall interconnected power system. For large power systems this constitutes a heavy task, from the CPU time-consuming viewpoint, even if the above mentioned computing architectures are adopted.
with L o = c(w,y, Zl . . . . . ZN) + ~tg(w,y,P, Zl,... ,Zj . . . . . ZN) N
+ pth(w,Y, zl . . . . . zN)- ~ p)Djzj j=l
+ I~hy(w~, x j, y j) + p~x~
Volume 15 Number 5 1993
277
In order to overcome this drawback, a suboptimal optimization procedure characterized by a two-level computational hierarchy can be suggested. At the first level, the optimal operating condition of each SD system is determined by solving independently the N optimization subproblems defined by equations (46)448) with p~=0, j = 1.... N. At this level, according to the assumptions made in Section II, the interconnection buses, for each SD network, have to be considered as generating buses with a fixed voltage, one of which is assumed as a slack bus. At the second level, the O P F problem (43)-(45) is solved, for the TG system, by assuming p j = 0,j = 1.... N, and considering the interconnection buses as load buses with fixed voltage magnitudes. At this level, the loads of the interconnection buses are the power flows which optimize the operation of the SD systems, as determined at the first level. Note that the suggested computational scheme can be deduced from the general method developed in Section III by using realistic assumptions. Namely, let wj,1 yj1 and x) (] = 1 ... N) be the solution of subproblems (46)-(48) for v = 1 and p) = 0
j= I...N
(51)
Figure 3. One-line diagram of the test system
The O P F problem (43)-(45) can be solved, for the TG system, by using equations (29)-(33) and assuming z) =xJ
j= I...N
(52)
Under the assumptions (51) and (52), equation (50) gives for v=2 Or2 - 0
j=l ...N
(53)
Thus the coordination phase proves unnecessary and the iterative solution scheme of the Decomposition Coordination method can be avoided. It should be noted that the suggested suboptimal procedure gives priority to the optimization of the SD systems. This is important because the variables Psi and Qsj can be properly taken into account. Namely, they have been explicitly considered as control variables. Moreover, it should be noted that, at present, these variables may represent the only available control actions in each SD system. On the other hand, since most of the control actions are concentrated in the TG system, it should not be difficult for the TG system to meet the operating conditions at the interconnection buses resulting from the solution of the decoupled optimization subproblems for the SD systems (see equation (52)). Finally, it is interesting to note that with the suggested suboptimal procedure the previously discussed distributed computing architecture can be still used.
Table 1. Operating conditions of SD-system generators, on 100 MVA base Generator
Real power (p.u.)
Reactive power (p.u.)
GS1 GS 2 GS 3 GS4
0.40 0.50 0.70 0.15
0.10 0.13 0.18 0.06
the generator nodes of the TG system; (ii) the operating conditions shown in Table 1 for the generators of the SD systems; (iii) the real nominal transformation ratio for all the interconnection transformers; (iv) a fixed voltage magnitude of 1.02 p.u. at the interconnection buses 8, 9, 25, 28, 32 and 36. In addition, only the network constraints have been considered for the dependent and control variables. Taking into account the above mentioned assumptions, the following cost function for the overall system is proposed
F=c+cl
V. Numerical example To demonstrate the effectiveness of the suggested approach, a 40-busbar modified version of the sample power system reported in Reference 34 has been used (Figure 3). This power system consists of three 132kV subtransmission networks SD1, S O 2 and SD 3 connected to a 275 kV transmission system through the interconnection buses 8, 9, 25, 28, 32 and 36. In this case there are three generators in the TG system and four generators in the SD systems. The network data are available from the authors. All simulations have been carried out by assuming: (i) a fixed voltage magnitude of 1.05p.u. for
278
with 3 C=Cp-~CQ-~- 2 Coi i-1 3 3
C1:
2 i=1
L'Ps,+ 2 CQ~, i=1
and
ep = ~ (ah + flhPch + 7.P~h)
oo: E
hG.CfG
2
E (eoh-eo,)2/2
hE,~a kE,CC°G
Electrical Power & Energy Systems
Z Q~+ ~_, ~ (QTj-QT,)E/2
cQ=
j~Jl i
i r~li
j~I
+ y' ~, (Qc.-QT)2/2 h~.SPG jeSF li
Cps/= ~
~
BjrPsjPs,
jES'ali rE~li
cos,--
J~,~b
Y
Je'~allr~.~f li
IQs,-Qs)2/2
where AeG is the set of the generating buses of the TG system; 6~, is the set of the interconnection buses between the ith SD system (i= 1,2,3) and the TG system; Cth, flh and 7h, hE'G, are the heat-cost coefficients of the generators of the TG system, the values of which are given in Table 2. The generating sources on the subtransmission systems are considered as dispersed generation, for which no cost curves are assumed. The terms B~ are the loss coefficients of the interconnection buses, considered as generating buses for each SD system 35. Thus, these coefficients are calculated by using only the line data of the SD systems (see Figure 3). The cost function F depends upon both real and reactive system variables and consists of two objectives. The function c represents the two goals to be achieved for the TG system: the minimization of the total operating cost and the optimization of the reactive power delivered by
all the reactive-power sources in the TG system3~. The function cx takes into account the specific goal for each SD system, which consists, in this case of the optimal management of real and reactive flows delivered to the SD systems. In particular, Cps' represents the power losses of the ith SD system aS. Table 3 presents the most significant results when the suggested decomposition procedure is applied for solving the problem of minimizing the function F subject to the network constraints of the overall power system. For purposes of comparison, Table 3 also shows the corresponding results when the same problem is solved by using the Decomposition-Coordination method. The comparison clearly reveals the validity of the suggested procedure. To illustrate better the importance of the presence of a specific goal dealing with the optimal operation of the SD systems, different cost functions have been considered and compared with the cost function F. To this purpose, the previously considered load condition has been assumed as daytime peak load and a typical daily load curve has been assigned to each bus of the test system. Figure 4 shows, for example, the load curves of buses 10
Table 2. Heat-cost coetticients for the generators of the TG system on 100 MVA base (heat cost f = ~ + flPc + ~t~c) Generator
~
fl
?
G1 G2 G3
0 0 0
170 170 85
2 2 2
OoT, 12 A.M.
' 4
8
12 N
4
8
12 P.M.
Figure 4. Daily load curves of buses 10 and 15
Table 3. Comparison of results of the Decomposition-Coordination method and the suggested non-iterative decomposition procedure
Total generation cost cp (£/h) CQ Jr- ZCQi ~,Cpsi ~CQs i CQ "q- ~,CQi -~ ~-dCpsi "~ ~ C Q s
i
Real-generation of TG system Reactive-generation of TG system Real-interchange between the TG System and the SD systems Reactive-interchange between the TG system and the SD systems Total real-generation Total active load Total real line-losses Real line-losses of TG system Real line-losses of SD systems Total reactive-generation Total reactive load Total reactive line-losses
Volume 15 Number 5 1993
DecompositionCoordination method
Suggested decomposition procedure
3611.421 28.832 1.057 4.222 34.111 26.336 4.142 16.220
3618.215 28.737 1.219 4.238 34.194 26.353 4.151 16.223
4.008
4.151
28.086 26.43 1.656 0.701 0.955 7.531 6.600 2.181
28.103 26.43 1.673 0.689 0.984 7.6111 6.600 2.265
279
and 15. Then, the following different cost functions have been defined F 1
AA
P(i IOI (p.u.)
= Cp~- CO
F 2 = Cp -~ C(~
24-
F 3 = Cp
22-
Note that these functions do not include any specific goal for the SD systems and deal with three different classical optimization problems for the overall test system. The cost functions F 1 and F 2 include optimum reactive power allocation. In particular, the function F 2 differs from F~ because the term c~ does not include the optimal management of the reactive sources at the interconnection buses. The cost function F 3 takes into account only the
2
201816 12 A.M.
4
8
12 N
4
8
12 t'.M.
Figure 8. T o t a l r e a l - g e n e r a t i o n r e q u i r e m e n t for the selected d a i l y l o a d pattern and for d i f f e r e n t cost functions: F , F~ • F2 ..... ; F3 ....
Cp (£/h) 3600-
s/~ ' i ,,/
260(]"
...',,
..,
\ i
',,,',,,
.;:. .f;"
"1 ....... 1600 12 A.M.
~I.
'8
1'2N
~1
I 2 P).l.
'8
l 2 A.M.
Figure 5. T o t a l p r o d u c t i o n c o s t of the o v e r a l l system for t h e selected d a i l y l o a d pattern a n d for d i f f e r e n t cost functions: F ; F~ ; F2 ..... ; F3 ....
4
8
l2 N
4
8
12 I'.M.
Figure 9. T o t a l r e a c t i v e - g e n e r a t i o n r e q u i r e m e n t for the selected d a i l y l o a d pattern and for d i f f e r e n t cost functions: F F~ • F 2 ..... ; F3 ....
pL/ (p.u.)
tat
2.o1
,
t
1.0 12 A.M.
iI
J,
'8
,
,,
%
*~V I
12~
%
/ .t
4
%%
12 I'.M.
8
Figure 6. Real losses of t h e overall system for the selected d a i l y l o a d pattern and for d i f f e r e n t cost functions: F ' F~ • F 2 ..... ; F3 ....
PI.
SD (p.u.) I'" ~ ~.~. ~,; / \ "%,
1.2"
/' /
,:al / /
0.8.
f / / /f
l, /
~,It
4X
"~ ,%
"x~/\~_'",,, \v /
\ ~ \ ' t\ ~\ \
6'/ /
0.0, 12 A.M.
"~'1,\
/..j/
0.4.
4
v..'~,
8
]2 N
4
8
I 2 P.M.
Figure 7. T o t a l real losses of t h e SD systems for the selected d a i l y l o a d pattern and for d i f f e r e n t cost functions: F " F1 • F2 ..... ; F3 ....
280
economic allocation of generation. Under these assumptions, the above defined optimization problems have been solved for the overall test system and the results have been compared with the ones obtained when the cost function F is adopted. Figures 5 and 6 show the total production cost cp and the real losses PL of the overall system, respectively, for the selected daily load condition. These figures indicate that dispatching the real and reactive power flows delivered to the SD systems, produces a remarkable saving of the operation costs and yields a considerable reduction of the total system losses. These results are confirmed by Figure 7, where the total real loss PL~o of the SD system are shown. Analogous conclusions can be derived by Figures 8 and 9 which show the total real and reactive power generations for the overall test system. Finally, it can be of interest to illustrate how the presence in the objective function of specific goals assigned to each SD system can have a remarkable effect on the network voltage profile. To this purpose, the following performance index can be used Jv = 2
PDIAV/2
iE,~D
where J D is the set of load buses of the overall system, PD, the real-power demand and AV~ the voltage magnitude deviation from the nominal value of the ith load bus. Figure 10 shows the values of Jv when the above defined cost functions are used for assessing the optimal operation problem of the power system.
Electrical P o w e r & Energy Systems
12
Jv
13
0.03s m,% t
0.02-
•, o / ,, ; r-':-'-r-'--J , ' `
It~
t S-°
aa i t
~ ....
•
I •
I
14
ss S
.... ~lwl ':;
',;,--
.
""~
15
0.0 | 2 A.M.
16 F i g u r e 10. V o l t a g e p r o f i l e i n d e x f o r t h e s e l e c t e d d a i l y load pattern and for different cost functions: F ; F1 ; F2 . . . . . ; F3 . . . .
17 18
VI. C o n c l u s i o n s In this paper, a generalized formulation of the OPF problem has been proposed for a large power system, consisting of a TG system connected to several SD systems. In addition to the classical objectives of the TG system, such as the minimization of the production costs, the approach has utilized specific goals for the SD systems which have been explicitly modelled with their control actions, inputs and coupling variables with the TG system. In this way a non-linear multiobjective optimization problem has been first stated. Then, to make this large problem more tractable, a Decomposition-Coordination scheme has been developed which uses the Goal Coordination approach. Finally, a non-iterative suboptimal procedure has been suggested which avoids the coordination phase of the Decomposition-Coordination method, with a remarkable simplification of the problem from the computational viewpoint. The validity of the proposed procedure and the accuracy of the results have been illustrated by a numerical example.
VII. R e f e r e n c e s 1
2 3 4 5 6 7 8 9 10 11
Willis, L H, Powell, R W and Tram, H N 'Computerized methods for analysis of impact of demand side management on distribution systems" IEEE Trans. Vol PWRD-2 No 4 (1987) pp 1236-1243 Runnels, J E "Impacts of demand-side management on T and D - now and tomorrow', IEEE Trans, Vol PWRS-2 No 3 (1987) pp 724-729 Yau, T S, Huff, R G and Willis, H L 'Demand-side management impact on the transmission and distribution system" IEEE Trans. Vol PWRS-5 No 2 (1990) pp 506-512 Morgan, M G and Talukdar, S N "Electric power load management: some technical, economic, regulatory and social issues', IEEE Trans. Vol PAS-67 No 2 (1979) pp241-313 Nordell, D E 'Principles for effective load management', IEEE Trans. Vol PAS-104 No 6 (1985) pp 1450-1454 Salehfar, H and Patton, A D 'Modelling and evaluation of the system reliability effects of direct load control' IEEE Trans. Vol PWRS-4 No 3 (1989) pp1024-1030 Warnock, V J and Kirkpatrik, T L 'Impact of voltage reduction on energy and demand: phase I1' IEEE Trans. Vol PWRS-1 No 2 (1986) pp92-97 Civanlar, S, Grainger, J J, Yin, H and Lee, H S S 'Distribution feeder reconfiguration for loss reduction' IEEE Trans. Vol PWRD-3 No 3 (1988) pp 1217-1223 Adibi, i i and Thorne, D K 'Local load shedding' IEEE Trans. Vol PWRS-3 No 3 pp 1220-1229 Sanghvi, A P "Flexible strategies for load/demand management using dynamic pricing" IEEE Trans. Vol PWRS-4 No 1 (1989) pp83-93 Edvinsson, M J 'Load management in Europe - marketing aspects' IEEE Trans. Vol PWRS-1 No 4 (1986) pp83-89
Volume 15 Number 5 1993
19 20 21
22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
Kirkham, H, Nightingale, D and Koerner, T 'Energy management system design with dispersed storage and generation"/EEE Trans. Vol PAS-100 No 7 (1981) pp3432-3441 Kirkham, H and Klein, J 'Dispersed storage and generation impacts on energy management systems' IEEE Trans. Vol PAS-102 No 2 (1983) pp339-345 Fernandes, R A, Bunch, J B, Chestnut, H, Easley, J H, Fiedler, H J and Rushden, F A 'Evaluation of a conceptual distribution automation system' /EEE Trans. Vol PAS-101 No 7 (1982) pp2024-2031 'The distribution system of the year 2000', Report by I EEE Task Group on Long Range Distribution System Design,/EEE Trans. Vol PAS-101 No 8 (1982) pp2485-2490 Lee, R E and Brooks, C L 'A method and its application to evaluate automated distribution control' /EEE Trans. Vol PWRD-3 No 3 (1988) pp1232-1240 Bose, A and Anderson, P M "impact of new energy technologies on generation scheduling" IEEE Trans. Vol PAS-103 No 1 (1983) pp66-71 England. W A and Harrison, L W "Integrating the load management function into the energy control center" /EEE Trans. Vol PAS-104 No 6 (1985) pp1281-1285 Chan, M L and Albujeh, F "Integrating load management into energy management system's normal operations - primary factors' IEEE Trans. Vol PWRS-1 No 4 (1986) pp 152-157 Bhatnagar, R and Rahman, S "Dispatch of direct load control for fuel cost minimization' /EEE Trans. Vol PWRS-1 No 4 (1986) pp 96-102 Vachtsevanos, G J and Kalaitzakis, K C 'A methodology for dynamic utility interactive operation of dispersed storage and generation devices'/EEE Trans. Vol PWRS-2 No 1 (1987) pp 45-51 Sun, D I, Ashley, B, Brewer, B, Hughes, A and Tinney, W F 'Optimal power flow by Newton approach' IEEE Trans. Vol PAS-103 No 10 (1984) pp2864-2875 Lee, K Y, M o h t a d i , M A, Ortiz, J L and Park, Y M 'Optimal operation of large-scale power systems" IEEE Trans. Vol PWRS-3 No 2 (1988) pp413-420 Papalexopoulos, A D, Imparato, C F and Wu, F F 'Large-scale optimal power flow: effects of initialization, decoupling and discretization'/EEE Trans. Vol PWRS-4 No 2 (1989) pp 748-758 Tinney. W F, Bright, J M. Demaree, K D and Hughes, B A "Some deficiencies in optimal power flow'/EEE Trans. Vol PWRS-3 No 2 (1988) pp676~583 Vaahedi, E and Zein-EI-Din, H M 'Considerations in applying optimal power flow to power system operation' IEEE Trans. Vol PWRS-4 Vol No 2 (1989) pp694-703 Singh, M G and Titli, A System: decomposition, optimization and control Pergamon Press, Oxford, USA (1978) Jamshidi, M Large-scale systems, modelling and control Elsevier Science, New York (1983) Bloom, J A 'Long-range generation planning using decomposition and probabilistic simulation" IEEE Trans. Vol PAS-101 No 4 (1982) pp797-802 Mansour, M O and Abdel.Rahman, T M "Nonlinear var optimization using decomposition and coordination" IEEE Trans. Vol PAS-103 No 2 (1984) pp246-255 W a d h w a , C L and Jain, N K 'Multiple objective optimal load flow: a new perspective', IEEProc. C Vol 137 (1990) pp 13-18 Peschon, J, Piercy, D S, Tinney, W F, Tveit, O J and Cuenod, M 'Optimum control of reactive power flow', IEEE Trans. Vol PAS-87 Vol 1 (1968) pp 40-48 D o m m e l , H W and Tinney, W F 'Optimal power flow solutions" IEEE Trans. Vol PAS-87 No 10 (1968) pp18661876 Shen, C M and Laughton, M A 'Determination of optimum power-system operating conditions under constraints" lEE Proc. Vol 116 (1969) pp 225-239 Bergen, A R Power system analysis Prentice-Hall (1986) Elgerd, O J Electric energy systems theory." an introduction McGraw-Hill, New York (1971)
Appendix I AI.1 Derivation of equation (50)
The computational task at the first level is to solve the N + 1 decoupled subproblems (43)-(45) and (46)-(48), for
281
fixed p j ( j = l . . . N ) . This is equivalent to solving the system of equations (29)-(33) and (34)-(38), respectively. At the second level, it is necessary to update pj such that the equation (39) is satisfied. Since pj does not appear explicitly in equation (39), it is necessary to introduce an iterative coordination algorithm for updating it. To this purpose, by differentiating equation (24) the following relationship is obtained 8L t d,~,q-(~L~ t dp (~L~tdw.-~-(~L~tdy.-~-(~)
\Sw)
\~y/ N [-f~L~t
N f~LX~t
+ E/~z./dzj+ j=~\ z j /
\~I~1
L//~..!
j=~L\
+l--ld~j+l
\O.~jJ
//~L"~ t
//¢~LX~ t
dwj"b{~v ,} dyj+ |\ -S-x]j J j/ \ yj/ Id~j÷
\8#j]
\~pfl
dxj
8L dpj = Kj~p~
(56)
j = 1... N
(57)
where Kj is a constant diagonal matrix. Since a solution is required which gives a minimum w.r.t, the variables and a maximum w.r.t, the Lagrange multipliers 27, it is necesssary to assume that the elements of Kj matrix (j = 1 ... N) are positive. Discretizing equation (57) gives AL Apj
Now, when the computational task at the first level is concluded, equation (54) reduces to
282
8L
--=0 j = 1...N ?Pj Then, let us assume
Apj = K j - -
dpj =0 (54)
do=o
or
(58)
j = 1... N
Finally, substituting equation (39) into equation (58) yields Apj = Kj~xj- Djzj)
j = 1... N
(59)
Thus, equation (59) allows the following iterative formula to be introduced p~' +' = p~ + K1(x ~ - Djz~)
j --- 1... N
(60)
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