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Centralized Generation Control of Real Power for Thermal Units by a Parametric Linear Programming Procedure
ECONOMIC LOAD DESPATCH
CENTRALIZED GENERATION CONTROL OF REAL POWER FOR THERMAL UNITS BY A PARAMETRIC LINEAR PROGRAMMING PROCEDURE L. Franchi, A. Garzillo, M. Innorta, P. Marannino and V. Marchese ENEL, Ente Nazionale per l'Energia Elettrica, Automatica Research Center, Via Valvassori Peroni 77, 20133 Milano, Italy Abstract. The problem of the centralized control of the active power is here considered as being subdivided into three levels with a hierarchy versus time: the scheduled hourly dispatching running the day before, the advance dispatching (AD), operating on the basis of the short-term load forecasting, the instantaneous economic dispatching (ED) integrated with the load frequency control (L.F.C.). This paper emphasizes AD and in particular the algorithm for its solution. AD modifies the trajectories of the thermal units scheduled the day before, taking into account the on-line load predictions and the constraints on the maximum allowable rate of the output changes of thermal generating units as well as the security of the networtk. Furthermore, the procedure is able to supply the trajectories which restore the security, should the system go into a non-secure or vulnerable status. Outputs of AD are also suitable participation factors to be used by ED. The linearization of both the objective function and the constraints allows the utilization of a parametric linear programming algorithm for determining at each instant the optimal operating point for the system. The system is supposed to evolve into an infinite number of steady state conditions. The procedure proposed for the AD results in a very fast solution of the problem and is, therefore, able to operate in a real time environment in large systems as well. Keywords. Large-scale systems; load dispatching; on-line operation; tion; linear programming; power system control.
optimis~
The computer procedure ARDIS (Innorta, Mara~ nino and Mocenigo, 1975) developed at ENEL is also a powerful tool for the scheduling of the active and reactive power which guarantees the system operating in security and with a desired quality of service.
INTRODUCTION The centralized control of the active power for the thermal units has been partitioned, according to a necessary hierarchy versus time into three subproblems (Quazza, 1976): 1) the hourly dispatching running 24 hours in advance; 2) the advance dispatching (AD) which oper~ tes in the framework of the online control utilizing the output of the on-line load forecasting (f.i. 30 minutes ahead); 3) the instantaneous economic dispatching (ED) which aims at minimizing fuel consumption and balancing the continuously varying demand from consumers In a timescale comparable to the one of the load frequency control.
Although the hourly dispatching programs can supply all the useful information about the optimal operating point in a given hour,they do not indicate how the power system must evolve during the time-interval [0, T] between two different states, while saving secu rity and economy. First of all, the rate of changes in MY] ou! put must be limited to a prescribed value for each generator, depending on the limitations of the boiler and combustion equipment response rates, if they persist for more than a few seconds (Concordia and others, 1966), (Bechert and Kwatny, 1972).
More or less general solutions of the problem of the hourly active and reactive dispatching have been available for many years (Carpentier, 1962), (Dommel and Tinney,1968), (Carpentier, 1972).
These constraints become more and more important as the size of thermal equipment be51
L. Franchi et al.
52
MODEL FOR THE PREDICTIVE SCHEDULING RELATED T?1)ICK-UP PERIODS-PARADIS PROCEDURE
comes large enough to be also used to follow the morning pick-up of the load. Furthermore, from an economical point of view it is not always convenient to join with a straight line the optimal points obtained by the steady-state hourly dispatching programs. Indeed, in the morning pickup for example, it can be convenient to manoeuver the MW output of the less expensive units at the maximum allowable rate and chan ge as late as possible the expensive ones. /
And finally, the conventional hourly dispal ching programs are based on steady-state operating cost characteristic and do not con sider the dynamic costs in passing from one operating state to another. In the last few years the problem of economic dispatching of thermal generating units in supplying a time-varying load has been formulated, taking into account dynamic costs, as a dynamic optimal control problem (Patton, 1972), (Bechert and Chen, 1977). The solutions proposed in (Bechert and Kwal ny, 1972), (Patton, 1972), (Bechert and Ch en , 1977) seem to require a great deal of comp~ ter time for large systems and do not consi der the security constraints on the network. The method here described takes into account the constraints on the maximum allowable ra te of change in MW output of thermal units (in long period evolutions as well as in a shorter security restoration procedure) and the "n" and "n-1" security constraints on lines and transformers. The parametric linear programming procedure utilized here allows us to quickly find the trajectories of the manoeuverable units which economically satisfy a load which slo~ ly varies with time (both for the preventive analysis of the morning pick-up and in the on-line environment of AD). In the proposed solution the costs associated with the act of changing the output of generators are not considered, mainly because no data are available at present on the fun~ tional forms of these dynamic costs, and further, because for slow changes the steady-state costs seem to properly represent the behaviour of the system from an economi cal point of view. Nevertheless, our model can be extended to deal with the dynamic costs even if the re suIting computational difficulties have not been considered as yet.
Given the electrical network with N nodes, the first n of them being of thermal gener~ tion, we want to satisfy the active load (C(t)) that is changing with a linear (or piecewise linear) law during the time interval [0, T] at the least possible fuel cost. Furthermore, it is required to satisfy the network constraints, the minimum and maximum limits both on the MW supplies and on their rate of changes for each generator. The proposed procedure requires the solution of the following optimization problem for each t £[0, T], n min P
i
r
f. (P. ) 1 1
n-1 JO
j
i N
r
+ i
= n+1
a .. (P. - P . ) + 1 1 1J a .. 1J
~C.
1
< J. J
jm
n-1
po
P n
0
r
+
(0)
n
+
i
r
1 1 = 1 b.(P.
- P
0
c
.)
+
1
N
r
+ i P. -.. P
= n+1 <
0
i
P. 1
- -.. v.t
b. 1
f::,
<
P. 1
C. 1
< P. < po. + v.t 1 1 1
i
1 ... n (3)
i
1 ••• n (4)
where: - the n-th node is the active power slack; - the objective function (0) represents the fuel cost which is, for every t, a function of the active power productions; - P. is the active power delibered at the 1 . node 1; - the constrants (1) are the linearizations of the current flows as functions of both the active generation at nodes 1,2, ... n and the active loads (or injection) at the other nodes (n+1 ... N)(2); - a is the set of indices of the line and t~ansformer currents which we control becau se they are above a certain percentage of (1)
(2)
. A' . . P".rametr1c Ctlve Dlspatchlng.
Among the last N-n nodes there are loads and hydro-~enerations which are assumed to change with a known linear law.
53
Centralized generation control their maximum allowable value in the optimal points found for t = 0 and t = T by means of the conventional dispatching pr~ gram ARDIS; - J~ is the value of the current j for t = 0; - J~ is the maximum allowable value for the J . current J. ( two d1fferent values are used for security "n" and "n-1"); - a .. is the sensitivity of current j with 1J . . . .1 ( both loads and respect to the 1nJect10n generations) ; p? is the value of the active power deliv~red at the node i for t = 0 calculated by the optimization program ARDIS; - /:; C. = K. t represents the changes in the lo~d i in the interval [0, t] with slope
MODEL FOR ADVANCE DISPATCHINGPARADISE(l) PROCEDURE Fig. 2 depicts the real-time environment and the main features of the advance dispatching procedure. In this case the evolution of the loads is determined by the on-line load forecasting procedure. Telemeasures (TM) and telesignals (TS) are used in the on-line state estimation program which supplies the network configuration at the time t = o. The operation of the system is economic and secure if in the previous interval [- T , 0] the following conditions has been verified:
K. - c6nstraint (2) represents the linearization of the balance equation with respect to active generations and loads; - b. is the sensitivity (similar to the a .. 1 1J furnished by ARDIS) of the active injection of the slack node with respect to the injection i; - constraints (3) represent the limits on the maximum and minimum deliverable power at node i; constraints (4) take into account the maximum rates of change of the MW output 1n the period of the increase or decrease of the load.
a) AD and ED have been active b) the configuration has remained unchanged. Otherwise the system at the time t = 0 can be far from the optimal point and the security requirements may be unsatisfied. (f.i. the security analysis reveals overloads on lines or transformers which do not cause the breakdown of the system provided that they are eliminated in an appropriate amount of time, about 10-20 minutes). If the security constraints are not fulfilled the first aim of the control is the restoration of a secure state. After that, AD changes the trajectories of thermal units, while pursuing the objective of minimum cost operation.
It will be shown in the following how to use t~e.con:traints (4~p~n or~er to satisfy the
1 < v .. -..----1
l1m1tat10ns v. <
dt Fig. 1 depicts the different modules utilized the day before for the scheduling of the active power.
The matl~ematical formulation of the optimization problem becomes: (1)p arametr1c . Act1ve . D'1spatc h"1ng and Em ergency.
optimal point at t-O and t-T
DATA FOR THE HOURLY DISPATCHING
• Oft line load. forecaatinq, unit cc.a1taent and hydroachedullng
ARDIS
-l\
-V
• Fuel coat. and un! t efficiency
PARADIS
Opt1Ju.l trajectories between
Active power optimization
t-O and t-T with aecurity
with aecurlty and voltage
and rate of ch&n9'e on MW out-
conaUalnta
put conatralnta
-ft-
__1 I1nearlzed network
• Data for and -n-1· aeeurl ty conatrannta
Fig. 1 - The d
1'Il
PARADISE
OPTIMAL MODIFIED TRAJECTORIES
ON-LINE LOAD FORECASTING STATE ESTIMATION
• MODIFIES THE TRAJECTORIES SCHEDULED THE DAY BEFORE
LIHEARIZED MODEL OF THE •
IIE'IWORJ<
RESTORES THE SECURITY IN
MINIMlIM TIME WHEN THE EMERGENCY TESTS
SYSTEM IS IN A NON SECURE OR VULHERABLE S'I'A'!'US
'rS
Fig. 2 - The advance dispatching
PARTICIPATION FACTORS
L. Franchi et aZ.
54
p,ep
J
N E i=n+l
a .. !:J.C. 1 1J n-l E i=l
0
J. +
J
N E i=n+1
~
(0' )
- constraint (6') removes the effects of the penalty term when a feasible point has been reached (ep = 1). LINEARIZED MODELS FOR THE PREDICTI-
pO +
n
(1' )
J.
J
ep (J~ J
n-1 E b.(P. 1 1 i=l
- J.) < 0
~
1
pO _ v. i --:L
t
CHING The solutions of the two problems become ve ry speedy and efficient if the non-linear terms of the objective functions (0) and (0') are approximated by piecewise linear functions.
(2' )
J
The following formulation applies to the AD problem but can be used for the predictive scheduling as well.
p~) + 1
b. !:J.C. 1 1
P. < P.
VE SCHEDULING AND THE ADVANCE DISPAT
a .. (P. - P'?) + 1 1 1J
a .. !:J.C. + 1 1J
N E i=n+l
--:L-
f. (P. ) - a.ep 1 1
n-1 E a .. (P. - P'?) + 1 1 1J i=l
J~ +
P n
for the current j).
N E i=l
min
The non-linear function f.(P.) is approximated by this piecewise lin~ar1function NTi r.(P.)'" f.(P.) + E 1 1 1 ~ h=l
(3' )
(4' )
P.
1
< P. < pO + v. t 1 i 1
where P.
(5' )
-
1
o 0<
(6' )
ep~l
< P
E
h=l
- P*
< P* ih i,h+l
i,h
N1.
The main differences are: the objective function (0') contains the penalty term- a.ep which forces the solution towards the feasible region for the current (a. 1S a large positive constant).
1
With the assumption of concave f.(P.) it is • 1- 1 . guaranted that at the opt1mal p01nt, 1f P. > 0, P. reaches its upper bound for 1r 1S each s < r.
- constrmnts (2') are the linearizations of the current flows that exceed their maximum allowable value J. at the initial point P'? (J~ - J. rep~esents the violation J
~
NT i +
The P.* h = 1... NT. + 1 are the abscissas 1,h . . 1 . . of the vert1ces 1n the chosen approx1mat10n (see Fig. 3), the Y' represent the slopes h of each piece and is the number of the pieces for the function f ..
The structure of this optimization problem is similar to the previous one.
1
= P.
With the previous assumption the AD model requires the solution of the following problem for each t £[0, TJ:
J
f. (P. ) 1
1
F" I
-~ I I
p* i1
p*
p* iNT.
i2
1
Fig. 3
p =p i
P
iNT.+1 1
1
Centralized generation control n
min P.
A.
~h,'f'
(Oil)
E i=l
n-1 E a .. ~J i=l
n-1 P < J - JO + E a .. (p~-P.)ih - j j ~J ~ ~ i=l
N
E a .. K.t ~ ~J i=n+1
it is necessary to update the vector p~ at each t. changing p~ into pti P.(t.) in (5"). ~ ~ i ~ ~ The trajectories obtained by solving the p~ rametric linear programming problem in[O, T] satisfy in each sub-interval [to ,t.] the necessary conditions for the optiJum ~f the following control problem.
( 1")
t.
n NT i -a,
~
NT.
n-1 E
a ..
~J
i=l
55
n-1 ~ JO. - J.)< E a .. (p~-P.)Pih + ,1,( 'f' J J -i=1 ~J ~ ~ h=1
f
min u ih
t.~- 1
NT i
N
E i=n+1
a .. K.t ~J
(2")
~
n-1 E b. ~ i=l
P ih
P
P ih ,::; P* ih + 1
0
P. - v.t ~ ~ ~
N
E
b.K.t
i=n+1 ~ ~
- P*ih
i h
(3")
1... NT.
The models previously described have some structural characteristics which are exploi ted in order to obtain an algorithm which operates quickly enough to be conveniently used in a real time environment.
~
... n
(5" ) ( 6")
This is a linear problem with the right-hand sides depending on the parameter t. Obviously for t = 0 there is only one feasible point po because of the limitations (5"). po is, of course, the optimum for the linear programming problem and so it is possible to immediately use the post-optimization techni ques (right-hand side parametrization) that provide the desired trajectories. In the resulting solution the interval[O, T] turns out to be partioned into a finite set of sub-intervals [to t.], where the MW ~-1, . ~ output of the thermal un~ts changes following a linear law. The time instants t. are characterized by . ~ . changes ~n the slope of the traJectory or by changes in the slope of the cost curve of at least one thermal unit. In order to satisfy the constrants d P. V. ~
~
~
dt
i
=
1 ••• n
THE PARAMETRIC LINEAR PROGRAMMING PROCEDURE FOR THE SOLUTION OF THE PREDICTIVE SCHEDULING AND THE ADVANCE DISPATCHING
(4")
1. .. n,
NT·~ + P. -< p~ + v.t E P ih ~ ~ ~ h=1 i
-
uih
L
h=1
with constraints (1"), (2"), (3"), (4"), ( 6") (see appendix I).
nh
n-1 E b.(P~-P.)+p ~ ~~ --n i=l
o~
~~
First of all, the constraints
o -<
P
P* ih
< P* ih - ih+1
i
1 ... n, 1. .. NT
h
are implicitely regarded by utilizing the well-known upper bound techniques (Dantzig, 1963). In this way the sizes of the matrices involved in the optimization algorithm are greatly reduced. Constraints (5") on the rate of change of M¥/ output are substituted by
NT.
E~
h=1
o~
P
ih
S.
~
+ s.
~
~
0
P. - P. + v. t ~
~
(v. + v.) t ~
( 7)
~
( 8)
~
where s. ~s a slack variable with upper ~ . bound depend~ng on the parameter t. After this the matrix of the constraints pr~ sents the structure shown below in the case of a system with four thermal units. In the illustrative example (fig. 4) the cost curve of each generator is linearized in three pi~ ces. Two constraints on current flows are
L. Franchi et aL.
56
violated and another one is taken under con trol.
01000000 00100000 00010000
o
00001000
&21 8 21
00000100
a 12 a 12 a 12 a 13 ·'3 a 13 0
·'1 ·'1 ."
a 21 a:z:z
b,
Fig.
b,
b2
b;z
b;Z
b]
b]
b] -1
-1
0
G0000001
-1
00000000
4 - Structure of Constraint Matrix for the Sample
8).
P0000010
&:22 &22 &23 &23 8 23 0
&31 a 31 &31 &32 &)2 8 32 a 33 &33 &33 0
b,
Assuming that this basis is also optimal in the interval [t., t. J it is possible to 1 ;1.+1 say that the generators 1, 3 and 4 have to be manoeuvered at the maximum speed (positi ve or negative) since the slack variables sl' s3' s4 relative to (7) are out of the basis at zero or at the maximum value (see
Model
where P. h'
Generator 4 is manoeuvered at the speed (wi thin the allowable maximum and the minimum) re~uired by the gradient of the total load. The current flows are feasible since the v~ riable
where:
et
- I is the identity matrix of order n (number of thermal units)
y F
A
B is a (n, m+l)- matrix, if m is the num A . . ber of flow constra1nts 1n the problem - FT and FA are matrices of dimension (m+l, n) and (m+l, m+l) respectively. Referring to the system with four units, if the set of basic variables contains P , s2' ll P33 , P42 , P21' °1' °2' °3 the above partit10n turns out to be:
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
a
0
a
0
a
0
b
a a a
b
11 21 31 1
A
with:
B
0
F -1
13 23 33 3
0
a
0
a
0
a
-1
b
12 22 32 2
- F B
TA
It will be shown that the different steps of the parametric linear programming algorithm (see appendix 11) can be executed by utilizing and updating at each iteration the (m+l, m+l) matrix F -1 instead of the comple te inverse B- 1 of order m+n+l. The larger the number of control units, the greater is the saving in storage re~uirement and computer time. For example, in a system with 50 control units and 10 contraints on current flows under control it is necessary to work with a matrix of order 11 instead of 61. Briefly, the 5 steps described in aE pendix 11 can be modified according to the followin~ rules: step 0 - for t = 0 the point p~ is the 0E . . .~. t1mal Solut1on, but some 1terat1on may be re~uired in order to obtain the optimal basis. step 1 - particular attention must be paid to the variables s. in (8) which have upper ..1 . bounds chang1ng Wlth the t1me t. The rel~ tion (1) of appendix 11 can be written x (t) = B- l d + B- l rt - B-1 R x (10)
B
R
57
Centralized generation control Where x is the vector of non basic variables an~ R is the matrix of the correspon~ ing activities. If in the relation (10) the above mentioned s. are displayed together with their activi ties R . and if R contains the remaining n;n basic ~~tivities the equation (10) can be rearranged in the following way: -1
-1
xB(t) = B B
-1
d + B rt
r:
R
f),
i E I"
(11 )
(v. + v.) t
si
~
1
From (11) it is easy to see that ln order to relate the basic variables x to the parame. . B ter t lt lS necessary to compute the vector
E
r -
R
si
i 8 the vector
(v. + v.) and then compute 1
~
-1~
Y
B r.
If the vectors Y and r are partitioned respec tively into IY , TA IT andlrT, r IT, withT Y and r containing the first n ~omponents T T ~ .• and Y and r contalnlng the last m+l of Y ~A. . A. . and r lt lS posslble to wrlte: Y T Y A
Cl
8
y
--1
F a
In this step the rules of appendix 11 must also be modified if a slack variable s. is a basic one. More precisely, if s. is the • . . . 1 l-th baslc varlable, the l-th component of (11) can be written as: s. + Y. t 10 1
s. 1
with s. 0 (since at each critical value o 10 . t. p. lS changed lnto P.(t.) in (7) and 1. 1. . th~_s lS equlvalent to s~(t~) = 0). 1
(8 is the set of the indices of the varlables s. which are out of the basis at their maximwi value).
r
(see appendix 11).
r r
1
If Y. > v. + v. + v., the variable s. becomes ~on-b~sic ~t m~imum value and tEe critical value of ~he parameter t is t = 0; if otherwise Y. < v. + v., the variable s. re• . 1 1 ~ 1 malns ln the baS1S for each value of t and consequently the critical value t lS dete~ c mined according to the standard rules descri bed in appendix 11. · Step 3 - The computation of the dual varla bles exploits the partitioning of the basis previously defined. Pursuing considerations similar to those in step 2 it is possible to write: Cl
S
y
T ( 12)
I t yields
A
and then
= (-C T
--1
r Y = Y r + F A T A A SUbstituting the expression of yof (9) in (13) we obtain
--1
( 15)
B + C ) FA A
A
and 'TT
T
( 16)
C - 'TT F TAT -1
For the computation of Y the left and right T hand sides of (12) are multiplied by B. This yields I
B A
Y T
r
FT
FA
Y A
r
Y = T
rT -
B A
T and
The computation of the r-th row of B can be performed in the same way. It is obviously that in this way step 3 can be performed by -l. utilizing only the inverse A · Step 4 - The choice of the entering varlable is made according to the rules of appendix II.
F
--1
A
( 14)
Y A T
• Step 2 - the vector =IY , Y I as computA ed in step 1 is used to o£tain the critical value t of the parameter t and the index r of the $ariable to be remouved from the basis
· Step 5 - The updating of FA only in some cases may require the computation of a SU1table row of 8 defined by (9). Since the columns of BA contain no more than one entry different from zero, this row of S --1 turns out to be a determined row of FA .
L. Franchi et
58
THE ON-LINE ECONOMIC DISPATCHING The knowledge of the trajectories furnished by AD can be utilized in the phase of Automatic Generation Control (AGC). In particular the slopes of the load diagrams of theE. mal units in each interval [to ,t.] can . . 1-1 1 . supply useful 1nformat10n about the partec1pation factors to the on-line economic dispatching. Concerning the interactions between AD and LFC, an appropriate study of the behaviour of the system subjected to the closed loops of the two regulation actions seems to be necessary. For this a considerable effort is required in a preventive simulation of the real time system operations in order to optimize the on-line control.
CONCLUSIONS Models for the predictive active dispatching and for the advance dispatching (AD) which both take into account the rate of change of MW output of thermal units have been prese~ ted. The connections between AD and the on-line economic dispatching have been pointed out. The solution algorithms have been implemented and tested for a large-size network with complexity similar to the Italian 380-220 KV transmission and production network. The test system consists of 300 nodes and 31 thermal plants. Fourteen constraints on line and transformer currents are taken into account. The piecewise linearization of the cost func tions yields 127 variables in the linear pr~ gramming problem. A load pick-up of about two hours has been simulated and the trajectories of the 31 ther mal plants have been obtained in about 15 seconds of the 66-60 Honeywell Computer,staE. ting from a feasible but not optimal point. The lower and upper rate of change of the MW output of the thermal units, in MW per minute, are supposed respectively + 0.025 P , . . -. n w1th P nom1nal power of the un1t. n The required computer time both for the predictive scheduling and for AD depends on the number of the critical values t. in the stu 1 died interval [0, T]. Very quick solutions are obtained in AD due to the shorter interval to be considered,
a~.
especially if the starting point po is optimal and feasible.
REFERENCES.
Quazza, G. (1976). Highlights on technological trends in the on-line optimization ·of power system operation. lEE Conf., London. Carpentier, M.J. (1962). Contribution l'et~ de du dispatching ~conomique. Bull. de la Direction des Etudes et Recherches EDF, Serie B, Aout. Dommel,H.W. and W.F. Tinney (1968). Optimal power flow solutions. IEEE Trans. on PAS, §I, 1866-1876. Carpentier, M.J. (1972). Results and extensions of the methods of differential and total injections. 4° PSCC, Grenoble, paper 2.1/8. Innorta, M., P. Marannino and M. Mocenigo (1975). Active and reactive power sched~ ling with security and voltage constraints 5° PSCC, Cambridge, paper 2.2/11. Concordia, C., F.P. De Mello, L.K. Kirchmayer and R.P. Schultz (1966). Effect of primemover response and governing characteristics on system dynamic performance. Proc. AIDer.Power Conf., 28, 1074-1085. Bechert, T.E. and H.G.~watny (1972). On the optimal dynamic dispatch of real power. IEEE Trans. on PAS, 21, 889-898. Patton, A.D. (1972).Dynamic optimal dispatch of real power for thermal generating units. Ph. Dissertation Texas A&M Univ., College Station, Texas. Bechert, T.E. and N. Chen (1977). Area automatic generation control by multi-pass d~ namic programming. IEEE PES Winter Meeting New York, February. Dantzig, G.B. (1963). Linear programming and extensions. Princeton University Press, Princeton, N.J., 368-384. Bryson, A.E. and Y. Ho (1969). Applied optimal control, Waltham, Mass. Blaisdell Publ.
a
~
Hadley, G. (1962). Linear programming. Addison Wesley.
59
Centralizated generation control
studied interval and satisfy the constraints of the dual problem:
APPENDIX I Characteristics of the traJectories obtained by the solution of the parametric linear problem The solution of the parametric linear programming problem defines K + 1 values of the parameter t (a, t , t '" t = T) that de1 2 [K] . . term1ne K sub1ntervals t. , t. where each . 1. act1ve power P. ( t ) changes1-1accord1ng to a . 1. . l1near law. In th1s append1x, we show that in each interval Et. , t.] the trajectories 1 satisfy the necessai conaitions for the optimum of the following problem P1. t.
y
n
1
J \-1
min u
P.1
E i=1
--0.
V.
1
i-l + P n
n
n H
i=n+l
1
1
b. (P. 1 1 )J~ {
J
i-l
n-l E
i=l
i=l
i-l b. (P. - P. ) + 1 1 1
i=l
+
- J.
*p. (P. --0. 1
E
J
~
a.
i-l A. u. + v*{p - P 1 1 n n N
E
i=n+l
N
E a. f:"C. + 1J 1 i=n+l
n *+ (P. - P. ) + E p. 1 1 1 i=l
P. ) 1
clH
A.
y. + V*b.
(lP.
1
1
1
E * j )J j
1
n E
P. < P. 1 1
*+ + p i
(4)
with A. (t. ) 1
i-l P. 1
. ( 1) f1xed .
In Et. , t.] the P.(t) satisfy the con.1-1.. .. stra1nts (1) ... (4) of Pl and the constraints i-l P. < P + v. (t - t. ) 1 - i 1 1-1
n E
a ..
1J
*p. 1
i=l
i=l
-P. < - P. 1 --0.
+
b.6.C.} 1 1
The functiollSA.(t) are the solution of the . . 1 adJ01nt system
b. 6. C. 1 1
P. (t. ) 1 1-1
1
i-l a .. (P. - p. ) + 1 1 1J
E
J
J
E
i-l P. ) + 1
-
i=l
N
E
n
y. P. +
E
i=l
J.
~ J.
P. +
1
The Hamiltonian funation of Pl is:
n a .. t,C. 1 1J
J
i
n-l
N
i=n+l
1J
+ - ---+ with )J. , p. , p. , p. , P. 1 1 1 1 J
E j
n-1 i-l i-1 ) + + E a .. (P. - p. 1 1 1J J i=l E
- P
i
E
1
J.
P
+
p
i=l
v. -< u. -< 1
u. 1
+ a .. )J. + p.
Vb. + ~ 1 J
1
n-l
y. p. dt 1
y.
1
a.
The necessary conditions of optimality for the problem Pl (Bryson and Ho, 1969) require that, if u(t) is its solution then v*, V~, *+ *-.. J P ., p. eX1st and sat1sfy: 1
1
*+ *1))Jj,Pi,Pi
~a
(5)
2) u.(t) satisfies the contraints 1
v. (t - t.
-1
1-1
)
(6)
+ --+ . Th e dual var1ables )J., v, P., P. ,P. , p.
. J constra1nts 1 . 1 (1) 1) 1 , ( 2, o f th e correspond1ng (3), (4), (5) and (6) are constant in the (1) Without loss of generality it is assumed that each fi(Pi) is approximated by a li near function.
v. < u. < V. -1- 1- 1
i
=1
... n
3) P.(t) satisfy the constraints (1), (2), (3), (4).
4) The complementary conditions are verified: that is, the product of each function * P*+., p. *- for the correspond1ng . V* , )J., const~aint~ is ~ero.
L. Franchi et al.
60
5) u(t) m1n1m1zes the Hamiltonian function computed in P.(t) and A.(t) in the region 1 1 V.
= \),
).1~ J
-
).1.,
J
*+
p.
1
Conditions (2), (3), (4) are verified by the trajectories obtained in the parametric linear programming solution. In order to verify condition (5)let us consider the adjoint system •
L
+
a .. + p. 1J 1
).1.
J
t
=0
with B the basis associated with the opti mal solution. -1
step 1 - Computation of the vector y B r which relates the basic variables x to B the parameter t.
1
A.
(8)
1
Conforming to the duality theory of linear it is possible to say that if p. < 0 the i-th constraint (5) 1S active and tfien u. = v.'
~ogramming
1
In this case we have p. = 0 and A. > O. 1
If y. > 0 for each i the basis B will be o~ timal for t E [0, + 00] and the solution of the problem is -1
1
The boundary conditions A.(t.) = 0 imply A.(t) ~ 0 in et. , t.] a~d ~onsequentely 1 . . . 1-1 1 .. . u = v. m1n1m1zes the Hamllton1an funct1on, i thus f~lfilling requirement (5).
P:
The same result is attained if P. = 0 the function A.(t) is zero in [t~ ,t~] and " 1-1:1 so con d 1't'lon (1). 5 1S ver1f1ed.
-1
xB(t) = B d + B rt
+ yt
t
E
[0, T] (1)
y. <
0
1
> T the problem is solved. cOtherwise, the variable x has to leave the Br basis.
If t
step 3 - Computation of the vector of dual -1 variables TI = C B where C are the costs of basic variab~es. Computat~on of the r-th -1 row of B . step 4 - Choice of the entering variable x K' The index K is computed according to the following rule (2).
APPENDIX I I
sidredepending on a parameter t can be formulated in the following way:
B
In this case the solution in [0, t ] is given by (1) while for t > t B is nof a fea. . c. . slble bas1s and some operat10ns are requ1red for obtaining a further increase of the parameter t.
K
-
C
K
max
j
A linear programming problem with the right~
o
min i
c
TIa
Outline of the parametric linear programming algorithm (Hadley, 1962)
x
step 2 - Computation of the critical value of t where 0fT)or more basic variables x Bi become zero : t
p.
which for (7) becomes
1
-1 -1 d + B B rt
p ..
1
Obviously with this choice the requirement (1) is satisfied.
A. = - y. + \) b. + . 1 1 1 J
step 0 - solution of the linear programming problem for t = 0 which satisfies the con dition
TIa
-
- j
C
j
y .
rJ
y . rJ
<
0
where C.and a. are respectively the cost and theJactiv!ty vector of the variable x.
J
and y K' Y . are the r-th elements of -1 r rJ-1 a .. B a and B K ~1 step 5 - Updating Band x and returning B to step 1.
s.t. Ax=d+rt
t
E[O, T]
x > 0
The solution techniques exploit the features of the dual simplex method. The main steps of the procedure are:
(1 )
When the upper bound techniques are used in the choice of the critical value of t we also must consider the basic variables that reach their maximum value. This rule is slightly modified in order to use the upper bound techniques.
CENTRALIZED GENERATION CONTROL OF REAL POWER FOR THERMAL UNITS BY A PARAMETRIC LINEAR PROGRAMMING PROCEDURE L. Franchi, A. Garzillo, M. Innorta, P. Marannino and V. Marchese ENEL, Ente Nazionale per l'Energia Elettrica, Automatica Research Center, Via Valvassori Peroni 77, 20133 Milano, Italy Abstract. The problem of the centralized control of the active power is here considered as being subdivided into three levels with a hierarchy versus time: the scheduled hourly dispatching running the day before, the advance dispatching (AD), operating on the basis of the short-term load forecasting, the instantaneous economic dispatching (ED) integrated with the load frequency control (L.F.C.). This paper emphasizes AD and in particular the algorithm for its solution. AD modifies the trajectories of the thermal units scheduled the day before, taking into account the on-line load predictions and the constraints on the maximum allowable rate of the output changes of thermal generating units as well as the security of the networtk. Furthermore, the procedure is able to supply the trajectories which restore the security, should the system go into a non-secure or vulnerable status. Outputs of AD are also suitable participation factors to be used by ED. The linearization of both the objective function and the constraints allows the utilization of a parametric linear programming algorithm for determining at each instant the optimal operating point for the system. The system is supposed to evolve into an infinite number of steady state conditions. The procedure proposed for the AD results in a very fast solution of the problem and is, therefore, able to operate in a real time environment in large systems as well. Keywords. Large-scale systems; load dispatching; on-line operation; tion; linear programming; power system control.
optimis~
The computer procedure ARDIS (Innorta, Mara~ nino and Mocenigo, 1975) developed at ENEL is also a powerful tool for the scheduling of the active and reactive power which guarantees the system operating in security and with a desired quality of service.
INTRODUCTION The centralized control of the active power for the thermal units has been partitioned, according to a necessary hierarchy versus time into three subproblems (Quazza, 1976): 1) the hourly dispatching running 24 hours in advance; 2) the advance dispatching (AD) which oper~ tes in the framework of the online control utilizing the output of the on-line load forecasting (f.i. 30 minutes ahead); 3) the instantaneous economic dispatching (ED) which aims at minimizing fuel consumption and balancing the continuously varying demand from consumers In a timescale comparable to the one of the load frequency control.
Although the hourly dispatching programs can supply all the useful information about the optimal operating point in a given hour,they do not indicate how the power system must evolve during the time-interval [0, T] between two different states, while saving secu rity and economy. First of all, the rate of changes in MY] ou! put must be limited to a prescribed value for each generator, depending on the limitations of the boiler and combustion equipment response rates, if they persist for more than a few seconds (Concordia and others, 1966), (Bechert and Kwatny, 1972).
More or less general solutions of the problem of the hourly active and reactive dispatching have been available for many years (Carpentier, 1962), (Dommel and Tinney,1968), (Carpentier, 1972).
These constraints become more and more important as the size of thermal equipment be51
L. Franchi et al.
52
MODEL FOR THE PREDICTIVE SCHEDULING RELATED T?1)ICK-UP PERIODS-PARADIS PROCEDURE
comes large enough to be also used to follow the morning pick-up of the load. Furthermore, from an economical point of view it is not always convenient to join with a straight line the optimal points obtained by the steady-state hourly dispatching programs. Indeed, in the morning pickup for example, it can be convenient to manoeuver the MW output of the less expensive units at the maximum allowable rate and chan ge as late as possible the expensive ones. /
And finally, the conventional hourly dispal ching programs are based on steady-state operating cost characteristic and do not con sider the dynamic costs in passing from one operating state to another. In the last few years the problem of economic dispatching of thermal generating units in supplying a time-varying load has been formulated, taking into account dynamic costs, as a dynamic optimal control problem (Patton, 1972), (Bechert and Chen, 1977). The solutions proposed in (Bechert and Kwal ny, 1972), (Patton, 1972), (Bechert and Ch en , 1977) seem to require a great deal of comp~ ter time for large systems and do not consi der the security constraints on the network. The method here described takes into account the constraints on the maximum allowable ra te of change in MW output of thermal units (in long period evolutions as well as in a shorter security restoration procedure) and the "n" and "n-1" security constraints on lines and transformers. The parametric linear programming procedure utilized here allows us to quickly find the trajectories of the manoeuverable units which economically satisfy a load which slo~ ly varies with time (both for the preventive analysis of the morning pick-up and in the on-line environment of AD). In the proposed solution the costs associated with the act of changing the output of generators are not considered, mainly because no data are available at present on the fun~ tional forms of these dynamic costs, and further, because for slow changes the steady-state costs seem to properly represent the behaviour of the system from an economi cal point of view. Nevertheless, our model can be extended to deal with the dynamic costs even if the re suIting computational difficulties have not been considered as yet.
Given the electrical network with N nodes, the first n of them being of thermal gener~ tion, we want to satisfy the active load (C(t)) that is changing with a linear (or piecewise linear) law during the time interval [0, T] at the least possible fuel cost. Furthermore, it is required to satisfy the network constraints, the minimum and maximum limits both on the MW supplies and on their rate of changes for each generator. The proposed procedure requires the solution of the following optimization problem for each t £[0, T], n min P
i
r
f. (P. ) 1 1
n-1 JO
j
i N
r
+ i
= n+1
a .. (P. - P . ) + 1 1 1J a .. 1J
~C.
1
< J. J
jm
n-1
po
P n
0
r
+
(0)
n
+
i
r
1 1 = 1 b.(P.
- P
0
c
.)
+
1
N
r
+ i P. -.. P
= n+1 <
0
i
P. 1
- -.. v.t
b. 1
f::,
<
P. 1
C. 1
< P. < po. + v.t 1 1 1
i
1 ... n (3)
i
1 ••• n (4)
where: - the n-th node is the active power slack; - the objective function (0) represents the fuel cost which is, for every t, a function of the active power productions; - P. is the active power delibered at the 1 . node 1; - the constrants (1) are the linearizations of the current flows as functions of both the active generation at nodes 1,2, ... n and the active loads (or injection) at the other nodes (n+1 ... N)(2); - a is the set of indices of the line and t~ansformer currents which we control becau se they are above a certain percentage of (1)
(2)
. A' . . P".rametr1c Ctlve Dlspatchlng.
Among the last N-n nodes there are loads and hydro-~enerations which are assumed to change with a known linear law.
53
Centralized generation control their maximum allowable value in the optimal points found for t = 0 and t = T by means of the conventional dispatching pr~ gram ARDIS; - J~ is the value of the current j for t = 0; - J~ is the maximum allowable value for the J . current J. ( two d1fferent values are used for security "n" and "n-1"); - a .. is the sensitivity of current j with 1J . . . .1 ( both loads and respect to the 1nJect10n generations) ; p? is the value of the active power deliv~red at the node i for t = 0 calculated by the optimization program ARDIS; - /:; C. = K. t represents the changes in the lo~d i in the interval [0, t] with slope
MODEL FOR ADVANCE DISPATCHINGPARADISE(l) PROCEDURE Fig. 2 depicts the real-time environment and the main features of the advance dispatching procedure. In this case the evolution of the loads is determined by the on-line load forecasting procedure. Telemeasures (TM) and telesignals (TS) are used in the on-line state estimation program which supplies the network configuration at the time t = o. The operation of the system is economic and secure if in the previous interval [- T , 0] the following conditions has been verified:
K. - c6nstraint (2) represents the linearization of the balance equation with respect to active generations and loads; - b. is the sensitivity (similar to the a .. 1 1J furnished by ARDIS) of the active injection of the slack node with respect to the injection i; - constraints (3) represent the limits on the maximum and minimum deliverable power at node i; constraints (4) take into account the maximum rates of change of the MW output 1n the period of the increase or decrease of the load.
a) AD and ED have been active b) the configuration has remained unchanged. Otherwise the system at the time t = 0 can be far from the optimal point and the security requirements may be unsatisfied. (f.i. the security analysis reveals overloads on lines or transformers which do not cause the breakdown of the system provided that they are eliminated in an appropriate amount of time, about 10-20 minutes). If the security constraints are not fulfilled the first aim of the control is the restoration of a secure state. After that, AD changes the trajectories of thermal units, while pursuing the objective of minimum cost operation.
It will be shown in the following how to use t~e.con:traints (4~p~n or~er to satisfy the
1 < v .. -..----1
l1m1tat10ns v. <
dt Fig. 1 depicts the different modules utilized the day before for the scheduling of the active power.
The matl~ematical formulation of the optimization problem becomes: (1)p arametr1c . Act1ve . D'1spatc h"1ng and Em ergency.
optimal point at t-O and t-T
DATA FOR THE HOURLY DISPATCHING
• Oft line load. forecaatinq, unit cc.a1taent and hydroachedullng
ARDIS
-l\
-V
• Fuel coat. and un! t efficiency
PARADIS
Opt1Ju.l trajectories between
Active power optimization
t-O and t-T with aecurity
with aecurlty and voltage
and rate of ch&n9'e on MW out-
conaUalnta
put conatralnta
-ft-
__1 I1nearlzed network
• Data for and -n-1· aeeurl ty conatrannta
Fig. 1 - The d
1'Il
PARADISE
OPTIMAL MODIFIED TRAJECTORIES
ON-LINE LOAD FORECASTING STATE ESTIMATION
• MODIFIES THE TRAJECTORIES SCHEDULED THE DAY BEFORE
LIHEARIZED MODEL OF THE •
IIE'IWORJ<
RESTORES THE SECURITY IN
MINIMlIM TIME WHEN THE EMERGENCY TESTS
SYSTEM IS IN A NON SECURE OR VULHERABLE S'I'A'!'US
'rS
Fig. 2 - The advance dispatching
PARTICIPATION FACTORS
L. Franchi et aZ.
54
p,ep
J
N E i=n+l
a .. !:J.C. 1 1J n-l E i=l
0
J. +
J
N E i=n+1
~
(0' )
- constraint (6') removes the effects of the penalty term when a feasible point has been reached (ep = 1). LINEARIZED MODELS FOR THE PREDICTI-
pO +
n
(1' )
J.
J
ep (J~ J
n-1 E b.(P. 1 1 i=l
- J.) < 0
~
1
pO _ v. i --:L
t
CHING The solutions of the two problems become ve ry speedy and efficient if the non-linear terms of the objective functions (0) and (0') are approximated by piecewise linear functions.
(2' )
J
The following formulation applies to the AD problem but can be used for the predictive scheduling as well.
p~) + 1
b. !:J.C. 1 1
P. < P.
VE SCHEDULING AND THE ADVANCE DISPAT
a .. (P. - P'?) + 1 1 1J
a .. !:J.C. + 1 1J
N E i=n+l
--:L-
f. (P. ) - a.ep 1 1
n-1 E a .. (P. - P'?) + 1 1 1J i=l
J~ +
P n
for the current j).
N E i=l
min
The non-linear function f.(P.) is approximated by this piecewise lin~ar1function NTi r.(P.)'" f.(P.) + E 1 1 1 ~ h=l
(3' )
(4' )
P.
1
< P. < pO + v. t 1 i 1
where P.
(5' )
-
1
o 0<
(6' )
ep~l
< P
E
h=l
- P*
< P* ih i,h+l
i,h
N1.
The main differences are: the objective function (0') contains the penalty term- a.ep which forces the solution towards the feasible region for the current (a. 1S a large positive constant).
1
With the assumption of concave f.(P.) it is • 1- 1 . guaranted that at the opt1mal p01nt, 1f P. > 0, P. reaches its upper bound for 1r 1S each s < r.
- constrmnts (2') are the linearizations of the current flows that exceed their maximum allowable value J. at the initial point P'? (J~ - J. rep~esents the violation J
~
NT i +
The P.* h = 1... NT. + 1 are the abscissas 1,h . . 1 . . of the vert1ces 1n the chosen approx1mat10n (see Fig. 3), the Y' represent the slopes h of each piece and is the number of the pieces for the function f ..
The structure of this optimization problem is similar to the previous one.
1
= P.
With the previous assumption the AD model requires the solution of the following problem for each t £[0, TJ:
J
f. (P. ) 1
1
F" I
-~ I I
p* i1
p*
p* iNT.
i2
1
Fig. 3
p =p i
P
iNT.+1 1
1
Centralized generation control n
min P.
A.
~h,'f'
(Oil)
E i=l
n-1 E a .. ~J i=l
n-1 P < J - JO + E a .. (p~-P.)ih - j j ~J ~ ~ i=l
N
E a .. K.t ~ ~J i=n+1
it is necessary to update the vector p~ at each t. changing p~ into pti P.(t.) in (5"). ~ ~ i ~ ~ The trajectories obtained by solving the p~ rametric linear programming problem in[O, T] satisfy in each sub-interval [to ,t.] the necessary conditions for the optiJum ~f the following control problem.
( 1")
t.
n NT i -a,
~
NT.
n-1 E
a ..
~J
i=l
55
n-1 ~ JO. - J.)< E a .. (p~-P.)Pih + ,1,( 'f' J J -i=1 ~J ~ ~ h=1
f
min u ih
t.~- 1
NT i
N
E i=n+1
a .. K.t ~J
(2")
~
n-1 E b. ~ i=l
P ih
P
P ih ,::; P* ih + 1
0
P. - v.t ~ ~ ~
N
E
b.K.t
i=n+1 ~ ~
- P*ih
i h
(3")
1... NT.
The models previously described have some structural characteristics which are exploi ted in order to obtain an algorithm which operates quickly enough to be conveniently used in a real time environment.
~
... n
(5" ) ( 6")
This is a linear problem with the right-hand sides depending on the parameter t. Obviously for t = 0 there is only one feasible point po because of the limitations (5"). po is, of course, the optimum for the linear programming problem and so it is possible to immediately use the post-optimization techni ques (right-hand side parametrization) that provide the desired trajectories. In the resulting solution the interval[O, T] turns out to be partioned into a finite set of sub-intervals [to t.], where the MW ~-1, . ~ output of the thermal un~ts changes following a linear law. The time instants t. are characterized by . ~ . changes ~n the slope of the traJectory or by changes in the slope of the cost curve of at least one thermal unit. In order to satisfy the constrants d P. V. ~
~
~
dt
i
=
1 ••• n
THE PARAMETRIC LINEAR PROGRAMMING PROCEDURE FOR THE SOLUTION OF THE PREDICTIVE SCHEDULING AND THE ADVANCE DISPATCHING
(4")
1. .. n,
NT·~ + P. -< p~ + v.t E P ih ~ ~ ~ h=1 i
-
uih
L
h=1
with constraints (1"), (2"), (3"), (4"), ( 6") (see appendix I).
nh
n-1 E b.(P~-P.)+p ~ ~~ --n i=l
o~
~~
First of all, the constraints
o -<
P
P* ih
< P* ih - ih+1
i
1 ... n, 1. .. NT
h
are implicitely regarded by utilizing the well-known upper bound techniques (Dantzig, 1963). In this way the sizes of the matrices involved in the optimization algorithm are greatly reduced. Constraints (5") on the rate of change of M¥/ output are substituted by
NT.
E~
h=1
o~
P
ih
S.
~
+ s.
~
~
0
P. - P. + v. t ~
~
(v. + v.) t ~
( 7)
~
( 8)
~
where s. ~s a slack variable with upper ~ . bound depend~ng on the parameter t. After this the matrix of the constraints pr~ sents the structure shown below in the case of a system with four thermal units. In the illustrative example (fig. 4) the cost curve of each generator is linearized in three pi~ ces. Two constraints on current flows are
L. Franchi et aL.
56
violated and another one is taken under con trol.
01000000 00100000 00010000
o
00001000
&21 8 21
00000100
a 12 a 12 a 12 a 13 ·'3 a 13 0
·'1 ·'1 ."
a 21 a:z:z
b,
Fig.
b,
b2
b;z
b;Z
b]
b]
b] -1
-1
0
G0000001
-1
00000000
4 - Structure of Constraint Matrix for the Sample
8).
P0000010
&:22 &22 &23 &23 8 23 0
&31 a 31 &31 &32 &)2 8 32 a 33 &33 &33 0
b,
Assuming that this basis is also optimal in the interval [t., t. J it is possible to 1 ;1.+1 say that the generators 1, 3 and 4 have to be manoeuvered at the maximum speed (positi ve or negative) since the slack variables sl' s3' s4 relative to (7) are out of the basis at zero or at the maximum value (see
Model
where P. h'
Generator 4 is manoeuvered at the speed (wi thin the allowable maximum and the minimum) re~uired by the gradient of the total load. The current flows are feasible since the v~ riable
where:
et
- I is the identity matrix of order n (number of thermal units)
y F
A
B is a (n, m+l)- matrix, if m is the num A . . ber of flow constra1nts 1n the problem - FT and FA are matrices of dimension (m+l, n) and (m+l, m+l) respectively. Referring to the system with four units, if the set of basic variables contains P , s2' ll P33 , P42 , P21' °1' °2' °3 the above partit10n turns out to be:
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
a
0
a
0
a
0
b
a a a
b
11 21 31 1
A
with:
B
0
F -1
13 23 33 3
0
a
0
a
0
a
-1
b
12 22 32 2
- F B
TA
It will be shown that the different steps of the parametric linear programming algorithm (see appendix 11) can be executed by utilizing and updating at each iteration the (m+l, m+l) matrix F -1 instead of the comple te inverse B- 1 of order m+n+l. The larger the number of control units, the greater is the saving in storage re~uirement and computer time. For example, in a system with 50 control units and 10 contraints on current flows under control it is necessary to work with a matrix of order 11 instead of 61. Briefly, the 5 steps described in aE pendix 11 can be modified according to the followin~ rules: step 0 - for t = 0 the point p~ is the 0E . . .~. t1mal Solut1on, but some 1terat1on may be re~uired in order to obtain the optimal basis. step 1 - particular attention must be paid to the variables s. in (8) which have upper ..1 . bounds chang1ng Wlth the t1me t. The rel~ tion (1) of appendix 11 can be written x (t) = B- l d + B- l rt - B-1 R x (10)
B
R
57
Centralized generation control Where x is the vector of non basic variables an~ R is the matrix of the correspon~ ing activities. If in the relation (10) the above mentioned s. are displayed together with their activi ties R . and if R contains the remaining n;n basic ~~tivities the equation (10) can be rearranged in the following way: -1
-1
xB(t) = B B
-1
d + B rt
r:
R
f),
i E I"
(11 )
(v. + v.) t
si
~
1
From (11) it is easy to see that ln order to relate the basic variables x to the parame. . B ter t lt lS necessary to compute the vector
E
r -
R
si
i 8 the vector
(v. + v.) and then compute 1
~
-1~
Y
B r.
If the vectors Y and r are partitioned respec tively into IY , TA IT andlrT, r IT, withT Y and r containing the first n ~omponents T T ~ .• and Y and r contalnlng the last m+l of Y ~A. . A. . and r lt lS posslble to wrlte: Y T Y A
Cl
8
y
--1
F a
In this step the rules of appendix 11 must also be modified if a slack variable s. is a basic one. More precisely, if s. is the • . . . 1 l-th baslc varlable, the l-th component of (11) can be written as: s. + Y. t 10 1
s. 1
with s. 0 (since at each critical value o 10 . t. p. lS changed lnto P.(t.) in (7) and 1. 1. . th~_s lS equlvalent to s~(t~) = 0). 1
(8 is the set of the indices of the varlables s. which are out of the basis at their maximwi value).
r
(see appendix 11).
r r
1
If Y. > v. + v. + v., the variable s. becomes ~on-b~sic ~t m~imum value and tEe critical value of ~he parameter t is t = 0; if otherwise Y. < v. + v., the variable s. re• . 1 1 ~ 1 malns ln the baS1S for each value of t and consequently the critical value t lS dete~ c mined according to the standard rules descri bed in appendix 11. · Step 3 - The computation of the dual varla bles exploits the partitioning of the basis previously defined. Pursuing considerations similar to those in step 2 it is possible to write: Cl
S
y
T ( 12)
I t yields
A
and then
= (-C T
--1
r Y = Y r + F A T A A SUbstituting the expression of yof (9) in (13) we obtain
--1
( 15)
B + C ) FA A
A
and 'TT
T
( 16)
C - 'TT F TAT -1
For the computation of Y the left and right T hand sides of (12) are multiplied by B. This yields I
B A
Y T
r
FT
FA
Y A
r
Y = T
rT -
B A
T and
The computation of the r-th row of B can be performed in the same way. It is obviously that in this way step 3 can be performed by -l. utilizing only the inverse A · Step 4 - The choice of the entering varlable is made according to the rules of appendix II.
F
--1
A
( 14)
Y A T
• Step 2 - the vector =IY , Y I as computA ed in step 1 is used to o£tain the critical value t of the parameter t and the index r of the $ariable to be remouved from the basis
· Step 5 - The updating of FA only in some cases may require the computation of a SU1table row of 8 defined by (9). Since the columns of BA contain no more than one entry different from zero, this row of S --1 turns out to be a determined row of FA .
L. Franchi et
58
THE ON-LINE ECONOMIC DISPATCHING The knowledge of the trajectories furnished by AD can be utilized in the phase of Automatic Generation Control (AGC). In particular the slopes of the load diagrams of theE. mal units in each interval [to ,t.] can . . 1-1 1 . supply useful 1nformat10n about the partec1pation factors to the on-line economic dispatching. Concerning the interactions between AD and LFC, an appropriate study of the behaviour of the system subjected to the closed loops of the two regulation actions seems to be necessary. For this a considerable effort is required in a preventive simulation of the real time system operations in order to optimize the on-line control.
CONCLUSIONS Models for the predictive active dispatching and for the advance dispatching (AD) which both take into account the rate of change of MW output of thermal units have been prese~ ted. The connections between AD and the on-line economic dispatching have been pointed out. The solution algorithms have been implemented and tested for a large-size network with complexity similar to the Italian 380-220 KV transmission and production network. The test system consists of 300 nodes and 31 thermal plants. Fourteen constraints on line and transformer currents are taken into account. The piecewise linearization of the cost func tions yields 127 variables in the linear pr~ gramming problem. A load pick-up of about two hours has been simulated and the trajectories of the 31 ther mal plants have been obtained in about 15 seconds of the 66-60 Honeywell Computer,staE. ting from a feasible but not optimal point. The lower and upper rate of change of the MW output of the thermal units, in MW per minute, are supposed respectively + 0.025 P , . . -. n w1th P nom1nal power of the un1t. n The required computer time both for the predictive scheduling and for AD depends on the number of the critical values t. in the stu 1 died interval [0, T]. Very quick solutions are obtained in AD due to the shorter interval to be considered,
a~.
especially if the starting point po is optimal and feasible.
REFERENCES.
Quazza, G. (1976). Highlights on technological trends in the on-line optimization ·of power system operation. lEE Conf., London. Carpentier, M.J. (1962). Contribution l'et~ de du dispatching ~conomique. Bull. de la Direction des Etudes et Recherches EDF, Serie B, Aout. Dommel,H.W. and W.F. Tinney (1968). Optimal power flow solutions. IEEE Trans. on PAS, §I, 1866-1876. Carpentier, M.J. (1972). Results and extensions of the methods of differential and total injections. 4° PSCC, Grenoble, paper 2.1/8. Innorta, M., P. Marannino and M. Mocenigo (1975). Active and reactive power sched~ ling with security and voltage constraints 5° PSCC, Cambridge, paper 2.2/11. Concordia, C., F.P. De Mello, L.K. Kirchmayer and R.P. Schultz (1966). Effect of primemover response and governing characteristics on system dynamic performance. Proc. AIDer.Power Conf., 28, 1074-1085. Bechert, T.E. and H.G.~watny (1972). On the optimal dynamic dispatch of real power. IEEE Trans. on PAS, 21, 889-898. Patton, A.D. (1972).Dynamic optimal dispatch of real power for thermal generating units. Ph. Dissertation Texas A&M Univ., College Station, Texas. Bechert, T.E. and N. Chen (1977). Area automatic generation control by multi-pass d~ namic programming. IEEE PES Winter Meeting New York, February. Dantzig, G.B. (1963). Linear programming and extensions. Princeton University Press, Princeton, N.J., 368-384. Bryson, A.E. and Y. Ho (1969). Applied optimal control, Waltham, Mass. Blaisdell Publ.
a
~
Hadley, G. (1962). Linear programming. Addison Wesley.
59
Centralizated generation control
studied interval and satisfy the constraints of the dual problem:
APPENDIX I Characteristics of the traJectories obtained by the solution of the parametric linear problem The solution of the parametric linear programming problem defines K + 1 values of the parameter t (a, t , t '" t = T) that de1 2 [K] . . term1ne K sub1ntervals t. , t. where each . 1. act1ve power P. ( t ) changes1-1accord1ng to a . 1. . l1near law. In th1s append1x, we show that in each interval Et. , t.] the trajectories 1 satisfy the necessai conaitions for the optimum of the following problem P1. t.
y
n
1
J \-1
min u
P.1
E i=1
--0.
V.
1
i-l + P n
n
n H
i=n+l
1
1
b. (P. 1 1 )J~ {
J
i-l
n-l E
i=l
i=l
i-l b. (P. - P. ) + 1 1 1
i=l
+
- J.
*p. (P. --0. 1
E
J
~
a.
i-l A. u. + v*{p - P 1 1 n n N
E
i=n+l
N
E a. f:"C. + 1J 1 i=n+l
n *+ (P. - P. ) + E p. 1 1 1 i=l
P. ) 1
clH
A.
y. + V*b.
(lP.
1
1
1
E * j )J j
1
n E
P. < P. 1 1
*+ + p i
(4)
with A. (t. ) 1
i-l P. 1
. ( 1) f1xed .
In Et. , t.] the P.(t) satisfy the con.1-1.. .. stra1nts (1) ... (4) of Pl and the constraints i-l P. < P + v. (t - t. ) 1 - i 1 1-1
n E
a ..
1J
*p. 1
i=l
i=l
-P. < - P. 1 --0.
+
b.6.C.} 1 1
The functiollSA.(t) are the solution of the . . 1 adJ01nt system
b. 6. C. 1 1
P. (t. ) 1 1-1
1
i-l a .. (P. - p. ) + 1 1 1J
E
J
J
E
i-l P. ) + 1
-
i=l
N
E
n
y. P. +
E
i=l
J.
~ J.
P. +
1
The Hamiltonian funation of Pl is:
n a .. t,C. 1 1J
J
i
n-l
N
i=n+l
1J
+ - ---+ with )J. , p. , p. , p. , P. 1 1 1 1 J
E j
n-1 i-l i-1 ) + + E a .. (P. - p. 1 1 1J J i=l E
- P
i
E
1
J.
P
+
p
i=l
v. -< u. -< 1
u. 1
+ a .. )J. + p.
Vb. + ~ 1 J
1
n-l
y. p. dt 1
y.
1
a.
The necessary conditions of optimality for the problem Pl (Bryson and Ho, 1969) require that, if u(t) is its solution then v*, V~, *+ *-.. J P ., p. eX1st and sat1sfy: 1
1
*+ *1))Jj,Pi,Pi
~a
(5)
2) u.(t) satisfies the contraints 1
v. (t - t.
-1
1-1
)
(6)
+ --+ . Th e dual var1ables )J., v, P., P. ,P. , p.
. J constra1nts 1 . 1 (1) 1) 1 , ( 2, o f th e correspond1ng (3), (4), (5) and (6) are constant in the (1) Without loss of generality it is assumed that each fi(Pi) is approximated by a li near function.
v. < u. < V. -1- 1- 1
i
=1
... n
3) P.(t) satisfy the constraints (1), (2), (3), (4).
4) The complementary conditions are verified: that is, the product of each function * P*+., p. *- for the correspond1ng . V* , )J., const~aint~ is ~ero.
L. Franchi et al.
60
5) u(t) m1n1m1zes the Hamiltonian function computed in P.(t) and A.(t) in the region 1 1 V.
= \),
).1~ J
-
).1.,
J
*+
p.
1
Conditions (2), (3), (4) are verified by the trajectories obtained in the parametric linear programming solution. In order to verify condition (5)let us consider the adjoint system •
L
+
a .. + p. 1J 1
).1.
J
t
=0
with B the basis associated with the opti mal solution. -1
step 1 - Computation of the vector y B r which relates the basic variables x to B the parameter t.
1
A.
(8)
1
Conforming to the duality theory of linear it is possible to say that if p. < 0 the i-th constraint (5) 1S active and tfien u. = v.'
~ogramming
1
In this case we have p. = 0 and A. > O. 1
If y. > 0 for each i the basis B will be o~ timal for t E [0, + 00] and the solution of the problem is -1
1
The boundary conditions A.(t.) = 0 imply A.(t) ~ 0 in et. , t.] a~d ~onsequentely 1 . . . 1-1 1 .. . u = v. m1n1m1zes the Hamllton1an funct1on, i thus f~lfilling requirement (5).
P:
The same result is attained if P. = 0 the function A.(t) is zero in [t~ ,t~] and " 1-1:1 so con d 1't'lon (1). 5 1S ver1f1ed.
-1
xB(t) = B d + B rt
+ yt
t
E
[0, T] (1)
y. <
0
1
> T the problem is solved. cOtherwise, the variable x has to leave the Br basis.
If t
step 3 - Computation of the vector of dual -1 variables TI = C B where C are the costs of basic variab~es. Computat~on of the r-th -1 row of B . step 4 - Choice of the entering variable x K' The index K is computed according to the following rule (2).
APPENDIX I I
sidredepending on a parameter t can be formulated in the following way:
B
In this case the solution in [0, t ] is given by (1) while for t > t B is nof a fea. . c. . slble bas1s and some operat10ns are requ1red for obtaining a further increase of the parameter t.
K
-
C
K
max
j
A linear programming problem with the right~
o
min i
c
TIa
Outline of the parametric linear programming algorithm (Hadley, 1962)
x
step 2 - Computation of the critical value of t where 0fT)or more basic variables x Bi become zero : t
p.
which for (7) becomes
1
-1 -1 d + B B rt
p ..
1
Obviously with this choice the requirement (1) is satisfied.
A. = - y. + \) b. + . 1 1 1 J
step 0 - solution of the linear programming problem for t = 0 which satisfies the con dition
TIa
-
- j
C
j
y .
rJ
y . rJ
<
0
where C.and a. are respectively the cost and theJactiv!ty vector of the variable x.
J
and y K' Y . are the r-th elements of -1 r rJ-1 a .. B a and B K ~1 step 5 - Updating Band x and returning B to step 1.
s.t. Ax=d+rt
t
E[O, T]
x > 0
The solution techniques exploit the features of the dual simplex method. The main steps of the procedure are:
(1 )
When the upper bound techniques are used in the choice of the critical value of t we also must consider the basic variables that reach their maximum value. This rule is slightly modified in order to use the upper bound techniques.