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Dynamic Optimization in Large Scale Power Systems
• DYNAMIC OPTIMIZATION IN LARGE SCALE POWER SYSTEMS
P.P.J. van den Bosch Laboratory for Control Engineering, Delft University of Technology, Delft, The Netherlands G. Honderd ' LabQratory for Control 'Engineering, Delft University of Technology, Delft, The Netherlands D.N. Vlugt Provinciale Zeeuwse Energie Maatschappij(PZEH}, Middelburg, The Netherlands
Abstract. Two topics in the dyn~ic optimization in large scale power systems with thermal units are discussed. First errors are considered in' the data with which the optimal load distribution and unit commitment arecalculated. Moreover their influence to the static and ftyDamic optimization problem is discussed. Second the possibilities to calculate an optimal production policy in.large scale power systems are considered.' It turns out that no method is able to determine in an acceptable amOunt of time the optimal unit commitment. Therefore a hierarchical approach is proposed to solve this optimization problem. ' Keyword,. Dynamic programming; hierarchical systems; optimization; power systems; error analysis. INTRODUCTION In The Netherlands ten independent companies provide for the production of electrical energy. In 1978 about 56.000 GWh. was delivered with a maximum demand of about' 9.500 MW. Except for two nuclear power plants, all production units are thermsl power plants fired with gas, oil and/or coal. The size of the units is between 20 and 630 MW net. Among these companies there is a cooperation within the SEP, i.e. the Coordinating Electricity Production Board for the long term planning (size and location,of new units), the power-frequency control, the common tie lines and the selling or buying of surplus or lack of power etc. However, due to their independence, there is up to now, no overall optimal production policy. In recent years an attempt has been made by the SEP to start and incorporate pr6duction,optimization on a national level, static as well as dynamic. This heuristic method will be considered later. in more detail. For smaller regions, Comprlslng about three companies, there is sometimes a cooperation to obtain an optimal production policy for all companies together. In general each company itself uses some kind of static optimization (load distribution) for its own units. In addition dynamic optimization
(unit commitment) is sometimes used to reduce the fuel costs. In close cooperation between the PZEM and the Laboratory for Control 'Engineering a feasibility study has been carried out to assess the possibilities of determining the optimal dynamic production policy for a period of, for example, one or two days for all companies in The Netherlands, or at least for some companies together. Besides the' profits that could be obtained by cooperation, some other factors have to be taken into account, e.g. ,the possibilities of calculatin~ the opti~l production policy, the influence of inaccuracies in the data that are the basis of the optimization, the limited amount of data that can be taken into account by a centralized optimization, the acceptance of the production policy by the local operators, possibility for accounting etc. Some aspects of this study will be discussed in this paper. This study is based on experiences in economic optimization projects as realized between the Laboratory for Control Engineering at the Delft University of Technology and several power plants in The Netherlands over a period of more than 13 years. In this paper, the mathematical 97
possibilities of determining a real optimum for the dynamic optimization problem and the influence of inaccuracies in the coefficients of the functions that describe the , fuel consumption are considered ~pecially. Most of the calculations concern the power' plants of the PZEM in Zeeland (The Netherlands), although the conclusions have a , much broader validity. '
which some quantities like flows, temperatures etc. are measured. When we
an~lyze
these factors for a thermal unit fired with gas and assume that the measurement errors can be described by a normal. distribution, then a standard deviation 0Q of about 0.6 % for Q. is obtained. In case oil is used, the J
flow measurement is somewhat more accurate,
2. ACCURACY CONSIDERATIONS
and the standard deviation for Q. then be-
Load Distribution, In general,. each thermal unit is described by some technical restriction (lower and upper bound) and by a function that relates the thermal energy Q of the fuel with the
J
comes about 0.5 % Ill.
Except for valve points, it is reasonable to assume that f will be a continuous function of P. So f is chosen as a polynomical of order n. With the aid of the least-square
net electrical power P. This function f {Q - f(P)} is generally valid only when there is one type of fuel and only when the unit is in steady state. The production cost Fp(P) is calculated out of f and the fuel price.
method the minimal error E m ) , 2 E (f(P.)-Q.) m j~) J J
--
m .1: If(~j)~Qjl/Qj ER- .! m J-) can be calculated as a function
~f
the number
of measurements m and the order n of the This function f is determined from some
polynomial f. In Tabel I the relative error
measurements. For several values P. of P,
ER is shown as it is calculated for a number'
which are fixed during an interval (.:!:. 2.-hrs).
of·thermal power plants.
J
the amount of fuel from which the Q. 's are ,
J
calculated is measured. Each pair (Pj,Qj)'
TABLE) Relative Error as a Function of.the Order of the Polynomial n and the Number of Measurements m.
is depicted as one point to describe the function f (Fig. I).
~
f
.4
25
7
%
%
Q
0
call hrJ
I
.7 %
1.0 %
2
.2 %
.5 %
%
.2 %
3
0
33
When we consider the staBdard deviation of· the accuracy with which Q is known, a choice for n larger than 2 is not meaningful, while a value for n less than 2 loses too much information. Consequently, the functicin f
P [MW] Fig. .1
Measur~ents
will be assumed to be of second order: 2 Q - f(P) • aP + bP + c In The 'Netherlands this second
(P., Q.) to determine J
J
Q ~ f(P). The accuracy with which P. and Q. are meaJ
or~er
poly-
nomial is nearly almost used to calculate J
sured depends on a number of technical factors, especially on the accuracy with
.
the fuel consumption as a function of the power. Once
~he
coefficients ai' b
i
and c
i
of unit i are known, the fuel cost F .
p1
98
follows by multiplying Q. with the price of
If we assume that these deviations belong to
the fuel. Summing the fuel costs of all n
a normal distribution, the standard devia-
1
units in operation gives the overall costs
tions for ai' b i , c i ' Popt and F can be In Table 2 averages of these
calculated.
F : p
F
deviations are shown. p
In order to obtain comparable results all
The propagation of the errors in (P.,Q.) to
costs are computed with the same cost
the coefficients a., b. and c., to·the op-
function F • Although these numbers depend
timal load distribution P and to the opt overall costs Fare analyzed in the fol-
on the units, the size of the units, the
J
. 1 1
J
p
1
.
number of units n, the fuels etc., these
p
lowing way. A simulation study has been
factors still have no major influence.
carried out. Starting with the measurements
However, there is one factor that has a
(P.,Q.) with standard deviation 0 J
J
p
• 0.3 %
dominant influence, especially on the
and 0Q • 0.6 %, we calculate each time the
deviation of the costs F • This factor is
optimal load distribution P t and F . Due op P to deviations in (P. ,Q.) of 'coefficients
the total demand D for electricity. This
J
J
p
dependence is illustrated in Fig. 2.
.
a., b.1 and c., Pop t and Fp have also some 1 1 deviations. TABLE 2 Propagation of Measurement Errors
measurements
o
coefficients of f
c1 • 50
% ; ob
load P
°P •
15
%
of
0.2 to 0.8 % ,
Costs F
P
P
• 0.3 % ; 0
a
.
P
0
'" 0.6 %
.
12
% ;
°c
• 25 %
t
f
,, , ,
• .5
I
I
,
I
I
I
I
I
i ,,, ; I
I
[~.
D
Fig. 2 Dependency of of p
_
II
of the demand D.
IT,,",
'L
3·
,
I I
'T", T.... I(.,
k
[hourS]
Fig. 4
1>
, I
p
~
t
Demand D for p periods in the k time interval.
In I the demand equais the sum of all lower
because its assumption that the cooling of
bounds of the units, while in 11 the demands
a unit can be described by means of a first
equals the sum of all upper bounds. In these
order system, is given by:
-th: i
points there ,is no freedom for the optimi-
F
zatiun to determine another optimum. So
S. 1
(t)
s
A. + B. (I-e 1
)
,1
errors in (P.,Q.) will 'then, not influence
There may be discussion about the validity
J
J
the load distribution. If there is more
of this model. In some studies another model
freedom, as is the case for a relatively
calculates F
low demand D, the errors
function of
OF
in~(P.,Q.)
J
J
increase
P
s.
(t) from a piece-wize linear the time t
1
121.
In general,
the values of the coefficients Ai' B and i are based on rules of thumb, in which Ai' i
Throughout the paper of will be assumed to , P be 0.5 %.
T
From Table' 2 it can be concluded that in the
unit., However, measurements that have been
neighbourhood of the optimum
th~
function
Fp is relatively flat in its variabl~s Pi' So
~
small deviation of the set points for
the power of the units from their optimal values will
generall~ not
very much. Still,
s~me
increase the costs
profits can be ob-
B.1 and
T.
1
are functions of the size of the
obtained for several thermal units show no nice fit With such a model. Therefore the accuracy with which the start-up costs can be
cal~ulated
is rather poor. Relating some
measurements of F
with the coefficients s. used in optimizatiAn procedures show errors
tained by mea~s of static optimization: An
of up to 50 %.
estimate of these profits, co~pared with the
Moreover, coefficients A. and B. are often
situation in which human operators determine
chosen to satisfy another requirement;
1
1
the load distribution, is about 1,% to 2 %.
Instead of only determining the extra amount
Of course this may be a considerable amount
of fuel, Ai is often used to avoid too many start/stops. This goal can be achieved by'
of money.,
making A. and B. larger than necessary to ,11
calculate the extra fuel costs. In section Unit CODIDitments
4 a method will be proposed to avoid.this
Besides the function
misuse of F • by the introduction,of the s on/off time 1 in the model.
~,
other information is
necessary to calculate the unit commitment for the optimal production policy for a' period of, for examrle, 24 or 28 hours
ahea~
This information is given by the costs for starting and stopping the production 'units. Many factors
dete~ine
these,costs. Certain-'
ly these costs of taking a unit into operation are,strongly dependent upon the time the unit has been out of operation. The
racies in the coefficients A., B. and
only, with fixed and precisely described procedures for starting up. Without these procedures it is impossible to determine the start-up costs accurately. Still, some models are used to calculate the amount of' fuel as a function of the time t a unit has fre~
quently used, because' of its simplicity and
a
T.
111
simulation study has been performed. Several times, for the same units and power demand D the' optimal production policy has been calculated with the aid of the dynamic programming method. First a reference situation has been solved. Then changes of 50 % in Ai' ,B and
start-up costs can be determined accurately
been out of operation. A model that is
In order to' study the influence of inaccu-
i
have been given, after which the
T
i optimal start/stop policy has been again determined. In order to make all runs comparable the ultimate costs have been calculated with the same, reference, cost function. Modifications of up to 50 % in A., B. 1
and
T.
1
1
result in maximal differences of 0.1
to 0.6 % in F, sometimes with a different unit commitment. So errors in the coefficients Ai' Bi and Tt may influence the 100.
selection of the units, but in the neigh-
.company determines its own optimum the over-
bourhood of the optimal
all costs will be F). When the optimum is'
trajecto~
this
offers no major increase in the costs, at
determined for all n companies together, the
least in the observed cases. Still, the start/stop costs is in the same order as the
costs will be F • If in both situations the 2 real optimum is ~alculated, F will be less 2 or at most equal to F). So the profit W, due
inaccuracy due to the coefficients of the
to higher level optimization, is
function f-
W=F) -F 2 Both F) and F2 , and so W, are stochastic variables. If we assume normal distributions
inaccuracy due to the coefficients.of the
If we assume a normal distribution for the errors in F and a 95 % uncertainty interval that contains the maximum error follows that
~F
~F,
it
•
- 2*T
F So we resume with results derived in this chapter: Errors in ai,bi,c i
~
Errors in A.,B.,T.
~
1.
1.
1.
OF
• 0.5 %;
~F
- 0.3 %;
~F
P
OF
p
- ) %
-.6 % .
with standard deviation OF ' ~F and respectively, it follows that 2
0w.2 -
OF
2 )
+OF
2 2
- 2 cov
0w
()
F).F 2
If F) and F are completely independent, it 2 follows that coy(F),F 2 ) - o.
Combining these errors of F and F. ·the • P errors in the overall costs F (F-F + F ) s
are given by (assuming these errors are
P
independent):
negative profit
i
OF - 0.6 % ; ~F - ).2 % These percentages give an indication of the order of magnitude of the errors in F. due·. to inaccuracies in the data which are used·to describe the fuel consumption of
~unit.
uncertainty of the Profits In the preceding section some indication is given of the influences of inaccuracies in
o
\11-
the function f and in the start/stop costs to the production costs F. As a consequence of the difference between the models and the data of the production costs and the real production costs; the optimum of the real production costs will generally not be found•. This offers a problem in determining the
Fig. 3 Probability distribution p(W) of the profi t w. However, F) and F2 have some common relations because both functtons are calculated
overall profits due to optimization. More-
from the same data. On the other·hand. they
over. this uncertainty makes it questionable
are not completely alike because they are
as to whether optimization at a high, regio-
the result of two different optimization
nal
~evel
offers real profits compared with
problems.
optimization at a low, local level. This wilt
So the real profit W is a normal distribu-
be illustrated with the following situation.
tion with mean Wand standard deviation
Suppose optimization is performed for an
This deviation depends on OF and on the
. area wi th n power companies. ·When each
0w.
cov(F),F 2 ). The profit W is illustrated in Fig. 3. 101
From Fig. 3 it can be concluded that, due to inaccuracies in the data with which the optimization is calculated, the profit W is a stochastic variable. Depending on the values of ~' of and cov(F 1,F 2 ) there is even a probability that optimization at a higher level offers a negative profit, although the dataassure a positive profit. If we use the-same optimization method in determining F as well as F2! cov(F ,F 2 ) will 1 I be large, through which 0w Ibecomes small. However, sometimes we cannot apply the same optimization method in determining F1 and F'r as will be explained in section 4. Consequently cov(F ,F 2 ) becomes smaller, while at 1 the same time Waecreases because for F 2 only an approxImate op~imum can be determined. -So in this case the possibility to realize a realistic profit decreases. 3. STATIC OPTIMIZATION Due to the uniform distribution of the main production and demand centers for electrical energy the losses due to tie lines need not be considered in The Netherlands. Only restrictions in the tie lines may be an extra constraint in the calculation of the optimal load distribution. In general, these extra constraints are not taken into account in the optimization procedure. If necessary. some c~rrections to the optimal solution are applied afterwards'. . , Another extra constraint is put on by the requirement for the spinning reserve R. One approach may be to distribute the spinning reserve proportional to the maximum power of each unit. Another method is to use by preference the nuclear and cheap thermal units without spinning reserve, while the older more expensive units have to provide for the spinning reserve. Always the sum all upper bounds of the units in operation has ro exceed the dema?d with the required spinning reserve.
9=
units n pose no extreme computer requirements. In case f. is a concave function (e.g. the power plafit of Amsterdam\21 ), or if there are units fired with two or more different fuels (oil and gas) the solution of the optimization problem becomes more complicated. Only timeconsuming iterative non-linear optimization_ techniques can be applied, e.g. the conjugate gradient projection method. This method has been developped for the power plant near the blast furnaces in IJmuiden (The Netherlands) 13\. In addition to gas from the blast furnaces, natural gas and/or oil also have to be fired in the boilers. . 4.' DYNAMIC OPTIMIZATION
In the dynamic optimization problem the selection of the units that contribute in the production wilf be optimized. Therefore. a timeinterval has to be defined (e.g. 24 or 48 hrs) for which the optimal unit commitment is calculated. This unit commitment depends on the production cost Fp' the start-up cost FS and the prediction of the demand D for each k k of the time-interval (Fig. 4).
~eriod
The dynamic optimization problem can be formulated for n units and for p periods of T hrs k each with P. k the power delivered-by un1t i in period k~' while y. k indicates the status. If y. k-0, unit i is 1,. shut down and if y. k1~ I unit i is running in period k. Time 1, t. k indicates in period k how long ago unit i 1 , has been shut down. There are only start costs F when unit i is start~d up S in peri04 k. j,k
F
P.1. k
n
Hin {1'~1 (a.P. P. 1 1
2
+ b.P. + c.).d.}
1 1
~
1
1
P.
<
-1 -
P.
<
1 -
P.
l'
n .1: P. - D 1-1 1
Analytical methods ,(e.g. the Lagrange method) are available to solve this problem quickly and accurately if a. > O. In general, a. is positive,' in which ~ase large numbers 1 of
F ,
+ F
Pi,k
S
)}
i,k
• (a.P.2 +·b.P. + c.).d .. T...,y. k
-F S. k1.
As a consequence of the choice of the quadratic function f to describe the production costs F , very elegant methods exist to solve p ~he static optimization problem. This optimization problem for n units is formulated as follows. with d. the price of the fuel of unit i: 1
n
p
Hin { t (1: , k k-I i-I y.1, k' P1, 1 1
1 1
1
1
A
1,
A. + B. ( I - e -to1. k/T.) 1 1
1
n i~1 P i •k - Dk k-I,p.
'So this problem is a mixed integer non-linear dynamic optimization problem. When F is linear" the solution can be calculated quite efficiently by means of the Branch and Bound method \4 \. However. f has to be chosen quadratic and F is also non-linear. So other S methods have to be considered. Besides the fuel consumption other considerations may make it desirable. due to technical or social requirements, to avoid too many stops and starts for a unit. This can be obtained by the introduction of some specific constraints in the op~imization, e;g. by means of the so-called on/off time, through which a unit has to be at least a certain amount of hours on or off, before its .tatus can be changed. This on/off time has turned out to be a valuable restriction.
102
Solutions for the Dynamic Optimization Problem As a consequence of the choice of a quadratic function f and the choice of a model for the start costs which depends on the time a unit is shut down. the solution of the dynamic optimization problem is quite complicated. In fact only the technique of dynamic programming as proposed by Bellman Is! is suited to solving this dynamic optimization problem in an acceptable amount of time, although. as will be shown. dynamic programming offers solutions in a reasonable amount of time for only a maximum of about 7 to 10 units.
The decision vector U may maintain or change k the status of each un1t. So. with n units . n l 2 - decisions can be made. Startin2 with one n-T state for k-O. we can calculA!T 2 new states for k-I. Agaiy each of the 2 states at k-I n generates 2 - new states for k-2 etc. This exponential growth of the number of states can be limited by several measures: 1- It sO,happens that in period k there are equal states (two'states are equal if all corresponding elements are equal), although' they have a different history. Then due to the Bellman's Principle of Optimality. only that' state that has the least overall cost up to period k needs to be stored. So if here are several paths to reach a state in period k, only that path with the least cost needs to be co'nsidered. 2- Not all decisions and not all combinations ·of units have to be.or can be considered, e.g. the demand. may be so large by which at least .3 units have 'to be in opereation. So a selection is very useful to reduce the number of combinations' and decisions and consequently the number of s~ates: With 6 units it may happen that the number of combinations is reduced from 63 to 13. However, it should be emphasized that as a consequence of this reduction the number of states decreases significantly, but at the same time not all situations are considered any longer. So this selection should be ~de very carefully. 3- Two states are equal when'the negative elements are the same. (positive values do not influence the costs). There is one exception to this, rule, namely the introduction of the on/off time.
Besides methods that can determine the optimal solution of the dynamic optimization problem. there are other. faster methods to calculate an approximate optimum. In The Netherlands two of these methods. among others. are used. namely a program based on the intuition and eiperience of a human operator and a program based on some heuristic algorithms. With the first program the operator can communicate in an interactive way. The oprator determines the optimal production policy on the basis of his own experience and insight into the economical and 'technical behaviour of the power ·system. The computer is only used to execute s~e static optimization calculations and to' com,pute the costs of the chosen overall production policy. This program is used to determine the optimal production policy for three companies. This way of'optimization turn's out to be very flexible because it can take into account additionally relevant data that are not included in the mathematical optimization problem. E.g. sometimes a unit is only partly available in. the time-interval or its upper bound is less for some periods due to measure~ When all states have been calaulated till the ments or maintenance. end of the interval (k-p), the optimal proThe second program uses a simple. heuristic duction policy can be easily obtained. by algorithm. Many times the algorithm will look. looking at the state at k-p with the least to see whether it is advantageous to stop. to . overall costs. The path from the beginning start or to maintain the current status of a (t-O) to this state '(k-p) is the optimal unit unit. Depending on their average costs (the cODlllitment policy. In fact forward dynamic cost per MW for P-P). all units are considered. programming has been applied. When no improvement 'can be found. the algor1thm concludes that an "approximate optimum" is Due to its character, dynamic programming is reached. In practice. many refinements have able to determine in an efficient way the been added to make this algorithm more reliable. optimal solution of the dynamic optimization problem as posed in this paper. However, its Dynamic Programming , application is limited by ita tremendous requirements for computer storage and computing So. only the dynamic programming method will time. In Table 3 estimates are given of the be able to determine in an efficient way the .computer storage ( in bytes) and computingreal optimum of the dynamic optimization ~ro~ time (in seconds for an IBM 370/158) as a function of the number of 'units. With p-15' blem. Therefore. the state of the system. the optimization is intended for 24 ,hours. which defines the status of all units. is These numbers are averages of many solutions introduced. With n units the state vector has n elements. The value of each element repre~ of the dynamic optimization problem. Depensents the time a unit is in operation (posi~ ding on the situation they may be different, especially for large n (nl8). Still Table 3 tive value). or the 'unit is shut down (nega~ shows clearly an upper bound for the applicative value). At the beginning of each period tion of dynamic programming. It may be questhe state for that period is calculated out of the state at the preceding period and a tionable as to whether this limit is for 7 or 10 units, but in any case for more than 10 decision vector U. This calculation or trans~ formation can be formulated as; units it is inefficient to calculate the solution with the digital computers of today.
TABLE 3 Estimates of the Computer-Storage and Computing-time Requirements as a Function of the Number of Units. n 5
storall;e 104 bytes 5
,, ,,
10 6 6 5.10 7 2.10
" ,"
6
6.10
7
7..10
8 9 10
4
policy. Still, such an approach is not possible for large-scale power systems, because , an operator cannot survey (too) many units at the same time.
cOllll)uting-time 7 seconds
5. HIERARCHICAL APPROACH
30 seconds 2 minutes 8 minutes 30 minutes 2 hours
Each extra constraint that limits the number of combinations of units to be considered at each period k will decrease the computer requirements. This is a nice feature of dynamic programming. Unfortunately, the earlier introduced on/off time increases these requirements. For not only the time a unit is shut down 'but ~so the time a unit is in operation now defiJ;les the state. 'In the next example the influence of the on/off time is illustrated. This ex~le deals with the company in Zeeland. Six units are available for the optimization. One unit, a nuclear power plant, always produces its maximumpower. Two units have upper bounds of about 80 MW and two units of 180 MW. Moreover, a gas turbine of about 20 MW is ava~lable. In Table 4 the number of states that has to be stored as well as ·the computing, time for an IBM 370/158 are given. Thes-e data are given for an optimization interval of 24 (p-13), for 48 (p~19) and for 96 (p a 40) hours.
In section 4 the possibilities of solving the dynamic optimization of large scale power systems have been discussed. It turns out that only one method can solve the problem accurately, but that because of the tremendous requirements for computer storage and computingtime dynamic programming can not be applied for large scale systems. On the other hand, there are more heuristic methods th~t are able to determine the solution in a~ acceptable amount of time, but this solution needS not to be optimal. So there seemS to be no way to solve the dynamic optimization problem, as defined in this paper, for large scale power systema. The situation is even more critical because 6f the small percentage of profits that can be obtained by applying dynamic optimization for a region ~f some companies. When at the low, local level the real optimum is found while at a higher level only an approximate optimum can be obtained, it may happen that the last solution is even more expensive.
For large dynamic systems a hierarchical approach turns out sometimes to offer a solution 161. Then the overall problem is decomposed, into subproplems that can be solved easily. A coordinator at a higher level then tries to converge the solution of the subTABLE 4 Number of States and Computing.,.Time 'problems to the solution of the overall proas Function of the On/oiftime ~nd blem. Therefore the coordinator asks for inNumber of Periods p. formation from the subproblems and gives information and directives to the subproblema. p-40 The power system itself is already decomposed On/off time p-13 p~19 into independent companies. Therefore the 970(1 Is) 2890(33s) 0 hours hierarchical structure of Fig. 5 seems to be 28.225(I8m6) 1774(20s) 4744(56s) 5 hours quite appropriate. 3961(52s) 6684(111127) 0 hours
-
A comparison made between the. dynamic programming and the heuristic program, shows the 'limitations of both approaches: first the timeconsuming character of dynamic programming and, second, the inability of the heuristic 'program to determine the optimal solution for all situations. In general, about the same s~lutions are found. However, in some situations the optimal cost diffe~s 1% to 2%, especial~y in situations'when it is advantageous to change ~he status of more than one uni t in the, same period'. Such a change can not be considered by the heuristic program. The "human-operator program" turns out to work satisfactorily for a region of three co~ panies with about 22 units. Due to the experience and knowledge of the human operator and the many restrictions that limit the choice of the units, the operators are able to determine an almost optimal production
Suppose each company has n. ( j-I,m) units. In general, not all units ~ill participate in the start/stop procedures. Some will continue on (e.g. nuclear plant) and others continue off. Once the number of active units is'bounded by about 5, the dynamic programming tech- . nique can calculate in an acceptable amount of time the optimal producing policy for one company with power demand D. k. On the basis of the optimal load distribdl10n and the optimal unit commitment ,for each company j, the coordinator can modify the power demand D. k
J,
into a new power demand D'. k' taking care that J, m E D. k ·
j_1
J,
m E D'. k
jal
J,
k-I,p
Moreover, the coordinator has to take care that some restrictions, due to safety considerations, are maintained, e.g. the requirement that each company must have each period k
104
coordinator calculates D! k for j=l,m; k=l,p J,
P. k
J,
D'
D!J, k
company 1
company j
calculates
calculates P. k J,
n. units J
m,k
company m n units m calculates P m,k D
D.J, k
m,k
Fig. 5 Hierarchical approach to solve the dynamic optimization problem of a power system. at least one (or more) units in operation etc. With this modified power demand D! J,k the optimal production policy for each company is again determined. This procedure continues till no more improvements 'can be obtained by the coordinator. Still, a method has to be·des~ribed that enables the coordinator to change D! in such
An ~dditionel feature, apart from the savings on fuel, of such a hierarchical approach is the possibility to determine the profits due to the cooperation. Moreover, it is easy to determine rules how to divide the profits among the participating companies. For the optimal costs are known when each company satisfies its own demand, and the costs are known when all companies cooperate.
J ,k
a way that a cheaper solution can be obtained. Three methods can be used to obtain a less expensive solution: - On the basis of the incremental costs of each company the coordinator can try to equalize them. The coordinator knows which D! k can be increased and which D! k dect~ased. However~ the stepsize of J , this modification is not known in advance. Therefore many iterations may be necessary. - On the basis of a recomputation of the static optimization for each period k. Only those units in each period that are selected at the lower level are considered, through which one step is necessary for each level. - On the basis of a computation of the dynamic optimization problem by the coordinator with the aid of a heuristic program. The unit commitment, as computed at the lower level with dynamic programming, can be used as a starting point for this heuristic program. For a heuristic program can deal with many units ()IOO), but gives only a local exploiration around the start point. Any of these three methods will give an improvement but no guarantee for an optimum. The third method seems especially promising, because many profits result not by starting a unit in one place but by buying some surplus from other companies. Research is going on to investigate the profits that can be obtained by cooperation between companies in the determination of an optimal production policy, with the aid of a hierarchical app~oach as decribed in this section.
CONCLUSION Errors in the data, which describe the fuel consumption of thermal power plants, set bounds to the possibility to det~ine an optimal production policy. These errors may even cause that optimization at a high level results in less profits compared with optimization at a low level. Another limitation is posed by the impossibility to solve in an acceptable amount of time the dynamic optimization problem, as formulated in this paper, for large numbers of units. Therefore a hierarchical structure in the solution is proposed. Although, with this approach, no guarantee can be given that the real optimum of the dynamic optimization problem will always be found, savings will be obtained compared with only using some kind of heuristic program or having only local optimization. •REFERENCES
2 3 4 5 6
Ottenhof, F.A. (1978). Economische optimali satie van een hierarchisch elektriciteits productie systeem. Lab. for Control Eng.(41). Honderd, G (1971). Dynamisch Programmeren Elektrotechniek, 4?, 112-116. Bosch, P.P.J. van den (1977) Optimalisering F.lektrotechniek, 55, 30-37 Taha, H.A. (1976), Operations Research. MacMillan Publishing Co., Inc. New York Bellman, R.E.,Dreyfus, S.E (1962). Dynamic Programming. Princeton Unv. Press. Princeton Singh, M.G, Titli, A. Systems, Decomposition Optimisation and Control. Pergamon International Library.(1978).
105
P.P.J. van den Bosch Laboratory for Control Engineering, Delft University of Technology, Delft, The Netherlands G. Honderd ' LabQratory for Control 'Engineering, Delft University of Technology, Delft, The Netherlands D.N. Vlugt Provinciale Zeeuwse Energie Maatschappij(PZEH}, Middelburg, The Netherlands
Abstract. Two topics in the dyn~ic optimization in large scale power systems with thermal units are discussed. First errors are considered in' the data with which the optimal load distribution and unit commitment arecalculated. Moreover their influence to the static and ftyDamic optimization problem is discussed. Second the possibilities to calculate an optimal production policy in.large scale power systems are considered.' It turns out that no method is able to determine in an acceptable amOunt of time the optimal unit commitment. Therefore a hierarchical approach is proposed to solve this optimization problem. ' Keyword,. Dynamic programming; hierarchical systems; optimization; power systems; error analysis. INTRODUCTION In The Netherlands ten independent companies provide for the production of electrical energy. In 1978 about 56.000 GWh. was delivered with a maximum demand of about' 9.500 MW. Except for two nuclear power plants, all production units are thermsl power plants fired with gas, oil and/or coal. The size of the units is between 20 and 630 MW net. Among these companies there is a cooperation within the SEP, i.e. the Coordinating Electricity Production Board for the long term planning (size and location,of new units), the power-frequency control, the common tie lines and the selling or buying of surplus or lack of power etc. However, due to their independence, there is up to now, no overall optimal production policy. In recent years an attempt has been made by the SEP to start and incorporate pr6duction,optimization on a national level, static as well as dynamic. This heuristic method will be considered later. in more detail. For smaller regions, Comprlslng about three companies, there is sometimes a cooperation to obtain an optimal production policy for all companies together. In general each company itself uses some kind of static optimization (load distribution) for its own units. In addition dynamic optimization
(unit commitment) is sometimes used to reduce the fuel costs. In close cooperation between the PZEM and the Laboratory for Control 'Engineering a feasibility study has been carried out to assess the possibilities of determining the optimal dynamic production policy for a period of, for example, one or two days for all companies in The Netherlands, or at least for some companies together. Besides the' profits that could be obtained by cooperation, some other factors have to be taken into account, e.g. ,the possibilities of calculatin~ the opti~l production policy, the influence of inaccuracies in the data that are the basis of the optimization, the limited amount of data that can be taken into account by a centralized optimization, the acceptance of the production policy by the local operators, possibility for accounting etc. Some aspects of this study will be discussed in this paper. This study is based on experiences in economic optimization projects as realized between the Laboratory for Control Engineering at the Delft University of Technology and several power plants in The Netherlands over a period of more than 13 years. In this paper, the mathematical 97
possibilities of determining a real optimum for the dynamic optimization problem and the influence of inaccuracies in the coefficients of the functions that describe the , fuel consumption are considered ~pecially. Most of the calculations concern the power' plants of the PZEM in Zeeland (The Netherlands), although the conclusions have a , much broader validity. '
which some quantities like flows, temperatures etc. are measured. When we
an~lyze
these factors for a thermal unit fired with gas and assume that the measurement errors can be described by a normal. distribution, then a standard deviation 0Q of about 0.6 % for Q. is obtained. In case oil is used, the J
flow measurement is somewhat more accurate,
2. ACCURACY CONSIDERATIONS
and the standard deviation for Q. then be-
Load Distribution, In general,. each thermal unit is described by some technical restriction (lower and upper bound) and by a function that relates the thermal energy Q of the fuel with the
J
comes about 0.5 % Ill.
Except for valve points, it is reasonable to assume that f will be a continuous function of P. So f is chosen as a polynomical of order n. With the aid of the least-square
net electrical power P. This function f {Q - f(P)} is generally valid only when there is one type of fuel and only when the unit is in steady state. The production cost Fp(P) is calculated out of f and the fuel price.
method the minimal error E m ) , 2 E (f(P.)-Q.) m j~) J J
--
m .1: If(~j)~Qjl/Qj ER- .! m J-) can be calculated as a function
~f
the number
of measurements m and the order n of the This function f is determined from some
polynomial f. In Tabel I the relative error
measurements. For several values P. of P,
ER is shown as it is calculated for a number'
which are fixed during an interval (.:!:. 2.-hrs).
of·thermal power plants.
J
the amount of fuel from which the Q. 's are ,
J
calculated is measured. Each pair (Pj,Qj)'
TABLE) Relative Error as a Function of.the Order of the Polynomial n and the Number of Measurements m.
is depicted as one point to describe the function f (Fig. I).
~
f
.4
25
7
%
%
Q
0
call hrJ
I
.7 %
1.0 %
2
.2 %
.5 %
%
.2 %
3
0
33
When we consider the staBdard deviation of· the accuracy with which Q is known, a choice for n larger than 2 is not meaningful, while a value for n less than 2 loses too much information. Consequently, the functicin f
P [MW] Fig. .1
Measur~ents
will be assumed to be of second order: 2 Q - f(P) • aP + bP + c In The 'Netherlands this second
(P., Q.) to determine J
J
Q ~ f(P). The accuracy with which P. and Q. are meaJ
or~er
poly-
nomial is nearly almost used to calculate J
sured depends on a number of technical factors, especially on the accuracy with
.
the fuel consumption as a function of the power. Once
~he
coefficients ai' b
i
and c
i
of unit i are known, the fuel cost F .
p1
98
follows by multiplying Q. with the price of
If we assume that these deviations belong to
the fuel. Summing the fuel costs of all n
a normal distribution, the standard devia-
1
units in operation gives the overall costs
tions for ai' b i , c i ' Popt and F can be In Table 2 averages of these
calculated.
F : p
F
deviations are shown. p
In order to obtain comparable results all
The propagation of the errors in (P.,Q.) to
costs are computed with the same cost
the coefficients a., b. and c., to·the op-
function F • Although these numbers depend
timal load distribution P and to the opt overall costs Fare analyzed in the fol-
on the units, the size of the units, the
J
. 1 1
J
p
1
.
number of units n, the fuels etc., these
p
lowing way. A simulation study has been
factors still have no major influence.
carried out. Starting with the measurements
However, there is one factor that has a
(P.,Q.) with standard deviation 0 J
J
p
• 0.3 %
dominant influence, especially on the
and 0Q • 0.6 %, we calculate each time the
deviation of the costs F • This factor is
optimal load distribution P t and F . Due op P to deviations in (P. ,Q.) of 'coefficients
the total demand D for electricity. This
J
J
p
dependence is illustrated in Fig. 2.
.
a., b.1 and c., Pop t and Fp have also some 1 1 deviations. TABLE 2 Propagation of Measurement Errors
measurements
o
coefficients of f
c1 • 50
% ; ob
load P
°P •
15
%
of
0.2 to 0.8 % ,
Costs F
P
P
• 0.3 % ; 0
a
.
P
0
'" 0.6 %
.
12
% ;
°c
• 25 %
t
f
,, , ,
• .5
I
I
,
I
I
I
I
I
i ,,, ; I
I
[~.
D
Fig. 2 Dependency of of p
_
II
of the demand D.
IT,,",
'L
3·
,
I I
'T", T.... I(.,
k
[hourS]
Fig. 4
1>
, I
p
~
t
Demand D for p periods in the k time interval.
In I the demand equais the sum of all lower
because its assumption that the cooling of
bounds of the units, while in 11 the demands
a unit can be described by means of a first
equals the sum of all upper bounds. In these
order system, is given by:
-th: i
points there ,is no freedom for the optimi-
F
zatiun to determine another optimum. So
S. 1
(t)
s
A. + B. (I-e 1
)
,1
errors in (P.,Q.) will 'then, not influence
There may be discussion about the validity
J
J
the load distribution. If there is more
of this model. In some studies another model
freedom, as is the case for a relatively
calculates F
low demand D, the errors
function of
OF
in~(P.,Q.)
J
J
increase
P
s.
(t) from a piece-wize linear the time t
1
121.
In general,
the values of the coefficients Ai' B and i are based on rules of thumb, in which Ai' i
Throughout the paper of will be assumed to , P be 0.5 %.
T
From Table' 2 it can be concluded that in the
unit., However, measurements that have been
neighbourhood of the optimum
th~
function
Fp is relatively flat in its variabl~s Pi' So
~
small deviation of the set points for
the power of the units from their optimal values will
generall~ not
very much. Still,
s~me
increase the costs
profits can be ob-
B.1 and
T.
1
are functions of the size of the
obtained for several thermal units show no nice fit With such a model. Therefore the accuracy with which the start-up costs can be
cal~ulated
is rather poor. Relating some
measurements of F
with the coefficients s. used in optimizatiAn procedures show errors
tained by mea~s of static optimization: An
of up to 50 %.
estimate of these profits, co~pared with the
Moreover, coefficients A. and B. are often
situation in which human operators determine
chosen to satisfy another requirement;
1
1
the load distribution, is about 1,% to 2 %.
Instead of only determining the extra amount
Of course this may be a considerable amount
of fuel, Ai is often used to avoid too many start/stops. This goal can be achieved by'
of money.,
making A. and B. larger than necessary to ,11
calculate the extra fuel costs. In section Unit CODIDitments
4 a method will be proposed to avoid.this
Besides the function
misuse of F • by the introduction,of the s on/off time 1 in the model.
~,
other information is
necessary to calculate the unit commitment for the optimal production policy for a' period of, for examrle, 24 or 28 hours
ahea~
This information is given by the costs for starting and stopping the production 'units. Many factors
dete~ine
these,costs. Certain-'
ly these costs of taking a unit into operation are,strongly dependent upon the time the unit has been out of operation. The
racies in the coefficients A., B. and
only, with fixed and precisely described procedures for starting up. Without these procedures it is impossible to determine the start-up costs accurately. Still, some models are used to calculate the amount of' fuel as a function of the time t a unit has fre~
quently used, because' of its simplicity and
a
T.
111
simulation study has been performed. Several times, for the same units and power demand D the' optimal production policy has been calculated with the aid of the dynamic programming method. First a reference situation has been solved. Then changes of 50 % in Ai' ,B and
start-up costs can be determined accurately
been out of operation. A model that is
In order to' study the influence of inaccu-
i
have been given, after which the
T
i optimal start/stop policy has been again determined. In order to make all runs comparable the ultimate costs have been calculated with the same, reference, cost function. Modifications of up to 50 % in A., B. 1
and
T.
1
1
result in maximal differences of 0.1
to 0.6 % in F, sometimes with a different unit commitment. So errors in the coefficients Ai' Bi and Tt may influence the 100.
selection of the units, but in the neigh-
.company determines its own optimum the over-
bourhood of the optimal
all costs will be F). When the optimum is'
trajecto~
this
offers no major increase in the costs, at
determined for all n companies together, the
least in the observed cases. Still, the start/stop costs is in the same order as the
costs will be F • If in both situations the 2 real optimum is ~alculated, F will be less 2 or at most equal to F). So the profit W, due
inaccuracy due to the coefficients of the
to higher level optimization, is
function f-
W=F) -F 2 Both F) and F2 , and so W, are stochastic variables. If we assume normal distributions
inaccuracy due to the coefficients.of the
If we assume a normal distribution for the errors in F and a 95 % uncertainty interval that contains the maximum error follows that
~F
~F,
it
•
- 2*T
F So we resume with results derived in this chapter: Errors in ai,bi,c i
~
Errors in A.,B.,T.
~
1.
1.
1.
OF
• 0.5 %;
~F
- 0.3 %;
~F
P
OF
p
- ) %
-.6 % .
with standard deviation OF ' ~F and respectively, it follows that 2
0w.2 -
OF
2 )
+OF
2 2
- 2 cov
0w
()
F).F 2
If F) and F are completely independent, it 2 follows that coy(F),F 2 ) - o.
Combining these errors of F and F. ·the • P errors in the overall costs F (F-F + F ) s
are given by (assuming these errors are
P
independent):
negative profit
i
OF - 0.6 % ; ~F - ).2 % These percentages give an indication of the order of magnitude of the errors in F. due·. to inaccuracies in the data which are used·to describe the fuel consumption of
~unit.
uncertainty of the Profits In the preceding section some indication is given of the influences of inaccuracies in
o
\11-
the function f and in the start/stop costs to the production costs F. As a consequence of the difference between the models and the data of the production costs and the real production costs; the optimum of the real production costs will generally not be found•. This offers a problem in determining the
Fig. 3 Probability distribution p(W) of the profi t w. However, F) and F2 have some common relations because both functtons are calculated
overall profits due to optimization. More-
from the same data. On the other·hand. they
over. this uncertainty makes it questionable
are not completely alike because they are
as to whether optimization at a high, regio-
the result of two different optimization
nal
~evel
offers real profits compared with
problems.
optimization at a low, local level. This wilt
So the real profit W is a normal distribu-
be illustrated with the following situation.
tion with mean Wand standard deviation
Suppose optimization is performed for an
This deviation depends on OF and on the
. area wi th n power companies. ·When each
0w.
cov(F),F 2 ). The profit W is illustrated in Fig. 3. 101
From Fig. 3 it can be concluded that, due to inaccuracies in the data with which the optimization is calculated, the profit W is a stochastic variable. Depending on the values of ~' of and cov(F 1,F 2 ) there is even a probability that optimization at a higher level offers a negative profit, although the dataassure a positive profit. If we use the-same optimization method in determining F as well as F2! cov(F ,F 2 ) will 1 I be large, through which 0w Ibecomes small. However, sometimes we cannot apply the same optimization method in determining F1 and F'r as will be explained in section 4. Consequently cov(F ,F 2 ) becomes smaller, while at 1 the same time Waecreases because for F 2 only an approxImate op~imum can be determined. -So in this case the possibility to realize a realistic profit decreases. 3. STATIC OPTIMIZATION Due to the uniform distribution of the main production and demand centers for electrical energy the losses due to tie lines need not be considered in The Netherlands. Only restrictions in the tie lines may be an extra constraint in the calculation of the optimal load distribution. In general, these extra constraints are not taken into account in the optimization procedure. If necessary. some c~rrections to the optimal solution are applied afterwards'. . , Another extra constraint is put on by the requirement for the spinning reserve R. One approach may be to distribute the spinning reserve proportional to the maximum power of each unit. Another method is to use by preference the nuclear and cheap thermal units without spinning reserve, while the older more expensive units have to provide for the spinning reserve. Always the sum all upper bounds of the units in operation has ro exceed the dema?d with the required spinning reserve.
9=
units n pose no extreme computer requirements. In case f. is a concave function (e.g. the power plafit of Amsterdam\21 ), or if there are units fired with two or more different fuels (oil and gas) the solution of the optimization problem becomes more complicated. Only timeconsuming iterative non-linear optimization_ techniques can be applied, e.g. the conjugate gradient projection method. This method has been developped for the power plant near the blast furnaces in IJmuiden (The Netherlands) 13\. In addition to gas from the blast furnaces, natural gas and/or oil also have to be fired in the boilers. . 4.' DYNAMIC OPTIMIZATION
In the dynamic optimization problem the selection of the units that contribute in the production wilf be optimized. Therefore. a timeinterval has to be defined (e.g. 24 or 48 hrs) for which the optimal unit commitment is calculated. This unit commitment depends on the production cost Fp' the start-up cost FS and the prediction of the demand D for each k k of the time-interval (Fig. 4).
~eriod
The dynamic optimization problem can be formulated for n units and for p periods of T hrs k each with P. k the power delivered-by un1t i in period k~' while y. k indicates the status. If y. k-0, unit i is 1,. shut down and if y. k1~ I unit i is running in period k. Time 1, t. k indicates in period k how long ago unit i 1 , has been shut down. There are only start costs F when unit i is start~d up S in peri04 k. j,k
F
P.1. k
n
Hin {1'~1 (a.P. P. 1 1
2
+ b.P. + c.).d.}
1 1
~
1
1
P.
<
-1 -
P.
<
1 -
P.
l'
n .1: P. - D 1-1 1
Analytical methods ,(e.g. the Lagrange method) are available to solve this problem quickly and accurately if a. > O. In general, a. is positive,' in which ~ase large numbers 1 of
F ,
+ F
Pi,k
S
)}
i,k
• (a.P.2 +·b.P. + c.).d .. T...,y. k
-F S. k1.
As a consequence of the choice of the quadratic function f to describe the production costs F , very elegant methods exist to solve p ~he static optimization problem. This optimization problem for n units is formulated as follows. with d. the price of the fuel of unit i: 1
n
p
Hin { t (1: , k k-I i-I y.1, k' P1, 1 1
1 1
1
1
A
1,
A. + B. ( I - e -to1. k/T.) 1 1
1
n i~1 P i •k - Dk k-I,p.
'So this problem is a mixed integer non-linear dynamic optimization problem. When F is linear" the solution can be calculated quite efficiently by means of the Branch and Bound method \4 \. However. f has to be chosen quadratic and F is also non-linear. So other S methods have to be considered. Besides the fuel consumption other considerations may make it desirable. due to technical or social requirements, to avoid too many stops and starts for a unit. This can be obtained by the introduction of some specific constraints in the op~imization, e;g. by means of the so-called on/off time, through which a unit has to be at least a certain amount of hours on or off, before its .tatus can be changed. This on/off time has turned out to be a valuable restriction.
102
Solutions for the Dynamic Optimization Problem As a consequence of the choice of a quadratic function f and the choice of a model for the start costs which depends on the time a unit is shut down. the solution of the dynamic optimization problem is quite complicated. In fact only the technique of dynamic programming as proposed by Bellman Is! is suited to solving this dynamic optimization problem in an acceptable amount of time, although. as will be shown. dynamic programming offers solutions in a reasonable amount of time for only a maximum of about 7 to 10 units.
The decision vector U may maintain or change k the status of each un1t. So. with n units . n l 2 - decisions can be made. Startin2 with one n-T state for k-O. we can calculA!T 2 new states for k-I. Agaiy each of the 2 states at k-I n generates 2 - new states for k-2 etc. This exponential growth of the number of states can be limited by several measures: 1- It sO,happens that in period k there are equal states (two'states are equal if all corresponding elements are equal), although' they have a different history. Then due to the Bellman's Principle of Optimality. only that' state that has the least overall cost up to period k needs to be stored. So if here are several paths to reach a state in period k, only that path with the least cost needs to be co'nsidered. 2- Not all decisions and not all combinations ·of units have to be.or can be considered, e.g. the demand. may be so large by which at least .3 units have 'to be in opereation. So a selection is very useful to reduce the number of combinations' and decisions and consequently the number of s~ates: With 6 units it may happen that the number of combinations is reduced from 63 to 13. However, it should be emphasized that as a consequence of this reduction the number of states decreases significantly, but at the same time not all situations are considered any longer. So this selection should be ~de very carefully. 3- Two states are equal when'the negative elements are the same. (positive values do not influence the costs). There is one exception to this, rule, namely the introduction of the on/off time.
Besides methods that can determine the optimal solution of the dynamic optimization problem. there are other. faster methods to calculate an approximate optimum. In The Netherlands two of these methods. among others. are used. namely a program based on the intuition and eiperience of a human operator and a program based on some heuristic algorithms. With the first program the operator can communicate in an interactive way. The oprator determines the optimal production policy on the basis of his own experience and insight into the economical and 'technical behaviour of the power ·system. The computer is only used to execute s~e static optimization calculations and to' com,pute the costs of the chosen overall production policy. This program is used to determine the optimal production policy for three companies. This way of'optimization turn's out to be very flexible because it can take into account additionally relevant data that are not included in the mathematical optimization problem. E.g. sometimes a unit is only partly available in. the time-interval or its upper bound is less for some periods due to measure~ When all states have been calaulated till the ments or maintenance. end of the interval (k-p), the optimal proThe second program uses a simple. heuristic duction policy can be easily obtained. by algorithm. Many times the algorithm will look. looking at the state at k-p with the least to see whether it is advantageous to stop. to . overall costs. The path from the beginning start or to maintain the current status of a (t-O) to this state '(k-p) is the optimal unit unit. Depending on their average costs (the cODlllitment policy. In fact forward dynamic cost per MW for P-P). all units are considered. programming has been applied. When no improvement 'can be found. the algor1thm concludes that an "approximate optimum" is Due to its character, dynamic programming is reached. In practice. many refinements have able to determine in an efficient way the been added to make this algorithm more reliable. optimal solution of the dynamic optimization problem as posed in this paper. However, its Dynamic Programming , application is limited by ita tremendous requirements for computer storage and computing So. only the dynamic programming method will time. In Table 3 estimates are given of the be able to determine in an efficient way the .computer storage ( in bytes) and computingreal optimum of the dynamic optimization ~ro~ time (in seconds for an IBM 370/158) as a function of the number of 'units. With p-15' blem. Therefore. the state of the system. the optimization is intended for 24 ,hours. which defines the status of all units. is These numbers are averages of many solutions introduced. With n units the state vector has n elements. The value of each element repre~ of the dynamic optimization problem. Depensents the time a unit is in operation (posi~ ding on the situation they may be different, especially for large n (nl8). Still Table 3 tive value). or the 'unit is shut down (nega~ shows clearly an upper bound for the applicative value). At the beginning of each period tion of dynamic programming. It may be questhe state for that period is calculated out of the state at the preceding period and a tionable as to whether this limit is for 7 or 10 units, but in any case for more than 10 decision vector U. This calculation or trans~ formation can be formulated as; units it is inefficient to calculate the solution with the digital computers of today.
TABLE 3 Estimates of the Computer-Storage and Computing-time Requirements as a Function of the Number of Units. n 5
storall;e 104 bytes 5
,, ,,
10 6 6 5.10 7 2.10
" ,"
6
6.10
7
7..10
8 9 10
4
policy. Still, such an approach is not possible for large-scale power systems, because , an operator cannot survey (too) many units at the same time.
cOllll)uting-time 7 seconds
5. HIERARCHICAL APPROACH
30 seconds 2 minutes 8 minutes 30 minutes 2 hours
Each extra constraint that limits the number of combinations of units to be considered at each period k will decrease the computer requirements. This is a nice feature of dynamic programming. Unfortunately, the earlier introduced on/off time increases these requirements. For not only the time a unit is shut down 'but ~so the time a unit is in operation now defiJ;les the state. 'In the next example the influence of the on/off time is illustrated. This ex~le deals with the company in Zeeland. Six units are available for the optimization. One unit, a nuclear power plant, always produces its maximumpower. Two units have upper bounds of about 80 MW and two units of 180 MW. Moreover, a gas turbine of about 20 MW is ava~lable. In Table 4 the number of states that has to be stored as well as ·the computing, time for an IBM 370/158 are given. Thes-e data are given for an optimization interval of 24 (p-13), for 48 (p~19) and for 96 (p a 40) hours.
In section 4 the possibilities of solving the dynamic optimization of large scale power systems have been discussed. It turns out that only one method can solve the problem accurately, but that because of the tremendous requirements for computer storage and computingtime dynamic programming can not be applied for large scale systems. On the other hand, there are more heuristic methods th~t are able to determine the solution in a~ acceptable amount of time, but this solution needS not to be optimal. So there seemS to be no way to solve the dynamic optimization problem, as defined in this paper, for large scale power systema. The situation is even more critical because 6f the small percentage of profits that can be obtained by applying dynamic optimization for a region ~f some companies. When at the low, local level the real optimum is found while at a higher level only an approximate optimum can be obtained, it may happen that the last solution is even more expensive.
For large dynamic systems a hierarchical approach turns out sometimes to offer a solution 161. Then the overall problem is decomposed, into subproplems that can be solved easily. A coordinator at a higher level then tries to converge the solution of the subTABLE 4 Number of States and Computing.,.Time 'problems to the solution of the overall proas Function of the On/oiftime ~nd blem. Therefore the coordinator asks for inNumber of Periods p. formation from the subproblems and gives information and directives to the subproblema. p-40 The power system itself is already decomposed On/off time p-13 p~19 into independent companies. Therefore the 970(1 Is) 2890(33s) 0 hours hierarchical structure of Fig. 5 seems to be 28.225(I8m6) 1774(20s) 4744(56s) 5 hours quite appropriate. 3961(52s) 6684(111127) 0 hours
-
A comparison made between the. dynamic programming and the heuristic program, shows the 'limitations of both approaches: first the timeconsuming character of dynamic programming and, second, the inability of the heuristic 'program to determine the optimal solution for all situations. In general, about the same s~lutions are found. However, in some situations the optimal cost diffe~s 1% to 2%, especial~y in situations'when it is advantageous to change ~he status of more than one uni t in the, same period'. Such a change can not be considered by the heuristic program. The "human-operator program" turns out to work satisfactorily for a region of three co~ panies with about 22 units. Due to the experience and knowledge of the human operator and the many restrictions that limit the choice of the units, the operators are able to determine an almost optimal production
Suppose each company has n. ( j-I,m) units. In general, not all units ~ill participate in the start/stop procedures. Some will continue on (e.g. nuclear plant) and others continue off. Once the number of active units is'bounded by about 5, the dynamic programming tech- . nique can calculate in an acceptable amount of time the optimal producing policy for one company with power demand D. k. On the basis of the optimal load distribdl10n and the optimal unit commitment ,for each company j, the coordinator can modify the power demand D. k
J,
into a new power demand D'. k' taking care that J, m E D. k ·
j_1
J,
m E D'. k
jal
J,
k-I,p
Moreover, the coordinator has to take care that some restrictions, due to safety considerations, are maintained, e.g. the requirement that each company must have each period k
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coordinator calculates D! k for j=l,m; k=l,p J,
P. k
J,
D'
D!J, k
company 1
company j
calculates
calculates P. k J,
n. units J
m,k
company m n units m calculates P m,k D
D.J, k
m,k
Fig. 5 Hierarchical approach to solve the dynamic optimization problem of a power system. at least one (or more) units in operation etc. With this modified power demand D! J,k the optimal production policy for each company is again determined. This procedure continues till no more improvements 'can be obtained by the coordinator. Still, a method has to be·des~ribed that enables the coordinator to change D! in such
An ~dditionel feature, apart from the savings on fuel, of such a hierarchical approach is the possibility to determine the profits due to the cooperation. Moreover, it is easy to determine rules how to divide the profits among the participating companies. For the optimal costs are known when each company satisfies its own demand, and the costs are known when all companies cooperate.
J ,k
a way that a cheaper solution can be obtained. Three methods can be used to obtain a less expensive solution: - On the basis of the incremental costs of each company the coordinator can try to equalize them. The coordinator knows which D! k can be increased and which D! k dect~ased. However~ the stepsize of J , this modification is not known in advance. Therefore many iterations may be necessary. - On the basis of a recomputation of the static optimization for each period k. Only those units in each period that are selected at the lower level are considered, through which one step is necessary for each level. - On the basis of a computation of the dynamic optimization problem by the coordinator with the aid of a heuristic program. The unit commitment, as computed at the lower level with dynamic programming, can be used as a starting point for this heuristic program. For a heuristic program can deal with many units ()IOO), but gives only a local exploiration around the start point. Any of these three methods will give an improvement but no guarantee for an optimum. The third method seems especially promising, because many profits result not by starting a unit in one place but by buying some surplus from other companies. Research is going on to investigate the profits that can be obtained by cooperation between companies in the determination of an optimal production policy, with the aid of a hierarchical app~oach as decribed in this section.
CONCLUSION Errors in the data, which describe the fuel consumption of thermal power plants, set bounds to the possibility to det~ine an optimal production policy. These errors may even cause that optimization at a high level results in less profits compared with optimization at a low level. Another limitation is posed by the impossibility to solve in an acceptable amount of time the dynamic optimization problem, as formulated in this paper, for large numbers of units. Therefore a hierarchical structure in the solution is proposed. Although, with this approach, no guarantee can be given that the real optimum of the dynamic optimization problem will always be found, savings will be obtained compared with only using some kind of heuristic program or having only local optimization. •REFERENCES
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Ottenhof, F.A. (1978). Economische optimali satie van een hierarchisch elektriciteits productie systeem. Lab. for Control Eng.(41). Honderd, G (1971). Dynamisch Programmeren Elektrotechniek, 4?, 112-116. Bosch, P.P.J. van den (1977) Optimalisering F.lektrotechniek, 55, 30-37 Taha, H.A. (1976), Operations Research. MacMillan Publishing Co., Inc. New York Bellman, R.E.,Dreyfus, S.E (1962). Dynamic Programming. Princeton Unv. Press. Princeton Singh, M.G, Titli, A. Systems, Decomposition Optimisation and Control. Pergamon International Library.(1978).
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