Optimal Management of a Large Power System The Problem of Refueling Nuclear Power Plants

OPTIMAL MANAGEMENT OF A LARGE POWER SYSTEM THE PROBLEM OF REFUELING NUCLEAR POWER PLANTS

1'. Lederer Department of General Economic Studies, Electricite de France 2, rue Louis Murat, 75008 Paris, France

Abstract. Scheduling the maintenance and refueling of Light Water Reactors sets a difficult dynamic optimization problem in random future. This paper aims at introducing an open-loop formalization concerning the optimal opera~ion of the French electrical generating system. Both technical and economic vital aspects of the problem are taken into account. Optimal .control of impulses with time-lags is used. KeYwords. Management systems,

~uclear

reactors, optimal control, delays,' impulses.

INTRODUCTION The main features of the French electrical system are : - a demand with pronounced variations between the seasons as well as during the day. Furthermore, this demand is subject to important random variations - diversified thermal plants. This means the cost per produced kWh is rapidly increasing with the thermal demand ; - a hydraulic system composed of about twenty. large seasonal reservoirs. Several of these reservoirs receive the ,greatest part of their inflows outside of the high demand periods. The complexity of the system makes it illusory to represent all the different decisionmaking situations. The different time-cycles inside of which thes~decisions take place suggest a decomposition in two cycles : - a yearly cycle. This cycle deals with the decisions concerning the dynamic regulation of the system : operation of the large reservoirs, maintenance and refueling of the thermal and nuclear plants. - a weekly-cycle where the discharges and the available thermal and nuclear capacities are those generated in the yearly-cycle. The yearly-cycle only is considered in this paper. In such a system, the hydraulic reserve plays a double role. On the year's time scale, the reserve must transfer energy, using its sto-. rage capacity in the best way, by concentrating its production at the time when demand is the highest. During the week, its

role consists in avoiding using the most expensive thermal plants at peak-demand ,hours and to obviate loss-of-load situations. Similarly, the schedule of maintenance and refueling of the thermal and nuclear plants should enable them to be available when the demand is the highest. Until recent years this has been easily achieved with conventional thermal plants (none of them had to be shut down during winter except for forced outages). Such a situation, however, is not to last long because of the development of nuclear capacity. Indeed, as far as the operation of the whole generating system is concerned, Light Water Reactors introduce new constraints, which did not exist with conventional thermal plants. These constraints are directly related to the characteristics of this type of nuclear plant: maintenance and refueling make it necessary to shut down the reactor for about two months each year. Moreover, due to the availability of maintenance staff, two units of the same plant cannot be simultaneously under maintenance,or refueling. 1 ' There exists, however, some flexibilities in the operation of these nuclear plants. These flexibilities are to be used in order to minimize the operating costs. When it happens that a reactor should be normally shut down at a time when the demand is high (or when another unit of the same plant is already being refueled), two kinds of "controls" are available in order to try and shift the refueling of the reactor out of this period : IMos t of the French nuclear plants include two or four units.

254

i) ,it is possible to anticipate the refue- . I - This problem can be set as an integer ling or to stretch-out the fuel (within non1inear program with the variable "O-reactor shut down" or ,,'I-reactor 'in given limits) ; operation". ii) it is possible to run the reactor only at partial power during some time, so 2 - The time at which the reactor wi1~ be shut that the burn-up of the fuel corresdown can also be taken as a variable and ponding to a normal shut down is it i~ possible to settle it directly. reached later on. This implies to follow in continuous time the evolution of a system which undergoes The problem is to control the system (using i) discontinuities and ii» so as to minimize the mathematical expectation of the operating costs (sum of the the reactor i is shut down at time 'i' nuclear fuel cost, fossil fuel cost and lossit will be put again into operatiPn at of-load cost) time 'i+ a where a rep~esents the THE PROBLEM The Difficulties The problem raises, simultaneously, all the difficulties for which a solution had to be found in order to operate the hydraulic reservoirs, and specific aspects which call for a special formalization and the use of adequate mathematical methods. It is a difficult problem for reasons due to : - the dynamic naturel'of generating equipment utilization to meet a rime-varying demand - the randomness of the system : electrical demand, hydraulic inflows and forced outages of the thermal plants ; - the large scale of the problem - moreover, the decisions to be taken are discrete. As' for the hydraulic reservoirs, it is a matter of dealing with a large scale stochastic system of which an aggregated representation (i.e. . here, the aggregation of several reactors) can only be considered with great difficulty. The constraints involved in reactor pi10ting, in their yearly shutdown for refue1ing, are added'to the usual operating constraints so that it becomes necessary to follow the operation of each reactor during several years, taking into account its availability (which is a random variable), so as to make sure that i) the operation defined, for each reactor, is feasible with respect to its own constraints ; ii) the nuclear fuel cost is correctly computed. A Fitting Formalization The very nature of the decisions which are to be optimized impresses its particular character to the problem :

duration of maintenance and refue1ing. The time 'i of the shutdown "impulse" is then a control variable of reactor i, whioh undergoes, with a delay a, a start-up "impulse". This second approach has been adopted. The problem is set as an optimal control problem with impulses and delays. The algorithm requires only usual gradient methods and it is not necessary to cope with a large no~-linear integer program as it'wou1d be .the case if the first approach had been chosen. Moreover this formalization makes it possible to consider the variations of the nuclear fuel cost due to the state of the reactor at the end of each campaign. FORMALIZATION The Variables For each reactor i (i are availab1~ :

1 to n), two controls

- " 1. is the time at which the reactor i will be shut down for maintenance and refueling. The reactor will then be put into operation again at time 'i + a where a is the duration of maintenance and refue1ing ; - U.(t) is the. load factor at time t (i.e.1. the fraction of the available power of reactor i used at time t) ; Xi(t) is the state variable of reactor i. It represents the energy which could be delivered by the reactor i from time t till maximum stretch-out. Using the .state variable X., it is possible 1. to compute the maximum power of the reactor (when it ..is available) P(X i ). We have the following diagram :

lIt is indeed a question of managing and renewing a stock (energy in the reactor's core) which use provides time-varying economical advantages.

235

Equations of Motion

maximum power available P(X } i

~

l out

I I

'X.

variable X

I

, I

I

energy "in the reactor"

'-+-_......_ _--I.I

Xmin

I I

The problem being set in "open-loop". the control variables are the same for all the time-series i. but the state variables depend on i through the equations of motions. ~ndeed for given controls T (shutdown'of reactor i) i and Ui(t}. the equation of motion of the state

normal shutdown

~~

u

is

Xmax

Notations

t:/= T i

Xmax ,is the state of a reactor just after refueling ;

t=/::Ti+Cl

Xmin is the limit of a possible stretch-out. In the following. we shall use the convention :

Thus. the state of the reactor remains constant when I (t) • 0 (forced outage). It underii goes two discontinuities in T and T +Cl : i i

Xi(t} - 0 means that the reactor i is under maintenance and refue1ing ; then P(X } i - P(O} = o. The following assumptions will be made for the sake of simplicity. They are unnecessary for treatment of the cOmplete problem : - the evolution of the state of reactor i. Xi(t}. does not depend on demand or hydraulic inflows. Therefore we shall omit (in our notations) the mathematical expectation on these random variables. though it is ' taken into account in the economical criterion, - the range of the study. [O.T] • covers several years and several shutdowns of each reactor (about a third of the fuel is renewed each time). It will here be assumed that each reactor undergoes only one shutdown in this interval. - again, in order to simplify. it is assumed that U. is a continuous function of time.

With the convention that if :

,P (O) = 0, it follows

Note: For given values of controls T. and U.(t). the state of the reactor at th~ time ~ of the shutdown, Xi(T i } is a random variable and its value might correspond to an antici-' pation. a normal shutdown or a stretch-out. The Constraints a} Impossibility of simultaneous refueling of two' units of a plant During the iterative search of an optimum. we might meet the following situation. where i and j are the indexes of two units of a same plant

~

T.+n

In fact U.~ will be a piece-wise function. constant on periods of fifteen days or· even a month.' This'is closer to physical operating constraints. Random Availability of the Reactors From the probability law of the forced outages of the reactors. time-series will be generated. Several time-series will be taken so as to get a good image of the theoretical laws Each time-series i (i - I to L) will be made of p functions Iii(t} : IU(t}, -

f

We know that unit j, shut down at time Tj • can only be put under maintenance and refueling at time T + Cl and will only be put i into operation again at time T + 2Cl. i We just have to consider i} that two delays are following control T i : in Ti +Cl. unit i is put into operation again.

I if reactor i is available at time t o if 'not.

I

in Ti + 2Cl • unit j is put into operation again. ii) that no delay is following control T ••

J

The gradients of the criterion with respect to T and Tj will take i) and ii) into account i for the next iteration. b) Operating constraints of the reactors (i = I to n)

c) Maximum anticipation and maximum stretchout Xmin

~

~

XiR, (T)

A

=

(i

MAXIMUM

MAXIMUM

STRETCH-OUT

ANTICIPATION

1 to n, 1 .. 1 to L)

The second term : nuclear fuel cost, additional-cost in case of stretch-out or economic value of the fuel recovered in case of anticipation is only a function of the state of the reactor at the time when it is shut down 1

L

L ..o..E

n

E K.(X·o(T:». I i"l 1 1.. 1

Summarizing, the problem has the following structure.

These constraints raise a "difficulty which is linked with the open-loop formalization. Indeed, for given values of the controls T. and U., X.(T.) is a random variable depen- 1 111 ding only on the random availability of reactor i. The~tore it seems realistic to set the above constraints as :

i) Prob (Xi (T i ) ~ A) < El ii) Prob (X · (T.) ~ X . )<'E • 1

1

2

m1n

It can be seen that i) and ii) become

x.1

(anticipation)

(0) given

subject to the constraints

(stretch-out).

No simultaneous refueling of two units of a plant. (i .. I to n)

The Economical Criterion

.p, J:i Ui(t) dt • Ai

Formalizing the problem in continuous time and following the state Xi of each reactor enables us to build an economical criterion as a sum of two terms : total operating cost : (mathematical expectation of the fossil fuel cost and lossof-load 'cost)

"ii

(i

=

(modulation)

(anti'i,atin' I to n)

Ui(t) dt
("ratth~ut

Necessary Optimality Conditions

+ (mathematical expectation of the nu-

clear fuel cycle cost). The first term is then an integral criterion which comes as a function of the total nuclear load:

t ~ I' g(.; 1=t}O

For our particular problem, these QPnditit are the following :

P(8 .(t». Ui(t) . l.1t) dt i

1=1"

I

or -L

The problem is set as an optimal control p blem with impulses and delays. In the appe Aix, the necessary optimal~ty conditions c such a problem are given.

-

~lg(X~o'U.(t»

i-I - 0

1..

1

dt.

- a necessary _conditiqn_ for the feasible solutions (T.) and (U.(t» to be optimun the existenc~ of multlpliers functions ~i1(t), solution of :

237

I

a

L aX~i

(Xii' Ui' t) dP(X (t» U ' Ui(t).Ia(t) dX

+ 1/Iu(t)

U

'1/I l.x. .• ('t-:-+ 1.

a)-O

1/1 •• (-r.) .. 1. x.

1.

dK(X('t-:-»

1. W•• ('t +1..) -. ---;;--~ ~x. dX

ii

Furthermore, if we set H(t,X.(t),U.(t),W .• (t» .. 1.

+

~

1.

1. x.

Wu (t) • P(X

i-I

U

REFERENCES

~ fII

L i-I VC

) . U (t) i

g(X .• (t),U.(t),t) 1. x.

• I

U

1.

(t)],

the f2llowing optimality conditions in Ui(t) and 't i must be satisfied a) Necessary optimality condition in U.(t) •

1.--

(for t

~

't ) i

/

aH(t,X. (t), u.1. (t) ,W.1. (t» , 1. aUi ( ) P'I a~ - P2 o~• t- .

b) Necessary optimality condition in 't

i

U.('t.) 1. ,1. where 1l
[

As far as. formalization is concerned, some points must be paid attention to. These concern the setting of the integral cost criterion, the generation of time-series to represent the availability of nuclear plants an~ above all the use of optimal control of impulses which seems to us particularly well' adapted. This method has only been applied to a few numerical problems of small scale and this work is probably the first where it is used in an economical matter of such importance.

1. • .. 0 "U.(t)dt - A.) 1. 1.

o

o.1.

~

0

CONCLUSION The problem studied in the present paper raises similar difficulties, to those that have already been mastered in the optimal operation of the hydraulic reservoirs. These difficulties concern essentially, on the numerical field, the important number of variables and'the multiplicity as well as the nature of the constraints.

[11 Bucher, Falgarone, F. and P. Lederer

(1978).

Optimal Operation of the large French hydropower system. Sixth Power Systems Computation Conference, IPC Science and Technology, p 568.

(2] Young, L.C. Calculus of variations and optimal control theory W.B. Saunders Company. Philadelphia, London, Toronto.

APPENDIX Optimal Control of Impulses in a Deterministic Process [2] Let ,.(i = 1 to n) be impulses; between the impul~es, the process is described by a differential equation :

Let a. be the discontinuity induced at time 'i. Tlien : (2)

X(,~) - X(,:-) + a. .• 1.

1.

1.

The problem is

+ .

_1: + X(, .) ~ X(,.) 1.

I

1.

K('.'X:-'X~»). 1.

1.

1.

Maximum Principle

with the adjoint system :

1/I(T) - 0 1/I('i) -

and

1/1«) (i - 1 to n).

The optimization in ,. is unconstrained. Therefore it is suffi!ient to know the gradient with respect to this variable •.

~ + + + dK - + - H(,.,X('.),1/I('.),U(,.»+ -"=-('.,X.,X.). 1.

1.

1.

.

1.

0·,.

1.

1.

1.

1.

239