Uncertainty in operational planning of power system generation

Uncertainty in operational planning of power system

generation Z Glanc

Computer Centre of Power Systems, Skr. 143,00-950 Warsaw 1, Poland

Short-term scheduling o f a power generation system is discussed with regard to the probabilistic character o f the load demand and the effects o f generation capacity outages. The probabilistic method and a practical algorithm o f optimal unit commitment and load distribution among power stations is presented. An example o f the computations performed for a large power system is given.

I. Introduction Power system operation is usually planned by solving a large and complicated optimization problem. In order to solve such a problem practically, various assumptions are necessary, the deterministic nature of parameters being one of the most important. In reality, however, some of the parameters or relationships are not known with absolute certainty and some parameters are predicted values biased by diversified random factors. Therefore, reality should be based on stochastic or probabilistic optimization models. Numerous papers have discussed the problem of data uncertainty. The References section is not comprehensive as a list of contributions but can be regarded as a set of illustrative approaches to solving the problem. References 1 and 2 deal with the theory of stochastic optimization, while References 3 - 7 are proposals for solving actual engineering tasks: however, accepted assumptions and formulated models vary with the structure and control policy of the power systems under COlisideration. In References 3 7, one or more random factors have been introduced, but studies in the latter case are restricted to small problems. Quoted contributions illustrate distinctive methods used to solve tire optimization problem including: an equivalent deterministic taska: stochastic approximation4: segmented linearizationS: an iterative procedure with deterministic optimization task and stochastic calculation of a spinning reserve6: and a Monte-Carlo method v. At present, there are no general methods or sufficiently effective algorithms that allow use of probabilistic methods in a large power system dispatch. The present paper outlines a model of. and a method of solving, the daily load planning problem for a system with a majority of thermal power stations 8. All random factors that significantly affect actual power system operation have been taken into consideration. A set of probabilistic calculations for an actual power system makes it possible to advance the simplified

Vol 2 No 2 April 1980

method of practical calculations for daily dispatching practice.

I1. Probabilistic model I 1.1 Statement of the problem Consider a power system with about 95% of the total generation capacity installed in thermal power plants. A small proportion of hydro power plants justifies the following method of daily load scheduling. Hydro plant operation is arbitrarily resolved and hydro power generation is subtracted from the predicted total power demand. Next, assuming a certain spinning reserve (established from engineering experience), a plan for thermal plant operation is specified. Planning is based on the solution of a deterministic task, which minimizes daily production costs. After applying time discretization, optimization is performed, step by step, for respective time intervals, rather than establishing a functional optinmm for the [0, T] period 9. In practice, for a definite time interval, the optimal commitment of a running unit is first obtained (by the confine times method), and then the optimal load schedule is calculated (by forming the Lagrangian function). Solution of all the examined time intervals defines an operation plan for thermal power plants.

During operation, however, task parameters usually differ from those assumed in the planning stage because of random factor actions, so that under actual conditions, the plan may not be optimal. In order to make it so, the plan must be adjusted for real conditions and the working unit loads must be operatively altered. In practice, this is done either by appropriate automatic control actions or by dispatcher decisions. However, unit commitment adjustment is limited by the time necessary for unit startup. Statistical analysis of past events shows that the total cost for real conditions can often be minimized by planning different unit commitments from those established by commonly used deterministic planning procedures. The question arises as to how the planning of unit commitment can take into account the random nature of predicted and planned parameters. In order to answer this question, a considered optimization task must be treated as the stochastic process. From theoretical and computational viewpoints, it is possible and advantageous to introduce certain simplifications into the problem. The assumptions s enable a set of stochastic process realizations in respective

0142-0615/80/020081--06 $02.00 © 1980 I PC Business Press

81

time intervals to be examined, instead of the stochastic process itself, so that the planning problem becomes a question of solving series o f probabilistic optimization tasks.

Note that with the help of the random variable U the probabilistic character of the set L units and vectors X m, XM(unit breakdown and random reductions of their generation) is modelled.

11.2 N o t a t i o n

Assumptions (a) (e) have been tested by statistical analysis and numerical calculations for the system under consideration.

[0, T] ti

examined time period (days) ith time interval; T = ~,t i i

For a given time interval-~ :

II. 4 M a t h e m a t i c a l f o r m u l a t i o n o f a p r o b l e m

L N

Consider the optimization problem for a given ith time interval. To solve the problem of finding the optimal load vector under specified q~th unit commitment and under certain realization z of the reduced power demand Z:

#

X, X m, X M

number of thermal units number of all possible unit commitment ,~ variants: N = 2 number of unit commitment variants number o f optimal variants obtained by deterministic task solution predicted total power demand planned hydro plants load column vector of a thermal unit load, vectors of capacity limit the units respectively, dimensions: L x 1 ' auxiliary diagonal matrix of z e r o - o n e elements, dimension: L x L.

(1)

Z=G-W

(Note that G and W are random variables ~ Z is a random variable.) The problem can be formulated as follows: ~(e: min fl(X)

X~B e

Be:{ X ' J X = z

Elements w# in matrix f2 allocate working unit variants: 1 ifjth unit incorporated =

wi]

where / = 1. . . . . L 0 otherwise

(2)

~2~,Xm ~< X ~< fZeXM where Be defines a set of feasible solutions and fx is a generation costs function which is defined in Appendix I. *

J = [1 ..... 11 auxiliary vector, dimension: 1 x L E(Y) Oy

expected value of Y standard deviation of Y

Solution Xe(z ) of equation (2) yields the actual load vector for a system with no outages. If, however, power outages u occur during system operation, then vector ) ~ ( z u) :~ ~ ( z ) is realized. Total unit load will then decrease to value a e, and a power deficit d e may result. There are

11.3 S i m p f i f y i n 9 a s s u m p t i o n s It is assumed that:

(a) the time period [0, T] is divided into intervals, in which deterministic parameters are constant and random parameters have a constant distribution function (b) reactive power and network losses are neglected (c) thermal unit startup and shutdown losses are linearly dependent upon unit standby time (d) in a given time interval: (i) G, the total power demand, is a random variable with known distribution (G = g + ~g, where g is the prediction and ~g is the random component); (ii) W, hydro plants load, is a random variable with known distribution (IV = ~ + ~w, where ~ is the planned load and ~w is the random component); (iii) U, over random generation capacity losses, is a random variable with known distribution (U= t~ ÷ ~u, where ~ is the prediction and ~u the random component) (e) random variables ~g, ~w ~u are mutually independent. "~Subscript i has been omitted except where essential, i.e. X = Xi, L = L i, etc.

82

(3)

J)~e(u, z) = J~e(z) -- ae(U, z)

and the power balance resulting under such circumstances will be

J:%(u, z)

+ de(U, z) = z

J~e(z)

ae(u , z) + de(u , z)

(4)

or

--

=

(5)

Z

Denoting capacity limits of the system for the q~th variant by M~n = J~2eX ra and M~ t = J ~ e X M respectively, the decrease of the load and power deficit can be specified as follows: "0

if

M~

u

if

z >M~

if

M~>~z>M~

a¢ = u - ( M y

z)

u '~< ~. z ~
.'

(z M~) i~ ~ z < M ~

.u'

if

z
0

if

m~-u'~z~M~

z

(M~.)

z

(M~

z
u

(6)

.'

i~ z > M ~

u') ~

u

u

(7) .'

Electrical Power & Energy S y s t e m s

where u' is the random reduction of the minimum capacity limit of the system: u' = f(u). In determining costs for a given z and u in the case of a power deficit, both production costs and financial losses due to deficit should be taken into consideration. As can be seen from equation (5), general costs s~ can be represented by: s, = f , ( ~ O )

0 z

(8)

f~(ao) + fB(do)

*

where f~(Xo) is the production cost for a system with no outages;f~(ao) is the cost reduction due to the decrease of production in the case of outages; and f~(d4~) is the loss due to power deficit. (Functions f~, fz, f~ are described in Appendix 1.) Let ?~ be a continuous random variable, the realizations of which are the solutions of problem (2) formulated above for realizations of the continuous random variable Z: ,

nence analysis of the daily power demand forecast; analysis of divergence between actual and predicted hydro plant load;and pertinence analysis of the forecast of power outages in daily planning. For the power system under consideration, variables G and W can be approximated by normal distribution. From equation (1), Z is derived as a convolution of variables G and W: ~ =~ - w; = (Oo~ "{- O2W) 1/2.

In practice, for a given confidence level, variables Z and U are approximated by restricted random variables which have discrete distributions of probability. A substantial part of the adopted algorithm is a reduction in the list of possible unit commitments proceeding from the list 05~ {1, . . . , N } up to the list 05~ {1. . . . . N'},

N'
*

x~ =

{X~(z)}~e_z

From equations (6) and (7), Z and U are random variables = A ~ , D o are random variables. From equation (8), ?~, A~, D~ are random variables = S~ is a random variable.

a is the curve of the system production cost for unit con> mitment 05 = ~2), which is the optimal unit commitment obtained from deterministic calculation of the expected value of power demand t

*

Therefore the average sum of costs for 05= constant can be obtained as an expected value of a variable S~:

t

i

b is the curve for 05= ~ z ), z = 2 - ~z c is the curve for 05= ~(z"), z" = e + ~

E(S¢,) = E{f, [?~o(Z)]} E{f=[Ac~(Z, U)]}

+ E{f~[De~(Z, U)]}

(9)

Thus the optimization task can be formulated as follows. Determine a variant of unit commitment so that the expected sum of costs is minimized:

~"

,rain ~tl

.....

=

E [S(05)] = N}

.....

A l'. 2', 3', 4', 5' is the curve of optimal systems production cost in the [z m, z M] period of power demand B is the probability distribution of the forecast of power demand (in practice, curve A can be defined with sufficient accuracy)

N}

[z m, z~ ] is the period obtained from the analysis of the technical condition of the system under consideration.

i * min,x}E~f~[×(05, Z) ] -f:[A(05, Z, U)] e~{~ ....

+ f3[D(05, Z, U)]} III.

E[S,(05)-S~(05)+Sa(05)]

min ~{1

d is the curve for 05 = ~(z"), z " = z" + ~;"

(1 O)

In this case, {1,

.,

.,N'}

=t*

*

/

~05(2), 05(z ), ~(z"), ~(z")}

Algorithm

The integer character of the problem formulated above determines the solution method. The simplest approach is an exhaustive review of possible unit commitment variants and a comparison of expected sums of the costs. However, since the size of the task for an actual power system is very large, such an approach is practically impossible. The adopted solution strategy s is a review of the reduced list of unit commitment variants. As can be seen from equation (9), the expected cost value for a variant 05can be determined by calculating expected values for respective components. In order to calculate the, first component, it is necessary to solve the optimization problem (2) for every realization of variable Z; to calculate the second and third components, the distribution of variables Ae and D e as a convolution of variables Z and U must be defined. The parameters of variables G. W and U (distributions) for a given time interval are obtained on the basis of statistical analyses of past events. The following are necessary: perti-

Vol 2 No 2 April 1980

/b 3

J x

.

d

?)

.~

"

5'

g

I

Z m

Z~

~

Z"

I

¢,

Zm ZM

Z

Figure 1 System production costs

83

Note that curve A is the inferior limit to the full set of system production costs, and that the set of curves (a- -d) is a finite sequence. Proceeding from 4>~ {1. . . . . N} up to 4>¢ {1,..., N'}, the optimal solution will not be omitted. The number of reviewed variants is then significantly reduced. Computations can be carried out as follows:

ecouomic load distribution is applied. Hence the oomph program essential in computations is a combination .d modified programs and those already tested and in us,

IV. Numerical example A thermal power station operation plan is specified fl~r ~ evening peak load in a working day. Task parameters hay, been derived from the real system data.

*

(i) The first variant is selected as 4>= 4>(f), and the next as 4>'= 4>(zt), where 1 = 2, zt = f + ~z(l). *

(ii) Vectors ~e(Z) and ~ d ( Z ) are obtained by solving problem (2) for 4>and 4>' respectively for all zi, i ¢ {1. . . . . n}. where n is the number of intervals for variable Z.

IV.1 Input data

(i) There are 139 units in 63 thermal power plants readx. T,, generate (L = 139). (Combined generation and heating power stations are substituted by equivalent, one unit characteristics.) The total capacity of all units is 15 740 M~. (ii) G : g = 15 500MW, Og = 295 MW:

(iii) Variables A~,, Ad, D,~, De,, are calculated according to equations (6) and (7) as a convolution of variables Z and U (see Appendix It). (iv) Expected cost values E(Se), E(Sd) are calculated in accordance with equation (9) (see Appendix III). (v) E(Se) and E(So,) are compared. Three results of the co:nparison are possible: E(Se') < E(S~). Therefore, the next variant is selected in the same direction of a coordinate ~z from the variant 4>'; in a new cycle of computations: !

t

*

4> = 4> ; 4> = 4>(21+1), 2l+ 1 = Zl +

~z( l + 1), l =

l +

1

W : # = 1 000 MW, o~, = 60 MW, hence Z : e = 14 500 MW, Oz = 300 MW. Distribution of Z is approximated within the confidence interval 0.997 by 65 discrete intervals. Distribution of U is approximated by 35 discrete intervals as is shown in Figure 2, in which ~u is expressed as a percentage o f M ~ . (iii) The set of generation cost characteristics is given as follows:

Calculations in steps (ii) (v) are repeated for new 4>'. E(Sd) > g(se). Therefore the next variant is selected in the opposite direction of a coordinate ~z from the variant 4>': in a new cycle of computations:

4>=4>:4>'=~(zt+O,z,+~=z,

~z(l+ l ) , l = l + l.

Calculations in steps (ii) (v) are repeated for new 4>'. E(Sd) < E(S~,) and the direction of a revision of variants has been changed earlier, or E(Sd) > E(S~,) and the list of variants is exhausted. This is the end of calculations.

k i =[(xj):eq;xj:x~n<~xj ~ = 1, 4>= 4. Note that:

(vi) The optimal solution is obtained as ~:min E [S(4>)].

rg = sos MW (3.S% (e)):

The algorithm may be illustrated by considering the example shown in Figure 1.

r~, = 913 MW (6.3% (2)); F

*

s4

¢

Assume that E{S[4>(z )]} > E{S[~(e)]} > ElS[~(z"')]} I * tl > F,{S[¢(~ )]} and the computations are beginning for

ii~ ! * t 4>= 4>(~), 4> = 4>(z )..Therefore: in the second cycle of computations, 4>= ~(y), 4>'= $(z"); in the third cycle of computations, 4> ~(z"), 4>'= $(z'"); the optimal solution is obtained in the third cycle:

~ = $(:") The proof that such an approach ensures rapid convergence of the algorithm has been discussed a. Note that to obtain the whole list of possible variants 4>, the deterministic program of unit commitment is used; and for calculation of the load scheduling (problem (2) for a given 4>and ai) the commonly used deterministic program of

84

i

~oi

~o~ ~ 50~i "~ ~ a. ~

%

I

40 :

~o~

20F

/

I0~ L

-4

Figure 2

~

-5

2

l

I

0

~.

2

5

4

5

6

7

8

Distribution of random capacity losses

Electrical Power & Energy Systems

~" *'/~':

----->~r'~ ~

-~

~

%Z o Figure3

,4

~

kj(xl)

~2

k~ (x2)

x3

k~(x3)

x4

kj(x4)

x5 x6

7

x? ~ ~;

xl

k~(xS) kflx~)

x~

Cost characteristic o f j t h unit

Ar = 6.3

3.5 = 2.8%

V I . Conclusions To comprehend uncertainty of input data in power system operation planning, it is necessary to formulate and solve probabilistic optimization tasks. Compared with planning based on deterministic models, probabilistic planning is more time- and effort-consuming. However, analysis of the problem and the presented example demonstrate that taking random factors into account can yield significant economic gains. It seems possible and advantageous to use the simplified method and algorithm presented in daily operational planning. The method incorporates the random nature of the parameters of the task and does not unduly complicate planning procedures already in use.

(z);

E[S(~)] = 2 081 252 ncu/h:

V I I . References

E [S(~)] = 1 992 475 ncu/h; a E = E[S(4;)

1 Hadley, G 'Nonlinear and dynamic programming' UK (1964)

S(4)1 = 88 7 7 7 n c u / h .

The total computation time needed to calculate ~ is 19 rain on a CDC 3 170 computer (data preparation and estimation of variable parameters are not included). Future improvements in the currently used experimental program will reduce this time.

Krumm, L A 'Conjugate-gradient methods for a control policy of power system' Novosibirsk (1977) (in Russian) Valdma, M H 'The uncertainty in one-stage optimal scheduling of power systems' Moskow (1977) (Reprint, in Russian)

V. Simplified method to comprise random factors On the basis of a series of experimental calculations, a simple method is proposed to adjust daily optimization computations.

4 Agarwal, S K 'Optimal stochastic scheduling of hydrothermal systems' Proc. lEE Vol 120 No 6 (1973) Muckstadt, I A and Wilson, R C 'An application of mixed-integer programming duality to scheduling thermal generation systems' IEEE Trans. Power Appar. & Svst. Vol PAS-87 No 12 (1968)

Periodically repeated probabilistic computations according to model (10) are the basis of a plan (~) for each time interval followed by finding Ar = r~ --r~.

Dillon, T S, Edwin, K W, Kochs, H D and Taud, R J 'Integer programming approach to the problem of optimal unit commitment with probabilistic reserve determination' IEEE Trans. Power Appar. & Syst, Vol PAS-97 No 6 (1978)

For a specified daily time interval, value Ar determines the modified spinning capacity reserve: k z ' = Az + At, where Az is a spinning reserve commonly used in deterministic calculations. •

A system operation plan is set up using a deterministic model in which power demand for each interval is computed as follows: E = ~

Filippova, T A and Schalnev, W G 'Optimal probabilistic unit commitment for the power station taking into account uncertainty of the data' Isvestia Syb. Div. A.N. No 3 (1975) (in Russian)

W + ~2'

Tire above procedure does not complicate a daily planning cycle and affords certain economic benefits.

Giant, Z B 'Optimal probabilistic short-term scheduling of power system_generation CIEEA Report Warsaw (1979) (in Polish)

To establish the repetition rate for the probabilistic computations, it is necessary to perform a statistical analysis of the stability of the random parameters in the definite tinre intervals.

kazebnik, A I 'Branch and bound method approach to optimal unit commitment' Kischinev (1970) (in Russian)

Table 1 Computational results for five variants ~

M•

1 2 3 4 5

11 12 12 12 12

M~ 973 094 182 219 376

Vol 2 No 2 April 1980

15 15 15 15 15

005 193 328 413 664

re

E(S,)

505 693 828 913 1 164

1 957 1 962 1 969 1 972 1 997

315 707 164 717 741

E(S2)

E(S3)

E(S)

14 139 7 248 4406 3 214 1 297

138 61 33 22 6

2 081 2 016 1 998 1 992 2 003

076 082 637 972 985

252 541 395 475 429

85

Appendix I

Deficit costs

Generation cost

It is assumed that the cost characteristic of the/th unit can be represented by a set of convexivity curves and that the probability p that each of them will occur is known. Therefore, the expected cost curve can be calculated as shown in Figure 3.

Estimates by various experts of f'mancial losses caused by a power deficit have given different results. If we acknow]edge that all estimates are equa]]y~ feasible, then equal probabilities can be attributed to relationships between losses and deficit depth. Relationship fa is therefore obtained as shown in Figure 4, where:

fa(dl ) = p'. •

v

v

.~.

vt

tt

r

In F~gure 3, ki(xl)=p .k~-(xl) p .k~.(xl);p,p probabilities Of curves k~,-k}'; and p' + p " = 1.

tv

f;(dl)

+ p " . f ~ ( a l ) + p". f~'(dl)

are the

p' +p" +p'"= 1

It is assumed (assumption (c), Section II.3) that B/= a/At/ where:B/is the startup and shutdown cost of the/'th unit; a/is the constant coefficient of t h e / t h unit; At/is the time during the [0, T] period when the jth unit is not scheduled for power generation. The point has been made 9, that such an assumption is admissible if At/<~ 20 h. Then, the total cost of production for L units is given by:

t

tt

/~t

t

t~

trt

p = p = p are the probabilities of curves f3, f3, f3 respectively.

Appendix

II

Convolution o f a random variable

U and Y are independent discrete random variables. L

P{U = ui}, P{Y = YJ} is the set of probabilities that yields

f , = ~ {k/[xj~#(4~)] + [1 - co//(4~)] c~/}

j=l

the distributions of variables U and Y. The cumulative distribution function of the random variable

Cost of decrease of production due to outages

f~ = X(~, z) a(~, z, u)

Z=U+Y

where X is the incremental cost of the power system defined as: ~

is obtained from the equation:

F(z)=e{Z
*

x - o~Arxrm,..~,z,,, 0[J~,C(z)]

~ P{U=ui}.P{Y=yl}, ui+yj
where the summation is realized for all i,/'.

~/

f'"~(dl]

)./~,,,

d=

f~(d~)

d2 d3 d4

f3(d2) f3(dg) f3{d4)

Appendix I II Expected value o f a function o f a random variable

Uis the discrete random variable. ~m

Y = f(U) is the function of U.

f~(dl', f"$(dl]

E(Y) = E If(U)] = ~. Pif(ui)

f'](dll

0

Figure 4

86

d=dl

i d

Relationship between deficit and financial losses

where Pi = P{U = ui}

Electrical Power & Energy Systems