Hydrothermal Optimisation by Refinement of Step-Loading Schedules

IFAC Symposium 1977 Melbourne, 21-25 February 1977

Hydrothermal Optimisation by Refinement of Step-Loading Schedules G.V. JONES Graduate Student, Department of Electrical Engineering. University of Tasmania. Hobart and

J.P. NORTON Senior Lecturer. Department of in Electrical Engineering. University of Tasmania. Hobart

SUMMARY A method is presented for the daily scheduling of hydrothermal power systems. Initially, hydro plant is committed by a step-loading schedule. Subsequently optimal load flow computations to schedule thermal generation and VAr allocation alternate with a gradient or Newton search to improve the hydro schedule. Realistic constraints are included. I

(a) transmission losses are incorporated by loss differentials, so the network must be separable into much simpler sections or must satisfy the assumptions made in deriving B-coefficients; (b) except in (Moskalev, 1963), only active generation is optimised, ignoring VAr distribution, because of the B-coefficient loss representation; (c) except for limits on active generation, no account is taken of transmission component ratings or security and stability limits. Unrealisable or unstable schedules may therefore result; (d) constraints on hydro plant discharge rates, in practice frequently state- and timedependent, are fixed or ignored. By arranging the optimisation around an optimal load flow (OLF) algorithm, the method presented here avoids these drawbacks.

INTRODUCTION

The increasing size and complexity of hydrothermal power systems have created a need for computerbased scheduling methods, while improvements in computing hardware and technique have brought optimal scheduling much closer to practical realisation. By optimal scheduling, we mean supplying all demands at minimum cost subject to physical, operational and statutory limitations. A family of problems distinct in character but interacting arises frum optimisation over a range of time-spans. The problems are instantaneous regulation, daily optimisation of the hour-by-hour schedule, medium-term optimisation, usually over a month or a week, annual optimisation and future planning over several years. The problems must be separated to limit computing requirements and to allow for differences in accuracy and detail of data. In annual optimisation, concerned with water resource allocation and maintenance scheduling, an accurate transmission model is superfluous as loads are not known accurately. In contrast, daily optimisation requires good load prediction and an accurate transmission model, but water resource allocation is largely specified in advance.

The inclusion of equality constraints describing the transmission system and inequality constraints imposed by equipment ratings in the minimisation of instantaneous operating cost was first treated in (Carpentier, 1963). The optimisation was posed as a nonlinear programming problem and a set of optimisation equations derived by Kuhn-Tucker theory. Several ootimisation technioues (Sasson, Vittori~ and Aboytes, 1971) for thermal power systems are based on this formulation.

In a hydrothermal power system, if the availability of plant is fixed the only cost directly scheduledependent is thermal fuel cost. However, hydro generation is limited by stream flows, storage and hydraulic interaction between plants, while fuel stocks for thermal generation can usually meet any short-term schedule within plant limits. Thermal generation may be scheduled on an instantaneous basis, but hydro or hydrothermal scheduling is a variational problem which must consider system dynamics and water use over a finite period.

Hydrothermal scheduling may be either split into two problems with independent goals or treated as an integrated problem. As separate problems, the hydro subsystem is scheduled once and for all to minimise e.g. total hydro plant losses or estimated cost of supplying the remaining demand thermally, then the best operation of the thermal plant and electrical network determined with reference to thermal generation cost. The schedule determined in this way is not in general optimal, but may be near-optimal when hydro-generation and the thermal generation near load centres form two distinct geographical groups.

Water inflow and load variation are stochastic, but in daily optimisation they may usually be regarded as deterministic without incurring significant penalty. In some cases very-short-term schedule updating to compensate for forecast errors may be worthwhile. This paper treats the deterministic daily problem, taking bus loads and reservoir inflows as known in advance. The problem has been tackled by several methods (Dillon and Morsztyn, 1972) including classical Lagrange mUltiplier theory, dynamic programming, gradient methods and Pontryagin's maximum principle. Solution methods proposed so far have many shortcomings , the most serious being that

Because the hydrothermal scheduling problem is variational, an integrated solution for a system of realistic size involves a very large number of variables (Dillon, 1974) and inequality constraint~ resulting in dimensionality difficulties. Alternatively, the integrated problem may be solved via a sequence of lower-dimensional problems, alternately rescheduling only hydro and only thermal plant, both with reference to the same goal. Our approach is to schedule hydro plant for maximum efficiency by a constrained global search, establi-

242

give new slopes. Reservoir depletion constraints are incorporated easily as shown below, and the OLF calculation observes all electrical constraints, so the optimisation may be stopped at any step to yield a feasible schedule.

shing unit commitment and providing a good starting schedule. A more accurate solution is then obtained by alternating OLF calculations to schedule thermal generation, VAr allocation and regulator s e ttings, observing equipment ratings and imposed l i mits, with a gradient or Newton search to improve th e hydro schedule. Computational experience so f a r suggests that this scheme will converge to the optimal schedule of the integrated system in spite of local minima characterising such problems. 2

The hourly discharge rates uk, k = 1,2, ... , K determining th e power flows-at the coordination busbars mu s t give the specified water use K

k~l ~k

PERFORMANCE CRITERION

(1)

The m1n1mum total cost, f, may be found by adjoining the total discharge constraints with Lagrange multipli e rs ~, giving

An overall performance criterion for a hydrothermal svs tem requires either that the reservoir depletion a t all hydro plants be specified in advance by a l onger-term schedule, or that a cost be assigned to wa ter wherever reservoir depletion is not specified. Th is cost has two components. One, reflecting the f uture value of water, depends on medium- and longt e rm stream flows and power demands. The other, i nd icating the short-term cost of supplying power fro m thermal sources instead, can be calculated for any plant and any time as a Lagrange multiplier in t he course of optimising the thermal schedule with hydro generation temporarily fixed. The longert e rm schedule could be modified in the light of such short-term costs. To avoid encroaching on the l onger-term scheduling problem, we shall consider as optimality criterion the minimisation of thermal cos t with specified reservoir depletion (Anstine and Ringlee, 1963). For a hydro system with tieline co nnections to adjoining systems, the object would be to minimise the cost of power purchases (Sokkappa, 1963). An isolated purely hydro s ystem mig ht use the criterion of minimising long-range sto rage depletion with specified depletion from its smaller pondage. Correspondingly, tielines or spe cified large-storage hydro plant would provide ' s lack-bus' powers and be scheduled in the same way a s thermal plants.

T

L

=

f

+~ (

K

L

~k - ~)

(2)

k=l then setting Vu L k

= gk +

~

= 2

k=1,2, ... , K (3)

where gk is the gradient of f with respect to ~k' At the-optimum, the co s t:discharge rat e gradients are the s ame for e ach time interval. In principle, th e optimum can be found by iterating A, solving (3) for uk a t each time and adjusting A until (1) is s a tisfied. In prac tice, efficient adj us tment of the dis c harge rates requires secondderivative information, obtained by further OLF solutions for small changes in coordina tion powers. Even then, to a vo id ite r ative adjustment of ~k' inver s ion of the Hessian ma trix for each k is required at every trial A unl e ss some simplifying a ssumpt i on can be made. -A simplified second-order method which calculates A in one step wi thout matrix inversion is de scribed later. Alternatively , A need not be found explicitly if (3) is rewritt e n IS g . = g/ - g. = 0 k , j = 1, •.• , K ( 4 ) ~ ~ For any choice of j, the discharge rates may be adjusted by a hill-climbing technique in conformity with (1) to bring th e c onstrained gradient s ~ to zero. The choice of t ec hnique is narrowed by the need to limit th e numb e r of OLF computations and to exploit known characteristics of feu). In particular, f is the sum of independent costs fk(~k)' near -quadratic and with strongly diagonal-dominant Hessians. Gradient and Newton hill-climbers taking advantage of these fe a tures have been examined.

From here on, we consider hydrothermal systems. 3

~

..::.b.

COORDINATION OF THERMAL AND HYDRO SCHEDULING

The electrothermal subsystem is represented by st atic load-flow equations and the power:cost-rate ch aracteristic of each thermal plant, with inequality constraints describing equipment ratings and st abilit y margins. Efficient computational techn i ques have evolved for OLF computation, optimising th ermal generation, VAr allocation and regulator se ttings for instantaneous loading conditions (S asson, Vittoria and Aboytes, 1971). Domme l and Ti nney (1968) use Newton's method to find the power flo ws, with first- and second-order gradient adjus tments of controls and external penalty factors to enforce inequalit y constraints. Dual variables a re generated, i.e. incremental costs of changes in co ntrols like thermal generation, bus volt age s and phase-shifter power flows. It is easy also to ge nerate dual variables for quantities which remain fix ed in the static optimisation, in particular power inflows at buses joining hydro plant to the r es t of the system, the coordination busbars. These du al variables interface the hour-by-hour OLF's to dynamic optimisation of hydrogeneration. Other OL F methods differ in how the y adjust controls and en force constraints.

In the gradie nt me thods, the step-length is calculated from se cond-derivative information yielded by an exploratory gradient step. A Taylor series for a change au in the ve c tor u of discharge rates in all intervals except j giv;s fl = f(u o + au) " fO + gO. dU + ~ duT.Ho. du (5) -

c -

c-

where g~ and H~ are the set of constrained gradients gkj-oVer the whole s c heduling period and the corresponding Hes s ian, a t the current discharge schedule u O• At the end of a gradient step 1

-

0

gc " g~ + Hc dU

0

=

0

(I - aHc)gc

(6)

If a is chosen to minimi s e fl , the new gradient is orthogonal to the search direction, so the optimum step size, a*, is given by

The hydro generation:discharge-rate characteristics co nvert incremental costs of power flows at the coordination busbars to cost:discharge-rate slopes aE ikl dUi for each hydro plant i and time interval k. Generation at all hydro plants and all times is va ried simultaneously using these slopes in a hillc limbing step. After each step, new OLF solutions

a

*

(7)

The denominator can be written in terms of either the gradient gg or the cost fO at the end of an exploratory gradient step of size a.

243

Using the gradient, (6) gives H~gg and a* = a oT 0 / oT (0 AO) (8) SC gc gc ~c - gc Alternatively (Adby and Dempster, 1974) from the Taylor series , 2(£0 _ fO + a·g oT .go)(a 2 c c

oT HO 0 gc c gc so that a*

a2

oT 0 /2(f0 gc gc

fO +

a gcoT gc0 )

then substituting in (3), solving for uk' summing over K and inserting the discharge con~raints gives u

(9) (20) As they stand, these expressions are notencouraginL involving the inversion of K+l matrices of the same dimensions as the number of discharge rates Nand K+2 matrix-vector products. However, the offdiagonal terms of the Hk's are often small. Replacing each Hk by its principal diagonal makes the inversions trivial and the second derivatives then require only one extra OLF solution per time interval. The expressions simplify to

(10)

Rather than minimising f directly, gradient T searche s can equally conveniently minimise gc.gc' The optimum is then known to be zero, provi~g-a measure of progress, albeit not in terms of the quantity ultimately of interest. Since Hg is symmetric, from (6)

= 2ag~T H~T H~ g~

(g!T. g!)

2 oT HO 0 gc c gc

K

T

~.gc

b ik h

2hiik

or a* -

, 0T 0) 2(f'0 _ fO + ag c gc

(13)

K

K b'k K -2(q . + E-~-)/ [_1_ 2 ~ k=lh iik k=lh iik

(14)

K

L.: {( gk -

L

k=l j=k+l

_

(15)

...:.J.

Minimising S by grad i ent steps gr

gOr - a HOg r r-E.

r=l, 2, ... ,K

(16)

in the unconstrained gradients, the optimal step lengths are given by

as

Table I:

aa r

= 0 at a

r

a

r

*,

r=l, 2, ..• , K

i=1,2, ... ,N (24)

Hill-climber performance was tested by Monte Carlo simulations, each of 16 4-step runs to refine 6hour schedules for 4 hydro stations. Deviations of the cost:discharge-rate relations from quadratic form were simulated by perturbing every exploratory cost and slope evaluation so that fO - fO and gg - gc were uniformly distributed between l±R times their correct values. The accuracy and robustness of hill-climbers A to F, using equations (8),(10),(13),(14),(22) and (23) respectively, were compared for a range of R. The starting schedule used generation at the hydro station with largest specified discharge to take up all the demand variation. Table I shows results for perfectly quadratic costs.

T

g.) (gk - gJ')}

iik

i=1,2, ••. ,N; k-l,2, ..• ,K (23)

where

A more symmetrical function of the gradients, whose minimisation satisfies (3) while treating all time intervals identically, is S =

(21)

may be minimised by sett i ng

Ai --2- -

which ' 0) gc

i=1,2, .•. ,N

i=1,2, ... ,N; k=1,2, •.. ,K (22)

(ll)

(12)

a*

1

and

Similarly,

* _ oTHO 0 / oTHoTHo 0 ge c cgc - gc cgc

K

b' k

-(q'+L-~-)/ E ~ k=lh iik k=l hiik

so the step size minimising g~T.g! is a

(19)

where

The choice between these two expressions for a * depends on their relative susceptibility to quadratic-approximation errors and the expense of computing gradients as well as costs in the exploratory OLF.

a:

_~l(~k + A)

z

k

(17)

Method

1

A,B C,D E

32.14 36.20 22.34 2 . 77

F

These equations are linear in the unknowns ar,* and all the Hg terms needed are supplied by Knew 0LF solutions-after exploratory steps, the same number as for Hgc in the previous methods. Separate step lengths - for each time interval quicken the minimisation at the expense of inverting a KxK matrix at each step, and, perhaps more important, obtaining schedules which do not obey the discharge constraints until the hill-climber has converged.

COSTS ACHIEVED WITH R ZERO Cost after step 2 3 20.22 23.95 0.55 0.08

14.19 17.03 1.28

4 10.26 12.21 0.02

For moderate perturbation, gradient methods A and C work well but Band D do not. The reason becomes clear on examining the expressions for a*. In B and D perturbatio~ of fo in , the ill-conditioned small difference fO - fO + aggTgg of relatively , large numbers is d isastrous ~or-some step s izes a. In A and C, comditioning is much better and unaffected by virtually identical results being obtained over several decades of i. Method A is rather better than C for R below about 0.5. Above 0.5, the positions are reversed. Careful choice of a gives good results with Band D, but they were rejected as too vulnerable. For R below about 0.2

a,

An attractive alternative is to find the optimal discharge rates in one step by Newton's method. If , T T fk = ;: uk Hk u + bk u k + c k (18) k

244

If the plant operates only at two discharge rates, the average generation and average discharge rate lie on the straight line joining the two corresponding points. If Oa is tangent to the one-unit curve and bc is common tangent to the one- and twounit curves, energy per unit discharge is maximised by step-loading between 0 and a or between band c or discharging continuously at a rate between Ua and Ub or Uc and the upper limit. The average characteristic is the broken line Oabcm. Although ideally the number and timing of switchings do not affect hydro efficiency, the switchings must be timed to minimise thermal operating costs while leaving some thermal spinning reserve to cover forecasting errors.

and 0.5 respectively, methods E and F remain much pr e ferable to A and C, but larger perturbations cause the Newton methods to break down before the gradient techniques. They misbehave badly by R= 0 .8, where A is adequate. Method C performs vi r tually as well at R=1.2 as at low R. Table II summarises these results. Table II: Method

R

COSTS IN PERTURBED MINIMISATIONS cost after step 1 2 4 20.15 9.83 7841 1176 23.10 9.90 45.20 43.65 9.86 1. 47 0.59 0.01

A B C D E F

32.55 37463 36.71 47.92 30.34 7.93

0. 8

A C F

33.94 19.90 10.99 37.92 22.15 8.05 48.78 51. 36 21.83

1. 2

C

39.85 22.67

0. 4

7.94

s.d. of cost 1 2 4 0.3 0.9 0.5 >10 5 >10 4 3768 2.2 1.5 2.1 13.1 34.6 53.1 13.8 28.3 3.5 4.2 0.6 0.01

To reduce the dimensions of the preliminary scheduling problem, (Mantera, 1972) allowed each plant at most two discharge rate changes. As in the fourhydro-plant example of Figs. (2) to (4), such a schedule is generally less efficient than if either discharge rate is allowed in each interval, subject to the total discharge constraint.

1.7 1.7 9.1 4.0 2.6 2.3 43.9 110 78.0 3.6

5.1

2.1

To check the effects of diagonal dominance of the cos t Hessians, the cross-product terms of each f k( ~k) were halved, adjusting the line ar terms to kee p the same schedules optimal. The gradient methods are scarcely affected by the change. Predic tably, the Newton methods improve, roughly hal vi ng the cost after one step of method F, with l ar ger reductions at later steps. Method E now performs almost as well as F until the onset of di vergence as R is raised. Both methods now dive rge at R just below 0.8. The gradient methods remain more robust, but less efficient at low R. 4

Total load

400

PoW@!" (MW)

300

COMMITMENT OF UNITS IN HYDRO PLANTS 200

For the hydrothermal optimisation method to be effective, the costs must be locally convex. As generation:discharge-rate characteristics for multiunit hydro plants have discontinuous slopes, to ens ure convexity a preliminary schedule must fix subsequent hydro unit commitment for each time interval. A suitable and commonly-used hydro s ch eduling technique is step-loading, restricting the discharge rate of an m-unit plant to m+1 levels corresponding to efficiency peaks. With predetermi ned daily reservoir depletion, operating each hyd ro plant optima11y without reference to the others gives the overall optimum in the ideal case where transmission losses vary negligibly with generation pattern, thermal cost characteristics ar e linear, and each reservoir has negligible head va riation. A single plant operates optimal1y by st ep-loading between two discharge rates. To see why , consider the two-unit characteristic in Fig. (1). 60

one

E

~------1

Fig,. 2

8 12 16 Time (h)

20

Plant

4

')J.

Optimum step-loading schedule of the four-hydro-p1ant system

!'i)0

400

Total load

Power (MW)

300

40

Power

200

(MW)

••• • •••••••••1 .-_,

.... , i ~ " "~'-' I .

i

. . __ . ...J 20

L •••••

r'-:I

.

! j I L ...J I

Total hydro generat Ion

.

I

Thermal

L_ generation

Hydro step modes t--------lIPlant 1 1-------J·Plant 2 Plant 3

o Fig. 1

----I~ Plant 1 1--------1. Plant 2 1---1 Plant 3 4

two units

unit

Hyd"o step mOC»1

100

Lb

Ub Discharge rate

t:=!_-!:=::;:::=;:::::::;;::~I Plant 4 4 8 12 16 20 24

Urrox

Tinw(h)

Generation:discharge rate characteristic of a two-unit hydro plant

Fig. 3

245

Best step-loading schedule obtainable using Mantera's method

5

3

4

any stage and still provide a feasible schedule. Sensitivity information, useful in longer- and shorter-term scheduling, is generated in the course of the computation.

6

500

400 6

The co-operation of the Hydro-Electric Commission of Tasmania is gratefully acknowledged.

Power (MW)

300 ············1..... _.............

200

Total hydro generation

!

. .,

. .·--1____ _ 7

L.-,

L -'"""l..... .. _

Thermal generation

~~~~--~--T--'--~

Fig. 4

8

12 16 Time (hI

20

Hydro step Plant 1 Plant 2 Plant 3 PICJ1t 4

ANSTINE, L.T. and RINGLEE, R.V. (1963). Susquehanna river hydro-thermal coordination. IEEE Trans. Power Appar. Sys. Vol.82, No.65, April pp .185-191. CARPENTIER, J. and SIRIOUX, J. (1963). L'optimization de la production a Electricite de France. Bull. de la Societe Francaise des Electriciens. March pp.12l-l29.

24

Optimum step-loading schedule obtained using a reordered load curve and restricted switching of hydro plants

DILLON, T.S. and MORSZTYN, K. (1972). New developments in the optimal control of integrated power systems including a comparison of different computational procedures. Proc. Power Systems Control Conf., Grenoble.

Here Mantera obtains thermal costs of $29136, versus $28766 by the less restrictive method. More important, unit allocation found by Mantera's method may lead the subsequent schedule refinement away from the global optimum.

DILLON, T.S. (1974). Problems of optimal economic operation and control of integrated hydro and thermal power systems. Thesis (Ph.D.), Monash Uni~

A compromise which usually gives the same unit allocation as the less restrictive method is to restrict switching like Mantera but reorder the loads hour-by-hour to form monotonic load duration curves. An important incidental benefit is fewer local minima to locate; local minima make a global search necessary in any case. Optimal step-loading schedules using reordered loads are characterised by each plant having at least one switching time in common with one or more others. Schedules constrained in this way require significantly less computation. The saving increases as the scheduling time-span is more finely divided. Switching times in a particular system may well be constrained further without endangering convergence to the global optimum. 5

REFERENCES

ADBY, P.R. and DEMPSTER, M.A.H. (1974) Introduction to optimization methods. London, Chapman and Hall.

• .--..._

100

4

ACKNOWLEDGMENT

DOMMEL, H.W. and TINNEY, W.F. (1968). Optimal power flow solutions. IEEE Trans. Power Appar. Vol.87, No.lO, Oct. pp.1866-l874.

Sy~

MANTERA, I.G.M. (1972). Optimal load scheduling of hydroelectric power stations. Thesis (Ph.D.), Univ. of Tasmania. MOSKALEV, A.G. (1963). Principles of the most economical distribution of the active and reactive loads in the automatically controlled power systems. Elektrichestvo, Vol.24, No.12, pp.24-33. SASSON, A.M., VITTORIA, F. and ABOYTES, F. (1971). Optimal load flow solutions using the Hessian matrix. IEEE Proc. PICA, pp.203-209.

CONCLUSIONS

The hydrothermal scheduling scheme presented here takes advantage of existing OLF and hydro scheduling techniques. Computation may be terminated at

SOKKAPPA, B.G. (1963). Optimal hydro-thermal scheduling with a pseudo-thermal resource. IEEE Proc. PICA, pp.354-360.

246