A nonlinear programming approach to optimal static generation ...

IEEE Transactions on Power Systems, Vol. 4, No. 3, August 1989

1140

A NONLINEAR PROGRAMMING APPROACH TO OPTIMAL STATIC GENERATION EXPANSION PLANNING Andr@s Ramos

Ignacio J. PCrcz-Arriaga, Member

Juan Bogas

INSTITLTO DE hVJ3TIGACION TECNOLOGICA Universidad Pontificia Comillas Albcrto Aguilera, 23 28015 Madrid, SPAIN Abstract - This paper shows how a standard nonlinear programming approach can be applied to solve a sophisticated version of the static optimal mix problem in generation planning. The solution of the static mix problem can provide useful insights if the underlying model is able to capture the relevant features of the actual system. The model presented in this paper considers technical minima of thermal capacity, detailed operation models of storage-hydro and pumped-hydro units, a realistic model of capital costs for hydro plants, operating reserve and minimum demand constraints, and also capacity already in existence. The model formulation is in a format that can be directly handled by the well-known MINOS code and can be efficiently solved. The use of a generalpurpose nonlinear optimization program results in a great flexibility in making possible to easily modify the model formulation or to adapt it to the particular characteristics of an electric system. A realistic application to the Spanish generation system is presented.

The present paper contributes to this body of technical literature in two respects. On one hand, it extends the modeling capability of previous formulations in several directions: technical minima are considered for the thermal technologies. detailed operation models are used for storagehydro and pumped-hydro technologies, and realistic capital cost models are employed for future hydro projects. On the other hand, the model has been formulated in such a way that a production-grade nonlinear optimization code such as MINOS [8] can be directly used, thus avoiding to resort to ad hoc solution techniques that are difficult to justify and do not easily admit extensions to the model, see [3, 4, and 61 for instance. The use of a optimization code makes the optimization process to be considered independently from the specific problem statement.

K c v w o r d s - Power system planning, Power generation planning, Power system economics, Optimization, Mathematical modeling, Expansion planning, Deterministic, Static.

In the next section the model that permits the computation of fixed and variable costs of the generation system is described first and then stated mathematically. Section 3 presents the nonlinear programming solution that has been adopted. The application to the Spanish system is discussed in section 4. Finally, section 5 summarizes the paper and presents the conclusions.

1. INTRODUCTION

2. TIIE MODEL

Static optimal mix models determine the best set of capacity additions to the present generation system, yielding the minimum total cost while subject to certain constraints, when only a prescribed future year is considered. The specific schedule for attaining the optimal mix is ignored in these models, which only examine the final outcome of the planning process. They lack the possibility of considering financial or other constraints that depend on the schedule of plant installation. The subjacent production cost model is usually deterministic. In addition, power plants are not individually considered, with a continuous variable accounting for the total amount of installed capacity belonging to each type of generation technology being used instead. Therefore static models cannot compete in accuracy with fullyfledged capacity expansion models, sec [ l ] for instance. However they are easy to use, require an small amount of data and, when they include enough modeling detail, they can provide good insights on the main trade-offs of the planning problem. Since the pioneer work in [2], several authors have dealt with the static expansion problem, with [3 to 71 being some references of interest.

2.1. Ouernting considerations Technical minima of thermal technologies are the minimum percentages of the capacities connected to the grid below which plants cannot operate. Technical minima are an actual limitation to the economic operation of the Spanish electric system, since they can be the cause of deviations of the generation dispatch from the strict economic order during offpeak hours. This situation is likely to occur in systems with a high percentage of base-load units without much loadfollowing capability. Installed thermal capacity is derated to account for any unavailability effects, both forced and scheduled, averaged over a one year period. In the absence of any restrictions, the optimal mix of thermal technologies is given by the classical relations in [2], which are intended to minimize the sum of fixed plus variable costs. Special circumstances, such as those mentioned above, may require a change in the loading priorities for thermal technologies, which is non trivial in general and will require some kind of optimization procedure.

Variable costs are zero for storage-hydro units, which therefore would tend to be base-loaded. However, hydro energy is limited by the availability of water inflows and these plants 8 9 WM 182-7 PWRS A p a p e r recommended a n d a p p r o v e d cannot generally operate at full capacity on a continuous by t h e I E E E Power S y s t e m E n g i n e e r i n g Committee o f t h e basis. Only a fraction of the rated power must be generated at I E E E Power E n g i n e e r i n g S o c i e t y f o r p r e s e n t a t i o n a t t h e all times, because of environmental concerns, irrigation or IEEE/PES 1989 W i n t e r M e e t i n g , N e w York, N e w Y o r k , other reasons. This must-run fraction is therefore treated as the J a n u a r y 29 - F e b r u a r y 3, 1989. M a n u s c r i p t s u b m i t t e d technical minima of thermal technologies. Run-of-the-river J a n u a r y 6 , 1 9 8 8 ; made a v a i l a b l e f o r p r i n t i n g hydro plants are handled entirely in this way. The dispatchable November 17, 1988. energy of storage-hydro plants must be used so that the variable operation costs are minimized. This minimization is 0885-8950/89/0800-1140$01 .00 0 1989 IEEE

Authorized licensed use limited to: UNIVERSIDAD PONTIFICIA DE COMILLAS. Downloaded on July 9, 2009 at 22:30 from IEEE Xplore. Restrictions apply.

1141

achieved when the hydro plants operate at full capacity, since in this form they do their best at replacing the most expensive thermal technologies, see Figure 1. As it is well known, the pumped-hydro technology is used to transfer energy from off-peak hours, when inexpensive generation technologies are still available, to peak hours, when expensive technologies can be economically substituted, see Figure 1. A percentage of the transferred energy is lost, due to the limited efficiency of the entire pumping/turbination cycle. The relative variable costs of the involved thermal technologies determine the amount of pumped-hydro utilization. It is important to note that the use of pumpedhydro may be globally economical even if the substitution scheme mentioned above results in a negative balance. This may occur when pumping artificially increases the minimum system demand, which may alleviate technical minima constraints, therefore allowing for a more flexible use of the available technologies. The model to be presented in section 2.3 captures all these trade-offs in a precise form. CrpCilp

. . .......................... ... ... .

:

:

:

function. The curve is now a kind of fit to a bunch of points in this plane, each one corresponding to a hydro project.

$

T

Figure 2. Fixed cost of storage-hydro technology. It has also been assumed that there is an user-defined function relating the capacity of a hydro project to the average energy production during the considered time period, usually one year, see Figure 3. It is assumed that the amount of energy that can be produced grows linearly with the installed capacity till the change point Ph is reached. Beyond this point the cncrgy increases, also linearly, at a lower rate, reflecting the fact of the saturation of the water resources of the hydro project under consideration.

Dm

. . ............................,..... . .........................92.% ........ ........:................ i

ph

MBj

Figure 3. Energy-capacity curve for storage-hydro technology Figure 1. Production cost model (offloading constraining case)

2.2. Fixed cost structure and capacitv restrictions The fixed costs of any technology include the capital investment costs, that will be assumed to be proportional to the installed capacity of the corresponding technology. The existing capacity of each type of technology will be modeled as a separate technology and will be assigned a zero fixed cost. The storage-hydro technology requires to be considered as a special case. The specific physical characteristics (storage capacity, water inflow, head, amount of required civil work) of future hydro schemes will generally require individual technologies to model each hydro site or at most, each group of similar units. Figures 2 and 3 depict the basic features of the storagehydro technology that are considered in the model. It must be noted that variations over the shapes of these curves can be easily accepted by the proposed problem formulation and solution algorithm. In Figure 2 Ph represents the capacity already installed, and thus its fixed cost is zero. Beyond Ph, the output capacity can only be increased by adding new groups, requiring extra construction work. This part of the fixed cost curve is modeled by a quadratic function. The increasing difficulty in installing new hydro projects due to the decreasing availability of sites in the Spanish hydroelectric system it is represented here by a quadratic

2.3. Problem formulation The optimal static generation mix can be determined by minimizing a cost function Z that includes variable costs and equivalent fixed annual costs of all the technologies in operation during the prescribed target year: Nt

Z=

Nh

NP

C (ft Tdat + vt At) + C Fh + C (FP + Vp)

(1)

t= 1 h=l p=l where: t, h and p are running indices for thermal, storage-hydro and pumped-hydro technologies, respectively; t=l,..,Nt; h=l,..,Nh; p=l,..,Np TI, Th and Tp are the available installed capacities of each technology (MW), ft is the annual fixed cost of technology t ($/MW.year), at is the availabiliiy of technology t (P.u.), vt is thc variable cost of technology t ($/MWh), At is the area below the load-duration curve (LDC) representing the energy generated by technology t (MWh). Fh represents the fixed cost of storage-hydro technology h ($):

and analogously for Fp. The meaning of fh2. Afh and Ph can be directly inferred from Figure 2 (it is assumed an availability

Authorized licensed use limited to: UNIVERSIDAD PONTIFICIA DE COMILLAS. Downloaded on July 9, 2009 at 22:30 from IEEE Xplore. Restrictions apply.

1142

equal to one for the storage-hydro and pumped-hydro technologies); and

V p represents the extra cost (usually a negative number) associated to pumped-storage operation ($).

computed, as it was done with At before, t=l,..,Nt. Therefore, from Tp and the LDC where the At's have already been determined one can directly compute Apt and ATt, as needed in (1, 2, 3). Again a dedicated subroutine of the program easily implements the required calculations.

*Ts vs -..:.:.:.:.: . . . . . .:.:.: ...

.. .. .. .. .. .. .. .. .. .. .... .......... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. . .. . .. . .............. . .. .. .. . . . . . . . . ATs-l vs-l-;!. :::. .:::. r:: : ::': :.:.:.:.:.I. 1.1. ;.:.:.I. ... ...... ...... ...... ................................................................................................................... .. .... .... .... .................................................................. .. ... ... .. ... ... ... ... ... ... ... .. ... ... .. .. .. .. .. .. . .. ... .. ..... ..... .... ..... ..... ................. .................................... . .. . .. . .. . .. . .. . .. . .. . . . . . .. . .. ... ... ... ... ... ... ... ... . ... .... ... .... ... .... ................................................................................................................ , ...................................... .................. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ... ... ... ... ... ... ... . ..i , -:< v2 /'I. . 1..:.:.. :-:< . L.:: . :-:< . .1::. :-:; . .L:: . :-. ............ ' .. .. .. .. ........................................................... AP2.'I : ............ : ... .... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... ...... :

where: A p t is the area above the original LDC (see Figure 1) corresponding to the energy pumped by the t-th thermal techno logy (MWh) A T t is the area below the original LDC (see Figure 1) corresponding to the energy generated by the pumpedhydro technology that replaces the t-th peak load thermal technology (MWh). Note that Apt and ATt cannot be both different from zero for any t. The precise determination of Apt and ATt is described immediately.

I

1

The only decision variables in the cost function that has just been described are the available installed capacities Tt, Th and Tp in the target year. It will be shown now that the remaining variables At, Apt and ATt in (1, 2, 3) can be computed from a given set of Tt, Th and Tp. Ignoring pumped-hydro for a moment, filling up the LDC with the installed technologies in the target year is completely straightforward, see Figure 1. Technical minima of thermal technologies and run-of-the-river hydro are loaded first. Then remaining thermal technologies follow in strict economic order, till the maximum load plus some specified reserve is met. Storage-hydro technologies are inserted whenever their energy exactly fits into the LDC while the technology operates at the rated capacity Th. The resulting loading point may occur exactly before a thermal technology is fully loaded or in between a thermal technology, see Figure 1. Every thermal technology is completely used up before starting with the next one in economic order since, otherwise, the minimum cost would not be attained for the given set of T's. A simple subroutine of the program computes At, t=l,..,Nt, from the LDC, Tt and Th. Any unused capacity is wasted and adds extra fixed cost. Therefore, the optimal mix should not contain any unused capacity. T h e same rationale indicates that pumped-hydro technologies must be used at the given full capacity T which is assumed to be the same for both pumping and turcination. What must now be determined is the amount of pumping that is economically optimal. This is easily shown for Np=l with the help of Figures 1 and 4. Since pumping and generating energies are directly related by the efficiency 17 of the entire cycle, one can choose the turbined energy as the only variable to be optimized; then

AT^

APp = ATp /tl

vl,q

AT^

AT^

1

I I I IAp,, I I

Aopt

ATP Imi

Figure 4. Pumping energy optimization process. The process can now be repeated in the same fashion with a new pumped-hydro technology. Here the only assumption is that the prescribed order p=l,..,Np is optimal. Although this is now a solved issue, see [9] for a detailed discussion in a probabilistic setting, it was decided not to include in the algorithm this extra complexity. A number of technical and economical considerations constrain the minimization of the cost function (1). It follows a description and a mathematical formulation for each one of them:

( i ) The sum of all installed available capacities, thermal plus storage-hydro plus pumped-hydro, has to exceed the LDC maximum demand DM by a prescribed reserve margin r in order to securely accommodate changes in the available capacity or in demand: Nt

Nh

NP

1T t +

Th+ h=l

Tp - (l+r)DM 2 0 p=l

t= 1

(5)

( i i ) The sum of all technical minima plus run-of-the-river hydro must not exceed the LDC minimum demand Dm plus the pumped-hydro capacity. For an small amount of time (e.g., 5 % of the considered period) this constraint may be relaxed, what amounts to correct Dm by a factor s:

(4)

Figure 4 represents in the x-axis and variable costs (of generation used for pumping corrected by efficiency in the pumping curve and peak-load substitution costs in the turbination curve) in the y-axis. If the storage energy limit of must the reservoir Armax is not reached first, the optimal be given by the intersection of the two stair-like curves of variable costs (it is assumed the pumping costs of technical minima to be zero). These two curves must be consistent with the changes in pumping and substituted thermal technologies that are shown in Figure 1, from which Apt and ATt can be

;

Ap1.q

I I

(l+s) D,

-

Nt

Nh

NP

mtTt t= 1

mhTh+ h= 1

Tp 2 0 p= 1

c

(6)

where mt and mh are the corresponding ratios of technical minima capacity for each technology.

(iii) All the installed capacities T s must be non negative. This restriction prevents absurd solutions from occurring. Tt,Th,Tp 2 0

t=l,..,Nt; h=l,..,Nh; p=l,..,Np

Authorized licensed use limited to: UNIVERSIDAD PONTIFICIA DE COMILLAS. Downloaded on July 9, 2009 at 22:30 from IEEE Xplore. Restrictions apply.

(7)

1143

(iv) There may be restrictions on the total amount of capacity that can be installed for a given technology; this may occur because of limited availability of hydro sites or political considerations (e.g.. quotas on oil or on imported coal capacities). For any already existing capacity a limit, which is equal to the currently installed capacity, is set to the actual capacity of this type that can be used for meeting the load in the planning horizon year:

(v) The installed capacity of a storage-hydro technology Th is related to the corresponding average available energy Ah by the curve in Figure 3:

i

Ahmaxmhl

Thmax =

Ahmax 5 PhHhl (9)

ph

i(Ahmax-PhHhl)/Hh2

Ahmax > PhHhl

optimization process. The derivatives must be also evaluated at the same time. The specified names for the constraints and variables, and also for their bounds and initial values are supplied in a file defined in a standard format which is read by MINOS before anything else. An initial feasible solution is introduced into this file. This allows MINOS to reach the optimal solution more quickly. In this initial solution the existing and the storage-hydro technologies take their maximum values. and the remaining capacity that is required to complete the maximum demand plus a reserve is shared by the new thermal technologies. MINOS performs the computation of dual variables of the optimization problem ( l o ) , i.e.. shadow prices of the constraints ( 5 , 6, 7, 8). These values are very important in the assessment of the economic impact of relaxing the constraints; of particular relevance in the Spanish case are the constraints (6) and (8). This capability is one of the strongest assets of this static optimal mix generation problem.

for h=l,..,Nh and similarly for pumped-hydro technologies. 4. CASE STUDY

3. SOLUTION METHOD The optimization problem that was presented in the last section can be summarily restated as the minimization of a nonlinear objective function (1, 2, 3, Figures 1, 2 and 4) subject to a set of linear constraints (5, 6, 7, 8 and 9): subject to

minx Z(x) AxLb xlower 5 x 5 xuppex

(lO.a) (lO.b) (1O.c)

where x is the set of installed capacities of the considered technologies Tt, Th and Tp in the prescribed target year. Given a value of x there is a well established procedure to compute the nonlinear function Z, which is not written in explicit terms of x. This format can be accepted by well proven optimization packages such as MINOS [8]. MINOS is a large-scale optimization program, for the solution of sparse linear and nonlinear systems. The objective function and constraints may be linear or nonlinear. The nonlinear functions must be smooth. Stable numerical methods are employed throughout. Features include a new basis package (for maintaining sparse LU factors of the basis matrix), automatic scaling of all constraints, and automatic estimation of some or all gradientes. Upper and lower bounds on the variables are handled efficiently. File formats for constraint and basis data are compatible with industry MPS format. It uses a powerful combination of the reduced-gradient and quasiNewton algorithms. The efficiency of the optimization can be enhanced by providing MINOS with the partial derivatives of Z with respcct to the independent variables x, in closed analytical form. This has been done and the expressions of these derivatives are provided in Appendix A. The computation of the values of the objective function value and the partial derivatives with respcct to the installed capacities is done in an user-written subroutine. This computation requires to run a deterministic production cost model like the one presented in Figure 1, including pumping energy optimization, each time the subroutine is called in the

The program has been applied to the analysis of a realistic example based on the characteristics of the mainland Spanish system for the target year 2005. The LDC has been obtained by expanding the 1985 LDC assuming a constant cumulative load growth of 3.3 %. This results in a LDC with a maximum demand DM = 37089 MW, minimum demand Dm = 14094 MW and a total energy of 205558 GWh for the considered period of one year. The existing technologies in 2005 are obtained by retaining the existing technologies in 1985 with a remaining lifetime longer than 20 years. Only one storage-hydro and another pumped-hydro technology have been considered; a skillful use of the cost functions (depicted in Figure 2) has made possible to include both the existing and new capacities within a single technology. The values of Ph and Pp have been chosen to coincide with the presently existing capacity of each technology, and consequently fixed costs below these values have been set to zero. Beyond this point fixed costs are set to the estimated fixed costs of the new storage-hydro and pumped-hydro technologies. Fixed and ones escalated the first year, the remaining

variable costs were computed from the current to 2005. The considered inflation was 8.3 % for 5 % for the three following years, and 4 % in years.

A reserve margin coefficient of 4 % was adopted. To relax the technical minima limitation it has been allowed to exceed the minimum demand during a 5 % of the total period. The considered thermal technologies are described by the following data: Name

at

mt

Vt

($/MWh) New NIJC Old NUC Old LGP New CAI Old CAI New HLA Old HLA New LGN Old LGN Old FOL

0.69 0.69 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82

0.80 0.80 0.63 0.40 0.40 0.48 0.48 0.66 0.66 0.33

ft Ttmax ($/MW.yr) (MW)

15.68 15.68 7 1.04 7 1.20 72.88 75.12 75.44 83.60 83.76 149.76

Authorized licensed use limited to: UNIVERSIDAD PONTIFICIA DE COMILLAS. Downloaded on July 9, 2009 at 22:30 from IEEE Xplore. Restrictions apply.

119352 0 0 77720

4695 1950

0

1100

62056 0 601 12 0 0

m

eo

m

3656 m

1050 1753

1144

where NUC stands for nuclear, LGP for brown lignite, CAI for imported coal, HLA for domestic coal, LGN for black lignite, and FOL for oil. Data for hydro technologies are: Name Storage-H Pumped-H

(ik)

Hjl (h)

Hj2 (h)

($/dW.yr)

A'max (dWh)

9000 1847

3504 3504

3504 3504

109248 70880

35048 10512

f.2

requiring that technical minima of all involved technologies do not exceed the minimum demand, including pumping from available base-load units; global economic optimization of the operation of pumped-hydro technologies. This experience appears to indicate that a wide variety of other modeling extensions can be very easily accommodated into the proposed approach. The computational behavior of the algorithm has been very satisfactory, and it has been enhanced by the use of analytical closed form expressions for the gradient of the cost function, which have been also developed in this work. 6.REFERENCES

Other required data are: Afh = 109.28 ($/MW.yr)/MW,

[ 11 MIT. "Electric Generation Expansion Analysis System.

mh = 0.25 and q = 0.67

The following table summarizes the resulting optimal mix provided by the program. Notice that several ex'sting technologies are not completely used in the horizon year. Name

installed capacity (MW)

New NUC Old NUC Old LGP New CAI Old C A I New HLA Old HLA New LGN Old LGN Old FOL Stor-Hyd Pump-Hyd Total

energy (GWh)

fixed cost (kS)

variable cost (k$)

17248 101491 2058592 1591386 26430 414415 0 4695 0 0 0 0 9409 35552 731262 2531294 0 1100 3122 227558 0 0 0 0 2210 7384 0 557029 0 0 0 0 0 0 0 0 0 0 0 0 10002 35048 219277 0 0 3000 7043 81725 - __ __ __ - __________ ..-...._... 47664 216070 3090856 5321682

total cost (k$) 3649978 414415 0 3262557 227558 0 557029 0 0 0 219277 81725

[2]

[3]

[4]

[SI

[6]

8412538

The model has the following main building blocks: [7]

* data input * creation of

* *

the standard file to be read by MINOS call to MINOS program routines that compute the objective function and the derivatives, i. e., deterministic production cost model results output

This model has been implemented in a computer Data General model Eclipse MV/lOOOO with 4 Mb of main memory. The program has been written in FORTRAN 77, and it has 2200 lines of code, besides the MINOS package. The case study requires approximately 5 seconds of CPU time to run.

5.CONCLUSIONS This paper has presented a new formulation of the static optimal generation mix problem. This formulation and the subsequent solution method, based on the MINOS optimization code, can easily afford the consideration of detailed and complex realistic features of a generating system. This fact has been illustrated by including in the model several extensions that had not been studied before in other static optimal mix approaches. These extensions include: a detailed structure of fixed costs of hydro projects, that also accounts for technical considerations such as reservoir size, unit capacity and the extent of the required construction; an operation constraint

[8] [9]

Vol 1: Solution Techniques, Computing Methods, and Results. Vol 2: Details of Solution Techniques, Data of Test Systems, and Glossary of Terms". EPRI EL-2561. Final Report. August 1982. D. Phillips, F.P. Jenkin. J.A.T. Pritchard, and K. Rybicki, "A Mathematical Model for Determining Generating Plant Mix". Proceedings of the Third IEEE PSCC. Rome, June 1969. N. Levin, and J. Zahavi, "Optimal Mix Algorithms with Existing Units". IEEE Transactions on Power Apparatus and Systems, vol. PAS-103, No. 5, pp. 954-962. May 1984. N. Levin, and J. Zahavi, "Optimal Mix Algorithms with Limited-Energy Plants". IEEE Transactions on Power Apparatus and Systems, vol. PAS-104, No. 5, pp. 11311139. May 1985. A.B. Borison, and P.A. Morris, "An Efficient Approach to the Optimal Static Generation Mix Problem". IEEE Transactions on Power Apparatus and Systems, vol. PAS103, NO. 3, pp. 576-580. March 1984. A.J. Conejo, I.J. PCrez-Arriaga, A. Ramos, and A. Santamaria. "Evaluation of the Impact of Solar Thermal Generation on the Reliability and Economics of an Electrical Utility System". Proceedings of the Third IEEE MELECON, Volume IV: Solar Energy, pp. 167-173. Madrid, October 1985. N.M. Stoughton, R.C. Chen, and S.T. Lee, "Direct Construction of Optimal Generation Mix". IEEE Transactions on Power Apparatus and Systems, vol. PAS99, No. 2, pp. 753-759. MarcNApril 1980. B.A. Murtagh, and M.A. Saunders, "MINOS 5.0 User's Guide". Technical Report SOL 83-20. Department of Operations Research Stanford. University December. 1983. A.J. Conejo, "Optimal Utilization of Electricity Storage Reservoirs: Efficient Algorithms Embedded in Probabilistic Production Costing Models". M.S. Thesis. Massachusetts Institute of Technology. August 1987.

APPENDXA The efficiency of the optimization can be enhanced by providing MINOS with the partial derivatives of the objective function with respect to the independent variables T's. This computation is intended to improve the efficiency and accuracy of program solution. The derivatives calculated and presented in this Appendix correspond to a system with any number of thermal technologies and one storage-hydro and one pumped-hydro technology. The extension to any number of hydro plants appears to be conceptually straightforward. The derivatives are exclusively based on the loading points of the units and on

Authorized licensed use limited to: UNIVERSIDAD PONTIFICIA DE COMILLAS. Downloaded on July 9, 2009 at 22:30 from IEEE Xplore. Restrictions apply.

1145

the invariant properties of the dispatch of a hydro unit embedded within a thermal unit. Derivatives for the thermal technologies A prior study of partial derivatives with respect to thermal technologies can be found in [6]. Without considering the pumping -hydro technology, the derivative of the objective function with respect to the available capacity of the t-th thermal technology is (see Figure 1):

The derivative of the objective function with respect to the storage-hydro capacity, without considering the .~ pumped-hydro technology, is therefore: ZTh = kh + T mh

+T [

Nt-1 ZTt = fdat + T mt [

.C J=t

uj(Vj-Vj+l) + UNt(vNt-vl) 1 (All

Nt-1

where: &j is equal to mt if l l j d or is equal to 1 if tSjSNt, U, is the loading point of the j+l-th technical minimum, uj is the loading point of the j+l-th technology and T is the total time period. This derivative can be intuitively obtained by observing Figure 1 and making a little perturbation to the size of the corresponding technology. It is important to take into account that installed capacity of each technology is split into two parts: the first one is the technical minimum, and the second one is the remaining capacity till the installed capacity of this technology is completed. Then i T t is the sum of the annual fixed cost divided by the availability plus the change in variable costs. This variation considers a perturbation affected by mt of all the technologies preceding the t-th technology in the loading order, plus another perturbation not affected by mt for the remaining technologies under the LDC. Derivative for the storage-hydro technology In the case of storage-hydro technology the computation of the derivative is more involved since a perturbation in its capacity has also an influence on the available energy, due to the relation between energy and capacity given by the curve depicted in Figure 3.

+ UNt(VNt-Vl) ]

Nt-1 C RjUj(Vj-Vj+l) 1 - Vd Hh j=l

where: is the technology dljSNt, Hh the thermal

fib

Nt-1 uj(vj-vj+l) j= 1

+ T mh [Vd - VI]

derivative of the fixed cost of storage-hydro with respect to T h ; R . is mh if l l j < d or 1 if stands for Hhl or Hh2 depending on Th and d is technology split by the storage-hydro technology.

Consideration of the pumned-hvdro technolorn When the pumped-hydro technology is introduced into the model, small changes occur, that affect the previously computed derivatives. The existence of pumping and turbination produces changes in the LDC; therefore loading points of the thermal technologies change. On the other hand, a perturbation in thermal technologies produces a change in the optimal amount of energy to be pumped by the pumped-hydro unit. This change in the optimal energy is different depending on what is limiting the amount of pumping. Three cases are possible: C1: the limit is imposed by the most expensive pumping thermal technology, k, (charging constraining case). C2: the limit is imposed by the maximum amount of pumped energy, APmax. C3: the limit is imposed by the cheapest substituted thermal technology, 0 , (offloading constraining case). Taking into account these three cases, the new terms to be added to the partial derivative with respect to the thermal technology, ZTt, are:

The whole perturbation for storage-hydro technology will be the sum of that caused by the perturbation in capacity plus that caused by the increment in the associated energy.

As the storage-hydro technology usually will be dispatched splitting a thermal technology, the perturbation of its capacity will give a similar expression to (Al).

where: s and q are the indices for the most and least expensive

However, this increment in the capacity produces a certain increment in the energy available that is equal to the utilization hours of the installed capacity of storage-hydro technology. This increment in energy has to be divided between the technical minimum and the remaining capacity. Since the technical minimum is base-loaded, its corresponding energy is equal to the technical minimum coefficient times the length of the considered time interval. The remaining energy is left to the portion of the storage-hydro capacity that is loaded in intermediate hours. This storage-hydro energy substitutes an equal amount of energy produced by the thermal technology split by the hydro unit. Figure 1 helps in obtaining the derivatives intuitively.

thermal technologies that are spanned by Tp on the capacityaxis (see Figure 1). The new terms in the p t i a l derivative with respect to the storage-hydro technology, ZTh, are: d T h = - (fq/aq)Rq k- 1

+ (fs/as)Rs- T mh AUNt V1 s-1

+ T [ jC= lRjAUj(Vj-Vj+l) - j=o C RjA~j(vj-vj+l)]

Authorized licensed use limited to: UNIVERSIDAD PONTIFICIA DE COMILLAS. Downloaded on July 9, 2009 at 22:30 from IEEE Xplore. Restrictions apply.

1146

The partial derivative of the objective function with respect to pumped-hydro capacity is obtained by considering Figures 1 and 4. This partial derivative presents the following expression:

Z T ~ = F p - fq/aq+

AndrCs Ramos was born in Guadramiro, Spain, in 1959. He received the Degree of Ingeniero Industrial, majoring in Power Systems, from the Universidad Pontificia Comillas, Madrid, Spain, in 1982. From 1982 to 1984 he was a junior research staff at the Energy Group of the Instituto Tecnol6gico para Postgraduados. From 1984 up to now he is a senior research staff at the Instituto de Investigacih Tecnol6gica in Reliability and Planning Groups. His major areas of interest cover applications of operations research in reliability and maintainability engineering of power plants and optimization of operation and planning of electric power systems. Ignacio J. PCrez-Arriaga was born in Madrid, in 1948. He obtained the Ph.D., and M.S. in Electrical Engineering from the Massachusetts Institute of Technology and the B.S. in Electrical Engineering from the Universidad Pontificia Comillas. He is the Director of the Instituto d e Investigacih Tecnol6gica and Professor in the Electrical Engineering School of the same university. His areas of interest include reliability, control, operation and planning of electric power systems.

Juan Boras was born in Madrid, in 1961. He received the Degree of Ingeniero Industrial, majoring in Power Systems, from the Universidad Pontificia Comillas, in 1986. His area of interest is optimization of operation and planning of electric power systems.

Authorized licensed use limited to: UNIVERSIDAD PONTIFICIA DE COMILLAS. Downloaded on July 9, 2009 at 22:30 from IEEE Xplore. Restrictions apply.