IAEA-TECDOC-433 IMPROVEMENTS

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IAEA-TECDOC-433

IMPROVEMENTS

TO THE IAEA'S WASP-111

PROCEEDINGS OF AN ADVISORY GROUP MEETING

TO REVIEW PROGRESS IN IMPROVING THE WASP-111 COMPUTER MODEL

AND METHODOLOGY FOR NUCLEAR POWER PLANNING IN DEVELOPING COUNTRIES

ORGANIZED BY THE

INTERNATIONAL ATOMIC ENERGY AGENCY AND HELD IN VIENNA, 4-7 NOVEMBER 1986

A TECHNICAL DOCUMENT ISSUED BY THE

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IMPROVEMENTS TO THE IAEA'S WASP-IAEA, VIENNA, 1987

IAEA-TECDOC-433 Printed by the IAEA in Austria

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PLEASE BE AWARE THAT

ALL OF THE MISSING PAGES IN THIS DOCUMENT

WERE ORIGINALLY BLANK

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The IAEA does not normally maintain stocks of reports in this series. However, microfiche copies of these reports can be obtained from

IN IS Clearinghouse

International Atomic Energy Agency Wagramerstrasse 5

P.O. Box 100

A-1400 Vienna, Austria

Orders should be accompanied by prepayment of Austrian Schillings 100, in the form of a cheque or in the form of IAEA microfiche service coupons which may be ordered separately from the INIS Clearinghouse.

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FOREWORD

An Advisory Group Meeting to review progress in improving the

WASP computer code and methodology was held in Vienna during November 1986.

The purpose of the present report is to include the principal

presentations made at the meeting as well as the conclusions and recommendations of the working groups and of the discussions.

It was generally felt that WASP remains an appropriate

methodology for generation expansion planning. Given the Agency's

objective to promote soundly based decisions on nuclear constraints, there is a clear role for the Agency to continue to support WASP, to continually review developments in the electric planning methodology and to improve WASP to make it more applicable to the needs of

Member States.

The introduction of VALORAGUA, a hydro simulation model, developed by Electricidade de Portugal, which allows for optimal management of hydro/thermal systems for electric generation systems

with large hydro components, should make WASP more useful for

electric system planners. In addition, the implementation of a PC version of WASP should stimulate wider interest in system planning.

By reaching a wider audience as that in the Advisory Group Meeting it is hoped that this report would be a useful guide for WASP users and electric system planners.

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EDITORIAL NOTE

In preparing this material for the press, staff of the International Atomic Energy Agency have mounted and paginated the original manuscripts as submitted by the authors and given some attention to the presentation.

The views expressed in the papers, the statements made and the genera! style adopted are the responsibility of the named authors. The views do not necessarily reflect those of the govern-ments of the Member States or organizations under whose auspices the manuscripts were produced. The use in this book of particular designations of countries or territories does not imply any judgement by the publisher, the IAEA, as to the legal status of such countries or territories, of

their authorities and institutions or of the delimitation of their boundaries.

The mention of specific companies or of their products or brand names does not imply any endorsement or recommendation on the part of the IAEA.

Authors are themselves responsible for obtaining the necessary permission to reproduce copyright material from other sources.

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into the WASP package a hydro optimization module that could serve two purposes: first, to determine the characteristics of the hydro system to be used to find the WASP optimal solution; and second, once the optimal solution is found, to check that the operation of the proposed hdyro-plant system leads to optimal use of the

generating stations.

Consequently, the VALORAGUA program developed by Electricidade

de Portugal (EDP), was selected as the external module, which

allows, together with WASP, to obtain optimum economic advantage of hydro/thermal systems with a large hydro component. EDP was hired under contract with the Agency to provide a fully documented version of rtASP-VALORAGUA. In addition, EDP was asked to make some further

improvements to WASP as follows: (i) evaluation and printing (in REPROBAT) of the expected fuel consumption per plant per year and per hydrocondition, (ii) printing (in REPROBAP) of the investment -and construction costs, IDC -and fuel inventory of the committed units, (iii) optimal selection of investment cash-flow schedule,

(iv) inclusion of capital escalation factors in investment costs, (v) preparation of output files for graphical display and (vi) evaluation and printing (in REMERSIM) of installed capacities, energy generation, capacity factors, fuel consumption and operating

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Another priority recommendation was to implement a

microcomputer version of WASP-III (PC-WASP) at the Agency. The PC-WASP is part of a larger energy/electricity planning package

ENPEP (Energy and Power Evaluation Program) consisting of six

modules.- (i) macro-economic analysis, (ii) energy demand analysis, (iii) integrated energy demand/supply analysis, (iv) electric load forecasting, (v) electric system expansion (PC-WASP) and (iv) impact analysis of alternative supply systems which was developed with

USDOE funding by the Argonne National Laboratory and it is a useful

tool in energy, electricity and nuclear power planning studies. One basic motivating factor for the development and implementation of a PC-WASP and, in fact, for eventually offering all the Agency's planning tools as PC versions, is the frequent inability of energy and electricity planners in developing countries to obtain access to mainframe computers which are usually required for sophisticated energy planning models. A dedicated microcomputer would allow

planners to carry out planning studies without interferences from

day-to-day problems that usually have priority on mainframe computers.

II. OBJECTIVES OF THE MEETING

(a) To review progress with WASP-VALORAGUA. VALORAGUA is a hydro simulation model, developed by Electricidade de Portugal (EDP),

which allows for optimal management of hydro/thermal systems

with a large hydro component. In WASP-III, generating systems with interconnected hydro reservoirs or large reservoirs are

not adequately modelled to obtain full economic advantage.

EDP, under contract, provided the WASP-VALORAGUA version

together with other improvements (i.e. output files for graphic displays, external cash-flow schedule selection, etc.).

(b) To review progress with the implementation of a PC version of

WASP. This PC-WASP is part of a larger energy/electricity

planning package called ENPEP (Energy and Power Evaluation Program) developed by the Argonne National Laboratory (ANL) for the Department of Energy in the United States.

(c) To discuss other possible improvements to the WASP methodology including: pumped storage, composite generation/transmission

system expansion planning, etc.

(d) To provide recommendations on future activities

III. CONCLUSIONS AND RECOMMENDATIONS

3.1 WASP VALORAGUA

The methodology and computer program should be thoroughly tested before the Agency's release to Member States. The USER'S GUIDE should include a thorough description of the VALORAGUA model, and its iterative mode of application with WASP, the constraints made and interpretation of the results, keeping in mind the pedagogical component of such a Guide. In this context it was mentioned that the opportunity to test VALORAGUA will be possible with Turkey in collaboration with the World Bank. The project would be funded through IBRD's loan proceeds to Turkey.

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3.2. PC-WASP and ENPEP

The micro computer version of WASP should make the WASP

methodology more accessible to system planners. It was recommended

that PC-WASP be released as soon as deemed appropriate once it is thoroughly tested in-house. It was also felt that the entire ENPEP package can be a very valuable tool in assisting developing Member States in the economic assessment and planning of their electric power systems including the nuclear option. It was proposed that the Agency's MAED be made available as a PC version and replace the first two modules of ENPEP (MACRO and DEMAND) since these do not have the same level of sophistication of MAED. In releasing the PC-WASP it was recommended to provide the micro-computer hardware along with the software. The release of the ENPEP package, suitably

tested and modified, should be given priority since it gives in one

package the possibility of performing integrated planning.

3.3 Pumped Storage

The proper simulation of pumped storage is cumbersome if account is to be made of the hour-by-hour operating decisions. In the planning mode some simplifications are possible and still maintain reasonable realism and accuracy. Perhaps a daily/weekly and even seasonal time frame may be useful. The work done by EPRI may provide a fertile ground for suggestions. A basic study on pumped storage simulation should be initiated and discussed in

subsequent meeting.

3.4 Composite Generation/Transmission Planning«

The matter is of importance since the transmission system, specially radial systems, have a profound effect on the decisions regarding the size, type and location of new unit additions. The difficulty is the simultaneous expansion of the system taking into account single and multiple outages of the generation/transmission systems in the calculation of production cost. The matter should be further discussed in subsequent meetings.

3.5 Other Improvements to WASP

The introduction of a financial planning algorithm, appended to WASP, was emphasized. Progress on FINPLAN should be reported by the

next AGM. The implementation of other suggested improvements (as described by Working Group No. 2) should be considered and reported

by next meeting. One useful improvement would be the calculation of the merit order of loading of the generating units within WASP.

3.6 Release of WASP to Consultants

The arguments pro and con for the release of WASP to consultants» that is, the commercialization of WASP, are aptly summarized by Working Group 2. The point was made that the release of PC-WASP will likely increase the "illicit" use of the program. It was strongly felt that the quality of power system planning and decisions on nuclear power would improve by releasing WASP to consultants and provide greater consistency in the power system

planning techniques as well as stimulate wider interest in power

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consultants were that they may likely make a profit out of using WASP and that consultants may not use WASP correctly and accord their "incorrect" results to the use of the code, to the Agency's discredit. The matter should be thoroughly discussed in-house before a final decision is made.

3.7 Scope of Work, Organizational Set-up and Framework for Future Activities

The importance of integrated energy, electricity and nuclear power planning in developing countries is sufficiently well

established. In this context, WASP remains an appropriate

methodology for generation expansion planning. Given the Agency's objective to promote soundly based decisions in nuclear investment, there is a clear role for the Agency to continue to support WASP and to continually review developments in electric planning methodology.

The AGM recommended that the Agency continues to adapt and make available to developing countries state-of-the-art tools and

appropriate methodologies for energy, electricity and nuclear power planning. In this context, the AGM recommended a further

strengthening of the Agency's collaboration with the World Bank.

The Agency, in collaboration with the World Bank, should

continue to provide leadership in improving and expanding the set of

planning tools available for energy, electricity generation

expansion and nuclear power planning, in the context of overall

national energy planning, to provide realistic, useful and objective

studies of the future economic role of nuclear energy in these countries.

The work of the AGM should be continued and should meet at regular intervals since the AGM has proven to provide an efficient forum to foster, pursue and guide the advancement of electric planning methodologies. In this context, the IAEA should continue to provide leadership and support since no other organization offers

the combination of methodology development and expertise, technical assistance and training in power system planning for developing

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VALORAGUA — A MODEL FOR THE OPTIMAL MANAGEMENT OF A HYDRO-THERMAL POWER SYSTEM V. BAPTISTA, M.N. TAVARES Departamento de Planeamento de Centres Produtores, Electricidade de Portugal, Porto, Portugal Abstract

This paper describes the VALORAGUA model for the optimal management of the operation of hydro/thermal systems. From a mathematical

standpoint, the optimal expansion of an electric power system is a multidimensional stochastic sequential decision problem of high

complexity. The objective is to minimize a discounted sum of investment

plus generation costs over a prescribed time horizon. In each period of the time horizon, the decision maker has to choose between the value of

immediate use of water measured by the corresponding economy of "fuel", as against a later use of the water measured in terms of future

benefits. The paper also presents a mathematical formulation of the

problem as well as a case study description and some applications of the

model.

I. INTRODUCTION

The importance of the electric power system in what concerns demand and supply and its relationship with the rest of the energy sector, in p a r t i c u l a r , and the whole economy sector in general (investment and balance of payments), justifies a careful analysis of its expansion over time and an accurate evaluation of each system state operating conditions. The scarcity of energetic, financial and other resources and the uncertainty of medium and long-range economic forecasts, strongly affect the complexity of those already difficult prob lems.

Our analysis deals with technical and economic variables. However, we cannot forget a lot of p o l i t i c a l , ecological and other Features (mostly not q u a n t i f i a b l e ) that weight on the decision m a k i n g process. So, the results of the models must be considered as mere a u x i l i a r y decision m a k i n g tools.

From a mathematical standpoint, the optimal expansion of the e l e c t r i c a l power system is a multidimensional sequential

decision problem of high complexity. The objective is to

m i n i m i z e the discounted sum of investment plus generation costs over a prescribed time horizon.

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Given a set of power plants a v a i l a b l e to meet the

r e q u i r e m e n t s of an expected demand, it is necessary to d e t e r m i n e some o p e r a t i o n decision rules in order to manage the power system in the most r e l i a b l e and economic c o n d i t i o n s , t a k i n g into account the stochastic nature of the e n v i r o n m e n t and the system itself, namely the water inflows, the e q u i p m e n t

a v a i l a b i l i t i e s and the u n p r e d i c t a b l e changes in demand.

Those two perspectives - optimal expansion p l a n n i n g and

o p t i m a l management of every of its system states in the

expansion path - are necessarily connected in the expansion p r o b l e m s . The management of a given c o n f i g u r a t i o n of the system

may be c o n s i d e r e d , however, as an autonomous p r o b l e m , its

degree of d e t a i l , and hence of complexity, depending on the feature to be analised. We must emphasize and we w i l l show it below that certain models for the system operation can provide useful information on the o p t i m a l i t y of the studied

c o n f i g u r a t i o n w i t h i n an o p t i m a l expansion perspective.

From a m e t h o d o l o g i c a l standpoint, it is useful to classify the power system p l a n n i n g models in two great groups a c c o r d i n g to t h e i r m a i n purposes:

"static" models whose purpose is the o p t i m a l management

of the system o p e r a t i o n once the investment decisions, and therefore the system composition, are Known;

- "dynamic" models, a i m i n g at the o p t i m a l expansion of the power system over time, thus j o i n i n g investment and management decisions.

This report describes a model V A L O R A G U A C 13, C 23 , L 33 -i n c l u d e d -in the f-irst group of models. In sp-ite of that, we can d e d u c e from it an easy-to-apply c r i t e r i o n , useful to check the o p t i m a l i t y of the system composition in a long-range perspective. This can be achieved by i n t e r p r e t i n g the solutions w i t h i n the scope of a g e n e r a l i z a t i o n of the m o d e l , so as to

include in the objective function, extended to a g i v e n horizon, a term of fixed capital costs, d e p e n d i n g on the c a p a c i t i e s of candidates to the system e x p a n s i o n C 4],

However, it must be emphasized that such c a p a b i l i t y of the m o d e l is l i m i t e d to the formal i n t e r p r e t a t i o n of the results of a "static" analysis of the system (1 e , w i t h a fixed configuration D, and the o p t i m a l expansion over time is not w i t h i n the scope of the formalized m o d e l . That c a p a b i l i t y is nowadays m a i n l y employed for c h e c k i n g purposes and/or small adjustments of the least cost system expansion path, and not for the "dynamic" studies of long-range expansion, w h i c h could be achieved by this way only at the cost of countless attempts and of i m p r a c t i c a b l e computing time.

For that purpose, the WASP model (Wien Automatic System P l a n n i n g Package, developed by T V A and 0 R N L and s u p p l i e d

by A I E A to Portugal L 53 C 63 ) is c u r r e n t l y employed. That

one is in fact clearly a p p r o p r i a t e d for the study of long-range o p t i m a l expansion of the power system. In the long-range

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p l a n n i n g studies and for isolated configurations of the optimal path generated by the WASP model, simulations are normally made with the model this report describes, aiming at obtaining results for the different purposes h e r e i n mentioned, and also at d e c i d i n g slight adjustments to the expansion programs based on the detailed analysis it provides.

Another important a p p l i c a t i o n of this model is h e l p i n g to p r e p a r e data concerning hydroelectric schemes, existing or to be integrated in the system, for expansion studies with the WASP model. As a matter of fact, requiring the characterization of hydroelectric power plants a c c o r d i n g to their c a p a b i l i t i e s for meeting load, the WASP model considers each plant as the aggregation of two or more blocKs, dealt with as separate plants, of both "base" and "peak" types, for instance. This decomposition obviously depends on which hydrological conditions and/or periods of the year are beeing considered.

The p r e p a r a t i o n of this data is obviously a tasK of some

complexity and without the help a formal model may include a

considerable m a r g i n of arbitrariness. The model this report describes is very helpful for this purpose because, from the simulation of the system for some a p p r o p r i a t l y chosen states with the h y d r o e l e c t r i c subsystem duly disaggregated, the r e q u i r e d data are easily prepared in rather objective terms.

The model is also c u r r e n t l y a p p l i e d , d e p e n d i n g on the purpose of the studies, to configurations w h i c h , due to exogenous constraints, are not situated on the o p t i m a l expansion path.

In this case we can point out the need for a d e t a i l e d

c h a r a c t e r i z a t i o n of the e l e c t r i c a l system operation in a m e d i u m

range perspective (up to six years), for p r e p a r i n g input to the annual p r o d u c t i o n and f i n a n c i a l plans of the e l e c t r i c a l ut11 ity.

The V A L O R A G U A II model w h i c h w i l l be presented below,

together w i t h some of its a p p l i c a t i o n s , is the c u r r e n t v e r s i o n

of^VALORAGUA model.

This version of V A L O R A G U A model is formed by two

i n t e r d e p e n d e n t m a i n modules! the former determines the m a r g i n a l v a l u e of water in the reservoirs, as a function of the storage level, and related to the mathematical expectation of future o p e r a t i o n costs, in a medium range perspective; the latter optimizes the short-term management of the system.

V A L O R A G U A is usually used interconnected w i t h an a u x i l i a r y model - MAINT C 7J , E 8] - whose objective is the o p t i m i z a t i o n of thermal u n i t maintenance actions (scheduled outages) a i m i n g at a more efficient o p e r a t i o n of the e l e c t r i c a l

power system.

The d e s c r i p t i o n of V A L O R A G U A model w i l l be made in

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Chapter 4 is fulfilled with a case study presentation and

in Chapter 5 some of the following additional capabilities of

the model w i l l be described:

the analysis and economic evaluation of hidroelectric or thermal projects;

the optimization of the main technical characteristics of an hydroelectric project;

studies on a simplified generation-transmission network enabling, for example, the d e l i m i t a t i o n of

"interesting areas" for future thermal power plants locat ion;

the evaluation of system operating costs associated to

delays on a new unit commissioning date;

settlement of energy marginal generation costs for tariff

ma K ing,*

the separate analysis of one or more hydroelectric cascades for a more accurate study of the water storage management in those cascades, ensuring the coherence of that analysis w i t h that of the r e m a i n i n g generating system;

multiobjective approach of optimal management of the system - for instance, by considering the multi-purpose character of some hydraulic resources.

II. GENERAL DESCRIPTION OF VALORAGUA MODEL

1. PROBLEM DEFINITION

The objective of the VALORAGUA model is:

to find the most economical o p e r a t i o n p o l i c y of a hydro thermal power g e n e r a t i n g system, t a K i n g into account the p h y s i c a l c o n s t r a i n t s and random c o n d i t i o n s of the system o p e r a t i o n .

It is a complex decision problem under a stochastic

environment in which the major uncertainties are associated with the hydro plants inflows, the load demand and the a v a i l a b i l i t y of the generating units.

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The hydro thermal power generating system consists of a non hydroelectric subsystem (shortly "thermal") and a

hydroelectric subsystem".

- the "thermal" power generating subsystem consists of thermal power plants and equivalent thermal plants simulating energy shortages and power imports.

- the hydro power generating subsystem consists of a network of hydro plants organized in cascades. A cascade or a hydro chain is a connected component of this network formed by hydraulically interconnected plants with their reservoirs and their turbining or pumping power station plants.

The hydroelectric plants are classified asî

run-of-river plants, if they have pratically no storage capacity and so water inflows are used as they become available and the water not immediatly utilized being spilled over.

storage plants, if they have a significant storage capacity with daily, weekly, seasonal or interannual regulation capability depending on the size of the reservoir and of the pattern of natural catchment inflows.

The management of regulating reservoirs is stochastic sequencial decision problem."

in each period, the decision maker has to choose between the value of immediate use of water (associated to a storage variation) measured by the corresponding economy on "fuel", and the expected value of future benefits (associated to a non immediate use of that water).

The VALORAGUA model was conceived to minimize the operation costs for a given configuration of an electrical power system. These costs are the sum of thermal plants operation costs, the cost of imported energy and the cost of energy not served minus the benefits from exports.

Eventual multipurpose use of water resources in one or more plants may be taken into account by imposing adequate constraints to the problem or, alternatively, by considering in the objective function additional terms corresponding to the benefits or losses associated to the non electrical purposes.

In this model we can also take into account the

electrical interconnection among different geographical

areas through a simplified electrical network, with the respective physical characteristics and with some

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simplifications in the load flow simulation. This feature may be used, for instance, to define locations for future thermal power plants.

In practice we must solve a rather complex decision problem, whose main complexities derive essencially from the following characteristics of the problem:

1. number of elementary periods to be studied 2. dimension of the hydro system

3. diversification of technical and economic thermal plants characteristics

4. adequate characterization of the load diaqram 5. dynamic characteristics of reservoir management 6. random characteristics associated with:

load

hydro plants inflows units availability

7. the intrinsic non linearities of the problem: head effects on hydro power output, thermal generation costs, hydro energy dependence on turbined water flow, etc.

2. CHARACTERISTICS OF THE ELECTRICAL POWER SYSTEM

2.1. General notions

The hydro thermal power system consists of a set of hydro cascades, with reservoirs, turbining plants and pumping plants, and a set of thermal plants with different operation costs (including equivalent thermal plants}.

The time intervals, t=l,. one year, the month

horizon is discretized in T time , TC*). Usually we analize a period of

being the elementary interval

The demand is deterministic for every elementary interval t and modelled by a staircase load duration curve with an adequate number of steps.

C») Most of the variables that we are going to introduce are dependent on time interval t, but we eliminate this explicit dependence whenever misjudgements are unlikely.

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2.2. Energy demand

The demand in the time interval t is modelled by a staircase load duration curve, with J steps, with time durations of A, , . . . ,A. .

step j

The area Cj represents the energy demand in the

A, A2

In the step j, the energy constraint is given by

where :

H : : is the hydroelectric power generation

P : : is the non hydroelectric power generation (thermal plants plus imports plus energy not served minus exports minus energy for special consumers)

Cj : is the demand

Usually the staircase load duration curve is modelled with five steps for each elementary time

interval .

2.3. Exported energy and special consumers

The unitary supply price for energy exports and for special consumers in the step J is given by a linear funct ion :

u . : fr., f+.l

rj L rJ rjj

-> IR

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where :

f . ! minimum output power f+. : maximum output power

a . >, 0 and b . ^ 0 characteristic parameters rj rj

r : index for exports or special consumers

2.4. Thermal plants, power imports and unserved demand

The thermal system consists of K thermal plants. The imports and the energy not served are simulated by equivalent thermal plants.

In a hydro thermal system the allocation in time of maintenance actions affects the outputs of the model. So we use the auxiliary model MAINT, described in Chapter 3, to define an optimal maintenance scheduling policy for the thermal units.

The random outages are taKen into account by

derating the capacities of power plants.

Every thermal plant k is characterized by the power output p in the step j subject to the constraint

pkj « pkj « p+kj

where :

p~. is the minimum power output in step j for kj thermal plant k

p . is the maximum power output in step j for kj thermal plant k

The unitary cost curve of the thermal plant k is a linear function of its output power p

v. : p~ . , p* . ———————————————————— > IR k Kk Kk

p ... a + b. • p .

Kkj k k rkj

where :

a, > 0 and b, >/ 0 are characteristic parameters of k the thermal unit k

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2.5. Hydroelectric system

2.5.1. General

We suppose that the hydroelectric system is composed by a hydraulic network with M nodes, corresponding to reservoirs with or without regulating capabilities, and the links consist of a set D of spillways, a set Q of turbining plants and a set B of pumping plants.

We assume that owing to the time discretization used by the model, the routing time of discharge between two consecutive hydroplants is negligeable.

2.5.2. Water balance equation and constraints

Let in the period t

d. - spilled volume, i e D

q - turbined or pumped water flow in the step j U" and i c QUB

and in the reservoir m

S C t D - initial storage

m

w - natural catchment inflow m

e - evaporated volume

m

S C t + l D - final storage (initial storage in t+1)

The relations to be satisfied are:

- water balance equation in each hydraulic node

S (t)+w ~e + Z d. ~ E d. , , Î eD i eD m m J J -Z ( E q. . A.)+ Z ( Z q. . A.)

ie(TuB

+

>i 'J J

i e D +

u - " '

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lower and upper bounds of the turbined or

pumped water flow

U (2.2)

ieQUB

- non negativity of the spills

d. >, 0 (2.3)

ieD lower and upper bounds of the reservoirs's storages

s~(t+0 * s

m

(t+i)

(2. The sets defined as'. m m=1,...,M Bt , B' , D+ D_m are

Q* - set of the turbining plants immediatly

upstream of plant m

Q~ - set of the turbining plants immediatly downstream of plant m

B+ - set of the pumping plants immediatly

m upstream of plant m

B^ - set of the pumping plants immediatly downstream of plant m

Dm - set of the spillways immediatly upstream of plant m

D - set of the spillways immediatly downstream of plant m

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The following diagram represents schematically the water balance equation.

Q*

}(Q+) q(B-) B" ***\ D* 1 rr S(t+1 d(D* i -S(t*J \ l d(D" D" ) ) /e B+ i q(B+ q(Q~

Q-2.5.3. Level-volume function of the reservoir

For each reservoir m of the system, we compute the variation of the water level with the corresponding volume by the level-volume curve

h (v) = h° + y

m m 'm • (v-V°)m 6m

volume, V

where :

is the minimum usable reservoir storage hft is the level corresponding to V

6m> are characteristic parameters of the reservoir m.

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We easily deduce the volume-level curve

2.5.4. Evaporation

The evaporated volume in the reservoir m during the time interval t is given by:

Sm(t) mi n

where:

nm is the average evaporation height during

the time interval t

2.5.5. Average level of water and level of the centre of gravity of the regulating volume

The average level of water in the reservoir m, during the time interval t is given by

h (s)ds !Sm(t) m

S (t+1) - S (t) m m

The level of the centre of gravity of the regulating volume in the reservoir m, in the end of the time interval t, is given by

h^(s (t+D) =

m m

s (t+D - v

u

m m • where!

u

Vm is the minimum volume for operation of the

reservoir m C«) («0 generally V° < Vm « SmCt+l))

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2.5.6. Hydro power output

We de-fine for turbining plant i

y. - global turbine/generator efficiency 3.- head loss coefficient

E. - minimum tailwater level ^i

During the time interval t for the turbining plant i between the upstream reservoir m and the downstream reservoir n, we define the average gross head

where :

^n = h"n(Sn(t)« Sn(t+l))

At the step J , the output power H .. for the turbining plant i fed by the water flow q — is given by

H*?. = y. (a. - ß.q?.) • q. . i j i i i i J U

The upper bound for turbined water flow, qt. , at the step j, is a function of the average gross head cu and the maximum installed turbining water flow (exogenously imposed} qT?x and given by

/ — : — + . / max o

q.j = min(q .. , q. •

where q? is the maximum turbined water flow at reference head a? . The values q? and a? are inputs to the model .

2.5.7. Power consumed by pumping plants We define for pumping plant i Vj- pump/motor efficiency

P.- pumping head loss coefficient

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During the time interval t for the pumping plant i with the upstream reservoir m and the downstream reservoir n, we define the average static head for pumping

hm- hn , i f h » ç ,

h

m

-e, . if

where:

At the step J, the power H.. consumed by pumping plant i, fed by the water flc/w q.. , is given

by IJ

H^. = —!— (a. + 0. q2. ) • q. .

i j y . i Mi MI J ' HI J

The upper bound for the pumping water flow,

q° , is a function of the average static head for pumping ex- and the maximum installed pumping water flow (exogenously imposed) qTfx , at the step j, and given by •*

q. . = mi n / max_{(q,j ,} _{- U?}

0

«i

where d is the maximum pumped water flow at the reference head a? . The values q° , a? , çj , çr and

Cj are characteristics of the plant and inputs to the model.

3. MATHEMATICAL FORMULATION

3.1. Introduction

The fundamental problem of system management is how to choose, among all feasible decisions, one, say D (t), that minimizes the sum of generating costs d u r i n g the time interval t and the minimum expected value, conditioned by D (t), of the future generation costs

(from t+1 onwards) associated with the storage S(t+l)C*) at the beginning of time interval t+1.

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Let <KSCt + lD> t+1)) the expected value, in the beginning of t, of future minimum operation costs

associated with a storage S(t+l) at the end of period t. Bearing in mind that the storage level in any reservoir can not go down a predefined minimum S~(t+l) and assuming that from t+1 onwards the operation of the system is optimal, the expected benefit due to the storage S(t+l)

at the end of t is given by

R(S(t+0, t+1) = J(s"(t+1), t+1) - *(S(t+1), t+1)

As the functions $ are not previously Known, the optimal management of the system requires the resolution of the two following subproblemst

ID Short term management of the hydro thermal system

In every period t, taking into account the state of the different system components and given the initial storage for each reservoir and the inflow that occurs in period t, we compute the final storage for each reservoir and the system generation schedule in order to minimize operation

costs in period t.

This is a non linear optimization problem

solved using an appropriated non linear programming

algor ithm.

In this problem the hydro thermal generating system may be completely disaggregated.

2)Medium term management of the water reservoirs

This subproblem determines the cost-to-go

functions, <|>CSCt+l, t+1)), related with the stored water value function, R(S(t+l), t+1), using a stochastic dynamic programming algorithm.

The hydro system is fully aggregated into an equivalent hydro plant, but the thermal system is considered with the same disaggregation level used

in the short term management problem.

3.2. Short term management of the hydro thermal system

_ For each time interval t, given the functions <|>C.,t+l) the initial storage SCt), the inflow w and the availability state of all components, the problem of short term management of the system in the variables

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SCt+lD, PC*D, q and d is

min nJT(S(t+1), t+1) + GtP,,...,?,)] (2.5)

S(t-H),P,q,d L JJ

subject to the following constraints:

- energy balance

A.( I Hq - Z Hb.) + P. >, C. (2.6)

J ieQ U" IcB U J J

j=1,...,J

- water balance equation in each hydraulic node

S (t+1) = S (t) + w - e - E d. + Z d. (2.7) in ni m m . ,.- i . _+ i i eD ieD m m \s ^ T. ( E q. . A.) + E ( E q. . A.)

ieQ~UB+ j=1 I J J ieQ+UB" j=1 I J J

m m J ^-m m J

m=1,...,M

lower and upper bounds of the turbined or pumped water flow

(2.8)

- non negativity of the spills

(2.9)

d. ^ 0

1

- lower and upper bounds of the reservoirs's storages

. (2.10) S (t+1) $ S (t+1) « S (t+1)

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Let <KS(tD* w, t) be the minimum value of the objective function of this problem and G(P^,..., Pj) the minimum operation costs associated with non hydro productions P^ , . , , , Pj defined as the minimum of the following non linear convex optimization problem in the variables p , . and f .

K.J rj

J K R

min E ( E v (p ) - p - E u (f ) - f ) A ( 2 . 1 1 ) j = 1 k=1 r=1

subject to the following c o n s t r a i n t s :

- energy balance

K * ( 2 . 1 2 ) E p. . A. - E f . A. 5. P.

i i kj J 1 rJ J J i=1 J

k = 1 J J r = 1 J J J I » - - - , J

- maximum and minimum thermal power plants output

(2.13)

Pkj * Pkj « Pkj k=1,...,K

- maximum and minimum export power

This is separable in J non linear convex subproblems being G.-CP:) the minimum operation costs associated with the non hydro productions P: in the step j

-So the function GCP] > • . . , Pj D is defined by G : |RJ ——————————— > IR

J E G:

3.3. Medium term management of the water reservoirs

The objective of the medium term management problem is to calculate the values of cost-to-go functions < i > C S C t D , t), expected value of the future minimum

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operation costs, that are related with the value of stored water in the reservoirs.

Given initial reservoirs storages S(t) and inflow w

to reservoirs during the time interval t, the expected minimum value of operation costs from t onwards

(including t) is given by <KS(t), w, t) .

If the probability space of hydrological conditions in period t is represented by L typical conditions w1 , w ... w with probabilities of occurrence probCw^), 1=1,... L, the expected value in the time interval t-1 of the minimum operation costs from t onwards, conditionned by the final storage S(t), is given by

L 5 J

4>(S(t), t) = L prob(v/) • «(.(S(t), w , t) £=1

So we obtain the following recurrence relation in the reverse chronological direction, typical of stochastic dynamic programming

L p -,

t)= Z prob(v/) • LinU(S(t+l),t+l)+G(P)) (2.15)

= L J

£=1

for t=T, . . . ,2,1

The value of storage S(t) at the end of the time period t-1 is given by

R(S(t), t) = *(s"(t), t) - *(S(t), t) (2.16)

supposing that R and <J> are di f f erent iable functions we have

I''ft' t! (2.17)

j \ >- / r—1 m

m=1,...,M

This equation gives the marginal value of storing

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3.4. O p t i m a l i t y conditions; economic interpretation of the dual variables

Let us consider the Lagrangian of the short term management problem and multipliers

X. , f , a.., u. , TT

j ' m i j i m

a s s o c i a t e d r e s p e c t i v e l y t o t h e constraints ( 2 . 6 ) , ( 2 . 7 ) ( 2 . 8 ) , ( 2 . 9 ) , ( 2 . 1 0 ) . The necessary conditions for a

m i n i m i i m A r o ! minimum 3 G . ^ - - A . = 0 ( 2 . 1 8 ) J 3Hq. X. . 'J A. - (Y -V ) A . + a . . - a . . = 0 ( 2 . 1 9 ) j 3 q . . j m n j ij ij U j = 1 , . . . , J i e Q X. . '•* A. + (^ -^ ) A . + a+. - a 7 . =0 ( 2 . 2 0 ) j 3q . j m n j ij ij U j = 1 , . . . , J i e B R J 3 H ? . 8 Hb. . ( q . . - q T . ) = 0 ( 2 . 2 2 ) i QUB j(q!j " qi j} = ° ( 2 - 2 3 ) i QUB di = ° ( 2 . 2 k ) i e D

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w+ (S* (t+1) - Sm (t+1)) = 0 (2.26)

m=1,. . ,M

The dual variables a , v , TT are associated respectively with the constraints on the turbined/pumped

water flows (2.8), the spills (2.9} and the reservoirs storage bounds (2.10).

The dual variable A: , associated with the energy balance constraint at step j (2.6), is called the

marginal production cost and, at the optimum, it represents the increase in the objective function if the

non hydroelectric generation P: is changed by one unit. In the same way the dual v a r i a b l e Vm , associated

with the constraint (2.7) is the m a r g i n a l value of water in the reservoir m d u r i n g the time interval t. At the opt imum

represents the m a r g i n a l cost of an extra unit of water to be supplied to the hydro power plant i, with the upstream reservoir m and the downstream reservoir n.

At the optimum

3H?. 'J

3q. . M

is the increase in the power output of the hydroeletric plant i, at step j, if the turbined water flow is changed by one unit. So

3H?.

X. A.

J J

represents the marginal benefit from an unitary change of the water flow and

8H?.

A. A. , 'J - (Y - Y ) A.

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represents the marginal rent of turbining in

hydropower plant i at step J. So

the

in the optimum/ if the water flow is not at any of its bounds, the turbining marginal rent is zero.

We can make identical consideration for pumping plants The e x p r e s s i o n E X . A . ( Z 3H?. ' J • i J J '• n 9S (t + 1) j=1 J J i eQ. m - Z 8H?. __LL . n as (t+D ieB m

represents the marginal benefit of an unitary change in the storage SmC t + l), in the time interval t.

R e w r i t i n g the equation [2.21) as -C = •——————-T————r-m 9S ( t + 1 ) m { Z X . A . :-i J J . U

u_

.D 3 S t + 1 i eB m - TT + TT m m we conclude that:

in the optimum, if the not at any of its bounds, of water in time interval marginal value of storage

storage Sp Ct+1) is

the marginal value t, is equal to the minus the marginal benefit of its immediate use

Thus, finding a mathematical expression for the trade-off p r i n c i p l e , just referred to in the b e g i n n i n g of this Chapter, between either an immediate use of water or storing it for future utilization.

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4. MAIN ALGORITHMS

4.1. Introduction

In this section we sketch the two main algorithms used to solve the short term management problem. The technical details are avoided as much as possible and the mathematics is not rigorous. Futhermore the algorithms are not stated in full detail.

We problem as structure :

can reformulate a mathematical

the short term program with the

management following min E f.(x.) i=0 ' ' s.t. I F.(x.) - C » 0 1=0 ' ' x. e X. , i = 0, 1, ..., 5 where : C f Cx0 0 X1 X]

primary power demand

thermal power subsystem decision vector set of feasible decisions for thermal power subsystem

thermal power subsystem productions thermal power subsystem operation costs power import subsystem decision vector

set of feasible decisions for power import subsystem

power imports

total power import costs

power export subsystem decision vector

set of feasible decisions for power export subsystem

- power exports

- gross revenue of power exports

secondary demand subsystem decision vector set of feasible decisions for secondary demand subsystem

- secondary demand

- gross revenue of energy sold to satisfy secondary demand

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VV

transportation network subsystem decision vector

set of feasible decisions for transporta-tion network subsystem

network power flow plus transportation losses

= 0

hydroelectric subsystem decision vector set of feasible decisions for hydroelectric subsystem

net hydroelectric production

hydroelectric subsystem related costs

(irrigation costs + flooding costs + ... + expected value of future minimum operation costs) Let x — \X-.J x .. , . . . ? ^c' X Y X X f(x) - Z f.(x i=0 ' F(x) = Z F.(x.) - C i=0 ' ' where :

N is the common dimension of the F. vector functions

Cl.e. F.-.X. ———————RND

Then the problem CPU can be written as;

min f(x) s.t. F(x) } 0 x e x 4.2. L a g r a n g i a n relaxation a l g o r i t h m ( h i e r a r c h i c a l decomposition, p r i c e coordinated a l g o r i t h m ]

Let X be the Lagrange m u l t i p l i e r vector associated with the constraint F(x) ^ 0. The restricted Lagrangian of problem C P D is defined as the function

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and the dual function of problem C P 1, at point \ , r(x), is defined as the value of the Lagrangian problem

min L(x,X) s .t. x e X

F is a concave function Dll] and if we denote by x(X) a solution to C P C X ) H we have that -F(xCX)) is a subgradient of r at X.

If x is a feasible solution to problem CPU and A > 0 than the saddle point theorem £111 asserts that

r(x) < f(x)

Moreover under the usual convexity conditions on the set X and the functions f and F we have that

max r(X) = mi n f(x) X ?> 0 x e X

F(x) > 0

and so, given that our

conditions solving problem C the problem

problem satisfies those

P D is equivalent to solve

max r(x) X V 0

henceforth the dual problem of C P 3 .

The dual program CDU is a linearly constrained convex program (-F is a convex function) and so it can be solved by any specialized algorithm for linearly constrained convex programs.

Assuming here the simplication that F is differentiable for all X > Q we see that if X* is an optimum solution to problem C D 13 than,by the Kuhn-Tucker theorem C113

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and (A*) * 0 if A* = 0 9X v ' " n or, in terms of F, and Fn(x(A*)) =0 if A* > 0 Fn(x(A*)) ^ 0 if A* = 0

Moreover, any A* that satisfies these conditions is, because of concavity of r, a global optimal solution to problem C D D . So these conditions are both necessary and sufficient for a global optimum.

Our algorithm solves the dual program by one of the four following algorithms depending on the slowdown of convergence rate, the dimension of the A -vector, and the differentiability of function r.

A. Ascent type algorithm

Here we assume that F is differentiable on A. A.l.Rosen's gradient projection algorithm

Cl 2D , Cl 3D .

A.2.A quasi-Newton algorithm Q4G combined with a conjugated gradient algorithm C15D .

B. PIES - type algorithm C16D.

C. The Wolfe decomposition algorithm C17J or its equivalent dual, the tangencial approximation algorithm of Geoffrion [18D as proved by Wolfe

[19: .

Solution of the Lagrangian problem

Anyway, whatever the algorithm we use to solve the dual problem we must always compute the value of r at

each point A, i. e., we must solve the Lagrangian

problem C PC A) 3 .

Remembering the definition of f, F, x and X we see

that C PC AD D is a mathematical program separable in the

variables Xj, i=0, 1,..., 5 and so, solving C P( A ) D is equivalent to solve the following independent mathematical programs'.

s.t. x. e X. i i

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which can be solved by different algorithms, for instance one for each subproblem, taking advantage of the

particular structure of each problem C P j C X ) H >

Interpreting X as the m a r g i n a l cost of producing energy Cthe common good to all subsystems) and assuming

that the price of energy is equal to its marginal cost and remembering that f;(x j) is the cost of decision Xj

and Fj Cxj ) is the production of subsystem i associated with decision x; we see that

XF. (x.) - f. (x.) i l i i

is the net benefit for subsystem i if he takes the

decisions x, and sells his product Cthe energy) at the price X.

So C P; CX) 3 can be interpreted as the problem that the manager of subsystem i faces with if he tries to

maximize the net benefit of his decision given that the

price of his produced good is X.

Futhermore any algorithm used to solve the dual problem can be stated in the following general framework:

0. Choose an inicial X >/ 0, set K = l. 1. Compute T C Xk)

2. Compute other relèvent information to the algor ithm

3. Check convergence conditions Ce. g. the Kühn Tucker conditions)

If no convergence was attained go to 4. Otherwise stop.

Xk is an optimal solution of C D 3.

k

4. Modify X using some a l g o r i t h m i c map

Xk+1—————————A(Xk,...)

5. Update the iteration counter,

k -<———————— k+1 and return to 1 .

Step 1 of this prototype algorithm is nothing more

than the solution of problem C PC Xk) D .

That is we have a two-level hierarchical price coordinated decomposition algorithm where'.

- at the bottom level, given the price system X each subsystem tries to maximize, independently of the other subsystems, his net benefit by solving the

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"local" subproblem C P; CX D 1 and returning to the top level (the coordinator or supervisor) his

production proposal Fj C x j ( X *) ) .

at the top level given the production proposals of the local subsystems F j C x j C X k ) ) , the supervisor modifies the price system as needed in order to enforce as much as possible the satisfaction of the constra int

F(x) >, 0

and return to the bottom level a new price system

This type of "tantonement" process continues until the supervisor agrees with the production proposals of the local subsystems C e. g., until the Kühn Tucker conditions are satisfied).

Po

<x

k

)

4.3. Feasible directions algorithm

Using the inequality

F (x ) >, C - E F.(x.) 0 O ._. I I

we can reformulate problem C P D as

min f* (C - Z F.(x.)) + l f (x.)

o . , i i . , 1 1

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where f* (P) = min f (x ) 0 0 0 s.t. F (x ) > P 0 0 x e X 0 0

is the minimum thermal operation cost we need to incur in order to satisfy a thermal power demand greater than or equal to P.

Let us introduce the following notation: X = l X - » X-,..., *r ^

X = X1 x X2 x...* X

5 5

f(x) = f*(C - £ F (x )) + Z f (x.)

i=1 i=1

for a neater presentation of the algorithm . Suppose that Xj are polyhedral convex sets, f is a convex function (this is true for the model stated in the first part of this chapter) and without too much loss of

generality suppose that f is continously d if f erent iable .

The prototype feasible directions algorithm used to solve the short term management problem is the following sucessive linear approximation algorithm C20D :

0. Initialization

Let x° a feasible solution to problem C P*3 that is x°=Cx°, . . ., x°) and x° e Xj , 1 = 1, . . . , S Compute f Cx° ) • Set K=0 1. Compute gradient of f at xk g =

2. Solve the direction finding problem

mi n g1 . z

s.t. z e X

and obtain a solution z K 3. Check convergence conditions

If

n k k.|

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or other convergence conditions are met stop Otherwise go to 4.

4. Line search

Find 9 > 0 and xk+1 such that

xk+1 = xk + ek(zk - xk) xk+1 e X

and

f ( xk + 1) < f ( xk)

5. Update the iteration counter

k <————— k+1

return to 1.

We note here that the direction finding problem C LP(x*) D can be decoupled into 5 independent problems!

k m i n g. . z.

s.t. z. e X. , i = 1,...,5

k \e

w h e r e g- is the component of the gradient V f ( x ) w . r . t. x j

Thus we use adequate algorithms to solve each of the problems C L P j C x ^ ) 3 taking advantage of the

structure of the polyhedral sets Xj, which in turn may be

decoupled into subproblems and these into sub-subproblems

and so on.

Moreover some of the subproblems C P\ (AD 3 of the Lagrangian relaxation algorithm are solved using feasible directions algorithms, thus using the C L P j C x ^ ) 3 problem

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III. GENERAL DESCRIPTION OF MAINT MODEL

1. INTRODUCTION

The objective of the MAINT model is

To define an optimal maintenance scheduling policy for thermal units in a hydro thermal system.

The foreseeable availability of thermal units during the simulation period is a particulary important parameter in the VALORA6UA model whose objective is the optimal management of a hydro thermal power generating system.

Different laws of distribution in time of the scheduled outages of thermal units significantly affect the output of VALORAGUA model, namely the minimum operation costs of the electrical power system , for the simple fact that if a given unit is out for maintenance purposes the system will have to meet the same demand with the available units not saturated, generally with higher variable costs. So we can associate an expected generation cost to every

maintenance scheduling and so the maintenance actions must

be established in order to minimize the global operation costs, under the given constraints.

The main problem is!

To set up the starting dates for the maintenance actions for every thermal unit of the power generating system, aiming at the minimization of the expected value of the sum of fuel costs, cost of energy imports and cost of energy not served.

This model iterates with VALORAGUA model until convergence of the management rules of the reservoirs is achievied, in the sense of the stabilization of the main parameters of its optimal operation.

The MAINT model was conceived only for the thermal power generating system. Indeed, taking into consideration!

l)the hydraulic interconnections and the advantage that we can take from them to attenuate the perturbations due to preventive maintenance;

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2)the nonperiodicity of long duration maintenance actions in some components (dams, for example) of a hydroplant;

3)the greater r e l i a b i l i t y of the components of hydroplants as compared to thermal plants one;

4}the possibility of preventive maintenance actions in

the purely run-of-river plants d u r i n g the dry season

without afecting the output of the plant;

5)the possibility of preventive maintenance actions in the electro-mechanical equipments of storage plants

during the periods where the reservoir is storing water ;

and with the exclusion of the aperiodic long duration repairs of dams, the maintenance actions in hydro plants may be included in the model, with adequate adaptations if reliable statistical data are available.

2. GENERAL NOTIONS

2.1. Characteristics of the model

The MAINT model main characteristics are: 1)the thermal units have two operation modes'.

totally available totally unavailable

2)the available capacity of each unit is constant during each time interval out of maintenance

per iods

3)the possible maintenance actions are: no maintenance

repair action

4)the constraints in the preventive maintenance are: continuity of each action

technical conditions

one maintenance action by year for every thermal unit

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5)the objective of the model is to choose the

starting dates to begin the maintenance actions on each thermal unit of the power generating system aiming at the minimization of the expected value of the global operation costs (sum of fuel costs, cost

of energy imports and cost of energy not served d u r i n g one year).

2.2. General definitions

2.2.1. Horizon

For maintenance purposes, the time horizon is

discretized in T time intervals, t=l,..., T.

Usually, as in VALORAGUA model, we analize a horizon of one year.

The annual simulation period is divided into

discrete values of weekly periods of time.

We note by Cl, T3 the time horizon Cin number

of weeks) and by Ct~ , t+l the time interval between

the week t~ and the week t+.

2.2.2. Power demand

The power demand Cjt is given by a staircase

load duration curve with J steps for every elementary time interval t (for every week).

2.2.3. Hydro power output

The hydro power output H:t is given for all

h y d r o p l a n t s at the step j for every elementary time interval t.

The hydro power output is dependent on the hydro condition 1, 1 = 1 , . . . L. So we can consider the hydro power output for certain specified h y d r o conditions (such as the dry year, the wet year and so on) or we can use the expected v a l u e of hydro power output for a series of t y p i c a l hydro cond it ions.

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2.2.4. Substitution cost

Let the thermal unit K be under a preventive

maintenance action in the time interval Ct~ , t+3,

and t+ « T. Let

t~f°Ck) the minimum generation cost, assuming that

the thermal unit K is not in preventive maintenance in the time period Cl, Tj .

V (K, t~ , t+ ) the minimum generation costs assuming that the thermal unit k is shutdown for maintenance purpose in the time interval Ct~ , t+D and totally available out of that

time interval.

The above defined two costs enable us to define the substitution cost of thermal unit K in

the time interval Ct~ , t+ J as

(3.1)

2.2.5. Continuity of maintenance action

A maintenance action of a given duration must

be performed from its starting date to its ending

date without interruptions.

So, once a preventive maintenance action

begins, it must have to be continued up to its end.

2.2.6. Resource constraints - maintenance crews

The maintenance actions on thermal units may be constrained, on a given date, by the a v a i l a b i l i t y of resources (materials, maintenance crews and so

on D .

The set of resources E, is partitioned into R

sets, Er, r=l,... R, that we call the maintenance

crew types with a time dependent response capacity,

given by the number of crews, ert > in the crew type

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3. MATHEMATICAL FORMULATION

For the complete description of the thermal unit K, k = l > . . . > K, we need to introduce the followinq variables

l]time independent

p^ installed capacity

s, duration of preventive maintenance action (in

weeks)

t|< initial time interval for the preventive

maintenance action 2)time dependent

i(<t operation state

=1 in service

=0 out of service, for reasons other than

ma intenance

m|<t maintenance state =0 in service

=1 in maintenance f, forced outage rate

ak ; coefficient of power l i m i t a t i o n s in the step jC akj e C 0, 13 )

For the description of the maintenance c a p a b i l i t y of crew type r, r = l , . . . R, we need to

introduce the v a r i a b l e

e t maximum number of thermal units that the crew

type E ,can simultaneously r e p a i r in the time interval t

So the p r o b l e m can be formulated in the v a r i a b l e s Pkjt' mkt and *k as C«D

min l ? £ \ (pkjt) ' Pkjt (3'2)

subject to the following constraints:

-power balance

(3-3)

£ P.J, » CJt - HJt J-,...J

L— I , . . • j I

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-lower and upper bounds on the power output of thermal units

(3.M

'"••'*

-availability of thermal power units

(3-5)

P+. = (1-m ).i .(1-f. ) . p. • a. . . . ,

kjt kt kt kt Hk kj j=1,...,J

-one maintenance action by thermal unit

I (3-6)

Z m. . = s.

t=1 kt k k=1,.,,,K

-continuity of the maintenance a c t i o n

W1 ( 3 - 7 )

tft mkt ' sk k = l , . . . , K

-availability of maintenance c r e w s

(3-8)

keE fc rt t=1 T

-limitations of maintenance periods

t" < t < t+ (3>9) k Vf \s

r \ r ^ . I l l /

k=1,...,K

-binary character of maintenance actions

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This is a mathematical programming p r o b l e m in mixed variables, the continuous v a r i a b l e s Pk ; t and the integer

variables mkt and t^ .

If we Know tk , satisfying to the constraint (3.93, we

can immediatly calculate mL). satisfying to (3. 6), (3.7),

C3.8) kt

mk t= 0 , V t < tk

mkt = 0 , V t > tk + sk-l

mkt = 1 , V tk « t < tk+sk-l

We are then confined to calculate the thermal units

generation schedule so that to minimize costs in the T time

intervals under constraints (3.3) and C3.4).

Therefore we can formulate the general problem only in the variables Pi,-., and t, .

For each maintenance schedule (t-,..., + ) there is an

associated optimum thermal power output P\,-t whose

corresponding value we call x Ct, , . . . , t.. ) . So there is a

« » . I K

function

X : {I,..., T}K ————————* IR

(t

lf

...,t

K

) -—————— x(t

r

...,t

K

)

whose minimum, over the feasible set of the maintenance

schedules, is the solution of the p r i m i t i v e problem.

4. MAIN ALGORITHMS

We have a large combinatorial programming problem and its resolution is d i v i d e d into two phases:

- the phase I consists in the determination, by a

heuristic method, of a feasible solution (t°,..., tj< )

that is, a preventive maintenance schedule,"

- in the phase II starting with that feasible solution

and using a sequencial m i n i m i z a t i o n process, determine an optimum preventive maintenance schedule (t^,...,t^) (that minimizes the expected value of the global

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The m a i n a l q o r i t h m s used are described in following

1. Phase I - Determination of a feasible solution

l.Let QK the set of K t h e r m a l units that must do

preventive maintenance in the time interval Cl, T3 . Make all units in QK a v a i l a b l e in all

the time steps in the horizon Cl, T3 . C mk t= 0 , V k e QK , V t e C 1 , T3 )

Let n=K

2 . Déterminât ion of the thermal unit z that is in preventive maintenance

2.1 For every unit K e Qn calculate

- the substitution cost

, t, t + s - l ) for

the time interval Ct^, t^+s^ -1 ) where the m i n i m u m substitution cost is located

<P(k, tk, tk+sk-l) = min If (k, t, t+sk-l)

2.2 Choose the thermal unit z that must do

maintenance in the interval Ctz, t2+sz-l) which is the thermal unit with maximum

substitution cost in that time interval

= max (k,

3. Retire the thermal unit z from Q

Qn~1 = Qn - (zl and set t°2=tz .

4 . n=n-l

If n=0, terminate phase I of the algorithm with

a feasible maintenance schedule (t°,..., t£) with an associated m i n i m u m generatinq cost

xCt°, . . ., t°

K

).

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2. Phase II - Sequencial m i n i m i z a t i o n process

Initialize the iteration counter, let n = l

l.For each thermal unit K determine t such that

n-1 .n-1

where :

is the tk feasible domain when

t +, . . , tß'lare the s t a r t i n g dates fortî. the m a i n t e n a n c e a c t i o n s of the t h e r m a l u n i t s 1 , 2, . , . , K-l , K + l, . . . , K r e s p e c t i v e l y . n-1 n-1, 2.If

lx(t"...,t")

-I |\ stop. Set (t* t*\ _ (tn tn\ V t1, . . . , i i / / - \\..,..., \.../ l l\ l N

Otherwise, set n=n+l and go to 1.

It can be shown that the process is convergent,

5. INTERCONNECTION BETWEEN M A I N T ANO V A L O R A G U A MODELS

The objective of the MAINT model is to minimize the thermal plants operations costs in the simulation period

being Known the hydro power output for each hydro condition.

This hydro power output is provided by the VALORAGUA model for a given maintenance scheduling policy. So we must iterate the two models until convergence of the management

rules of the system reservoirs is achieved. I VALORAGUA MODEL 1 1 1 H.t C tr / • . • > T » j MAINT MODEL 1 1 1

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This procedure normally converge in only two itérât ions.

IV. A CASE STUDY DESCRIPTION

After the theoretical description of VALORAGUA model and

its connection with the MAINT model, we will briefly present in

this Chapter the standard output and the main data requirements

for running the former.

1. CHARACTERISTICS OF THE SYSTEM CONFIGURATION

From an optimal long range expansion of the Portuguese electrical system, generated by WASP, we selected the configuration corresponding to the year 2001. The main

features of this configuration are!

a. Annual energy demand of 34 TWh with a peak demand of 6530 MW.

b. Special consumers demand up to 300 MW.

c. 29 hydroelectric plants associated in 10 cascades of hydro schemes (FIGURE 1).

d. 5 of those hydro plants are also provided with p u m p i n g capabilities C * D «

e. The m a x i m u m energy accumulated in reservoirs with

regulating capabilities is about 23% of average annual hydro generation.

f. 9 thermal power plants corresponding to the following

technologies: nuclear, coal-fired units with and without SOx removal, oil-fired units and gas turbine units.

g. l equivalent thermal plant to simulate night import.

h. 2 equivalent thermal plants for simulating two levels of unserved demand.

i. At this stage the installed capacity in thermal plants is 5145 MW and in hydro plants 4662 MW. At present, the corresponding values are, respectively, 2425 MW and 2900

(*) Hydro plants with turbining and pumping capabilities w i l l be named "mixed" hydro plants.

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Allô Rgbcqâo

LU

O

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MW. So, we are moving from an hydro dominated system to a thermal dominated one.

2. DATA REQUIREMENTS

In Chapter 2 the data needed to characterize an

electrical generation system were described. In ANNEX A all

the data required for this case study are presented in

detail.

Here we only point out some significant aspects of these data.

2.1. Study period

The model analyses a period of one year, being the

the month the unit of time usually considered for the

management purposes of the hydro thermal power system. However, with the only limitation of computing time, the basic unit period (month) can be reduced and the study period can be widened, if justified by the applications, supposing a well defined system composition w i t h i n that time span.

2.2. Load demand

Power demand at every time period is m o d e l l e d by a staircase load d u r a t i o n curve with an adequate number

(five in this case study) of steps. Coefficients that

establish the correspondence between each step power demand and the mean power value of the load diagram

enable us to consider different load shapes for

sensitivity studies.

In this case study it was also admitted a special consumers demand up to 300 MW with a limit supply price;

that demand w i l l be met only when the marginal generation

cost is lower than that price.

2.3. Thermal system

Several parameters are necessary to describe the

thermal units, splitted into two sets of data. We only refer here some of them:

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FOREWORD

EDITORIAL NOTE

CONTENTS

ie(TuB

>i 'J J

u - " '

s~(t+0 * s

(t+i)

Q*

h^(s (t+D) =

s (t+D - v

h

-e, . if

«i

u_

r(x) < f(x)

<x

)

(3.1)

(3.M

'"••'*

(3-5)

(3-8)

(t

...,t

) -—————— x(t

...,t

)

xCt°, . . ., t°

).

lx(t"...,t")