Annals of Operations Research 120, 45–57, 2003 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
A Goal Programming Model for Rescheduling of Generation Power in Deregulated Markets ∗ E. CENTENO, B. VITORIANO, A. CAMPOS, A. MUÑOZ, J. VILLAR and E.F. SÁNCHEZ-ÚBEDA
[email protected] Instituto de Investigación Tecnológica, Universidad Pontificia Comillas de Madrid, c/ Santa Cruz de Marcenado, 26, 28015 Madrid, Spain
Abstract. In deregulated electrical systems, production schedule for power plants is the result of an auction process. In the Spanish case, this schedule includes two main concepts: energy production (to be actually produced) and secondary reserve (to maintain available). The generation company faces the problem of converting energy schedule into a power schedule, respecting the reserve schedule as well as technical constraints, and trying to accomplish different goals: to minimise the production costs, to obtain smooth shapes for the power schedules and to optimise eventual compensation in schedules. A weighted goal mixed integer programming model with a real-size application to deal with this problem is presented. Keywords: electricity markets, power generation dispatch, goal programming, integer programming, shortterm scheduling
Introduction Electricity industry is experimenting a profound restructuring process in an increasing number of countries all over the world. The objective of these changes is to bring about competition in some of the business activities. Although the details of the deregulated marketplace may vary from one case to another it is generally assumed that electricity should be traded in a similar fashion to other energy commodities. Generation companies were traditionally subjected to regulatory policies in which short-term decisions about the amount of energy that each power plant should produce and the precise moments when the plant should start-up or shutdown were taken with weekly cost-minimisation unit commitment models used by a central operator. Unit commitment models have been widely studied and applied to power generation in regulated markets. The basic formulation of this problem can be consulted, for example, in the classical handbook [12]. Surveys of recent works in this field can be seen in [9,10]. More recently there are new applications of unit commitment models in deregulated market, some of them presented in [4]. General changes in utilities control centers including this kind of algorithms are described in [1]. Deregulation implies a change in the way that unit commitment is performed. Each firm has to decide the energy that it should offer as well as the capacity that should be ∗ The authors would like to express their sincere gratitude to the Spanish electrical utility Endesa for
providing funding for this work and for their insightful suggestions and assistance.
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reserved for ancillary services in successive daily markets. There are markets in which scheduling of power generation and reserve for the day are obtained for bidding units, taking into account demand forecast. Several daily markets may be summoned to adjust the energy to be generated by the bidding units. Afterwards each company elaborates the scheduling for the power plants included in the bidding unit. The complete description of the bidding elaboration process as well as the description of the computer system used to solve it can be found in [11]. This tool includes as a submodule the model described in this paper. Each generation company has to determine the so-called rescheduling of power plants from the scheduling of bidding units for the next day taking into account different goals. Among these goals are included the fulfilment of the reserve schedules of bidding units, of the energy schedules of bidding units and of the total energy schedule of the company. Besides, generation cost must be minimised. In this paper a multiobjective model to deal with this problem is presented. The detailed problem description is presented in section 1. Section 2 is devoted to present the developed goal programming model, and in section 3, two real size cases for a Spanish utility and computational results are presented. The main conclusions obtained are compiled in section 4. 1.
Problem description
This section describes the rescheduling problem as a part of the bidding elaboration process. The scheduling of electrical power plants in some deregulated systems, like the Spanish one, is performed through several successive markets that are convoked daily to schedule production for the next day. The schedule includes two main concepts. First, the energy production and second, the so called secondary reserve. Energy and reserve are traded in different markets. In energy markets, generation companies present bidding for their power plants, grouped into bidding units. A bidding unit may be formed by one or several power plants. The bidding for each hour consists of a set of different energy blocks, characterised by their energy quantities and their prices, corresponding to each bidding unit, and optionally some associated constraints. There is an auction process and, as a result of the matching between bidding and estimated demand, the production schedule for each bidding unit and for each hour is obtained. There is also a secondary reserve market where generation companies also present bids for the bidding units that have the capability of power regulation, i.e., to increase or decrease in real time the power. The purpose of this reserve, in brief, is to fulfil deviation between forecasted and real demand. Bids can be for positive or negative power reserve. Secondary reserve market is convoked once a day, meanwhile energy market is convoked several times. The first energy market (so-called daily market) establishes a provisional energy schedule and the following energy markets (so-called intradaily markets) ensure whatever adjustments may be necessary in the schedule due both to technical and economic reasons. The total amount of these adjustments represent less than 10% of energy
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traded in daily market. As result of the auction process for the successive markets of the day, each offer unit has a final energy schedule to be produced each hour as well as a secondary positive and negative reserve to maintain available. The model presented is aimed to determine the power to be produced with each power plant in each hour, assumed known the secondary reserve schedule for the bidding units and the provisional energy schedule provided by daily market. This model is run by the generation company before presenting its bids in one of the successive energy markets and it allows to determine: • A power schedule for each power plant and additionally, the available energy that can be bid in the next energy market. • The possible infeasibility of the program of any bidding unit, which could be compensated with other units of the same company. Each generation company must present bids in the next market to compensate infeasible schedules to avoid penalties. This is possible, while the bids which the company presents to the auction process, can include not only new energy blocks to add to the schedule but also blocks to withdraw from the previous one. These two results are used as input data for a separate module, not described herein, that elaborates biddings. These biddings are mainly oriented to fulfil the power schedule suggested by the model. Strategic bidding, oriented to profit maximization only concerns daily market and exceeds the scope of this paper. The aim of this model is to determine the power to be produced with each power plant in each hour minimising the total cost for the company. This cost comprises production cost, including start-up and shutdown plant costs, reserve cost, including cost of eventual fast changes in production if reserve is used, and some penalties for deviation from the schedule and for non-smooth scheduling. The relationship between produced energy and cost for each plant is represented using a convex linear piecewise function. Besides, the proposed production plan must respect in one hand the energy and reserve schedules and, on the other hand, the power plants technical constraints. Respecting energy and reserve schedule implies: • Energy schedule for each bidding unit must be fulfilled by the power plants it consists of. The total energy produced by the group of plants must be equal to the bidding unit program. • Schedules are defined as energy quantities while productions to be determined are a power value. Energy is defined as the time integral of power1 with respect to time. 1 Energy and power concepts are commonly misunderstood in the context of electrical energy production.
Electric energy must be interpreted as the produced good. The total produced quantity is measured in megawatts-hour (MWh). Power is the energy produced in a time unit and the corresponding unit is the megawatt (MW). If power is constant, energy is obtained as power multiplied by time. Otherwise, energy is the integral of power with respect to time. There is a third concept in this paper that can be easily confused with these ones. It is ramp-rate, defined as the maximum allowed variation of a plant power in one hour. The unit for this concept is Megawatt divided by hour (MW/h). Ramp-rate may be also interpreted as an upper limit for the absolute value of the second derivative of energy.
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Power variations of power plants are considered as linear within an hour, in order to simplify the model. In [7,8] the modelling in power terms of energy schedules for a single power plant is detailed. • The secondary reserve program must also be respected. Each bidding unit must be able to increase or decrease its production in the corresponding positive or negative scheduled reserve value for each hour, if necessary. Besides, technical constraints of the plants must be also included. The presented model is focused in thermal power plants characteristics, but it can easily be broaden for other kind of plants as hydroelectric ones, for instance. Technical constraints for power plants comprise: • Power plants can be committed or not for production, but when they are, they have to equal or exceed a minimum power, including a margin for secondary reserve. This minimum value can be different from one hour to another one. When the produced power of a plant is below the minimum and increasing, the plant is said to be startingup. There is a similar situation when the plant is shutting down, the power of the plant is below the minimum and decreasing. • Power plants have to respect a maximum power, including a margin for secondary reserve, that can be different from one hour to another one. • When a plant has a secondary reserve schedule, it must be committed for production, so it cannot be starting-up or shutting-down. • The so called ramp-rate: the variation of produced power for each power plant in an hour is limited. This limit is different for an increase or a decrease. Its value is also different if the plant is starting-up or shutting-down, or if it has a reserve schedule or not. The shape of energy programs guarantees that starting-up or shutting-down is performed as fast as possible. • Reserve has an upper limit that depends on generated power for each plant. Relationship between reserve limit and produced power is not continuous nor convex, but it is possible to divide power in a finite set of intervals such that upper limit for reserve is constant inside each of them. • Bidding units may have a maximum and minimum power due to security reasons. 2.
Model description
2.1. Data representation Index sets Hours, bidding units of the generation company, power plants in a bidding unit, intervals of piecewise linear energy costs, reserve limits intervals,
h ∈ H, u ∈ U, g ∈ Gu , t ∈ T, r ∈ R.
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Problem data • Bidding units data, u ∈ U : energy schedule for bidding unit u in hour h (MWh), puh pbsuh , pbbuh secondary reserve schedule positive/negative for unit u in hour h (MW), upper and lower limits for generated power for unit u in hour h l¯uh , l uh (MW). • Power plants data, g ∈ Gu : potgh , pot gh maximum/minimum power generation for plant g in hour h (MW) if committed, pbgh , pb gh maximum/minimum secondary reserve level for plant g in hour h (MW), increasing/decreasing ramp rate when plant g is committed for rsgh , rbgh production in hour h (MW/h), increasing/decreasing ramp rate if plant g is starting-up/shuttingrsagh , rbpgh down in hour h (MW/h), increasing/decreasing ramp rate if plant g has a nonzero rsrgh , rbr gh secondary reserve schedule in hour h (MW/h), unitary cost of energy of plant g for interval t in hour h (€/MWh) cveght (used for linearization of energy cost), etght , et ght maximum/minimum energy of plant g for interval t in hour h (MWh) (used for linearization of energy cost), potr ghr , potr ghr maximum/minimum secondary reserve of plant g for interval r in hour h (MW), bsghr , bbghr maximum secondary reserve level to increase/decrease of plant g for interval r in hour h (MW), variable cost of secondary reserve of plant g in hour h (€/MW), cvrgh start-up or shutdown cost in hour h (€). cparr h • Penalty costs data: penalty cost to avoid deviations in energy program in hour h in a unit cieh (€/MWh), penalty cost to avoid deviation in the total energy generated in hour h cih (€/MWh), penalty cost to avoid deviation in secondary reserve programs in hour h cirh (€/MW), csuaph penalty cost to smooth power changes in hour h (€/MW). 2.2. Decision variables In order to clarify the model, data have been noted with lower case letters, but first letter of variable names will be Greek or capital letters. Moreover, variables beginning with
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letters α, β will be deviation variables of goals, and other Greek letters will be used for binary variables. • Continuous variables: power generation over the minimum allowed power generation if Potgh committed (pot gh ) of plant g in hour h, Pparrgh power generation of plant g in hour h if it is not committed, Bsgh , Bbgh positive/negative secondary reserve level of plant g in hour h, power program variation in plant g in hour h, Suapgh Etgh energy generated by plant g in hour h, energy total cost of plant g in hour h, Cegh • Deviation variables: positive/negative deviation in the energy program of unit u in hour h, αuh , βuh positive/negative deviation in the total energy program in hour h, αeh , βeh αbsuh , βbsuh positive/negative deviation in the positive reserve program of unit u in hour h, αbbuh , βbbuh positive/negative deviation in the negative reserve program of unit u in hour h. • Binary variables: δgh νgh ϑgh δrghr
commitment generation decision (1/0) for plant g in hour h, secondary reserve schedule decision (1/0) for plant g in hour h, starting-up or shutting-down decision for plant g in hour h (1 if starting-up or shutting-down, 0 otherwise), assignment (1/0) to a reserve limit interval r of plant g in hour h.
Let it note that produced power of plant g in hour h is not represented as a decision variable. However, it can be easily computed as Potgh + δgh pot gh + Pparrgh . In the following when the power level is needed this expression will be included. 2.3. Goals Different and conflictive objectives are involved in this problem: to achieve the energy programs and the reserve programs, to minimise the total cost and to schedule smooth power changes in plants. So, the problem is a multiobjective one, and a multicriteria decision method must be applied to model and solve it. The selected method is weighted goal programming. Charnes et al. [3] and later Lee [6] and Ignizio [5] introduced goal programming, and it was the beginning for a lot of subsequent works, mainly by its power and flexibility to model situations with conflictive objectives. Different variants of goal programming have been developed from those years. Weighted goal programming has been selected in this case because its flexibility to aggregate in the objective function objectives formulated like deviation from the goal (schedules) with other formulated over his real
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value or deviation to the ideal and impossible goal zero (costs and smooth). Moreover, because the deviations to the schedules can be identified straightway in the optimal solution by the deviation variables, and this information is one of the most relevant in order to prepare bids for subsequent markets. The different criteria considered in this problem are related to the energy schedule of a unit, the total energy scheduled for the company, the positive reserve schedule for a unit, the negative reserve schedule for a unit, the total cost for the company and the smoothness of power changes. Penalties related to the goals representing these criteria are presented below, and goal constraints are presented in the next section. The penalty of deviations of energy schedule is formulated in two senses: there is a penalty when the schedule of a unit is not satisfied, but can be generated by another unit, (αuh + βuh ) · cieh (1) u,h
and another global penalty if the total energy scheduled for a company is not satisfied (αeh + βeh ) · cih . (2) h
The penalties of deviations in the positive and negative reserves are formulated by (αbsuh + βbsuh + αbbuh + βbbuh ) · cirh .
(3)
u,h
The total cost includes the energy generation costs, the reserve level costs and the startup and shut-down costs Cegh + (Bsgh + Bbgh ) · cvrgh + ϑgh · cparr h . (4) g,h
g,h
g,h
And the last goal is to achieve smooth power changes that is formulated in the following way Suapgh · csuaph . (5) g,h
In order to aggregate these objectives, the penalties of deviation variables are subject to the following relationships between them. Firstly, cieh (penalties for deviation in energy schedule) must be greater than any Cegh (energy generation cost) in order to compensate the schedule of a bidding unit with another one only when the unit itself is not able to accomplish it. Secondly, cirh (penalties for deviation in reserve programs) and cih (penalties for deviation in the total energy scheduled for the company) should be greater than cieh and they have a real extra-cost meaning for the company as these deviations will be covered by other company. Finally, csuaph (penalty for power changes) is greater than Cegh , but it can be greater or lower than other penalties depending on the importance given to have an smooth power shape, but it is difficult to evaluate accurately. In brief, relations between penalties and costs are Cegh < cieh < cirh , cih and Cegh < csuaph .
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Therefore the final considered objective function is the aggregated function of these objectives (αuh + βuh ) · cieh + (αeh + βeh ) · cih u,h
h
+ (αbsuh + βbsuh + αbbuh + βbbuh ) · cirh u,h
+
Cegh +
g,h
+
(Bsgh + Bbgh ) · cvrgh +
g,h
ϑgh · cparr h
g,h
Suapgh · csuaph .
(6)
g,h
2.4. Constraints 2.4.1. Goal constraints with deviation variables The goal of achieving the energy schedule as previously explained is modelled for each unit, with the following expression Etgh − αuh + βuh = puh ∀u, h (7) g∈Gu
and for the global generated energy by the company it can be expressed as a function of the previous deviation variables (αuh − βuh ) ∀h. (8) αeh − βeh = u
Positive and negative secondary reserve schedules must also be satisfied in the final scheduling. These goals are formulated through the following deviation variables Bsgh − αbsuh + βbsuh = pbsuh ∀u, h, (9) g∈Gu
Bbgh − αbbuh + βbbuh = pbbuh
∀u, h.
(10)
g∈Gu
2.4.2. Linearization of energy cost The energy cost in each plant is a nonlinear convex function; this function will be represented by a piecewise linear function taking into account that the cost will be minimised. Therefore, T intervals are defined for the linearization, and each one is defined as (et ght , etght ) with an unitary cost of the energy cveght . So the energy cost is modelled in the following way: (11) ( etght¯ − et ght¯) · cveght¯ + (Etgh − et ght ) · cveght ∀g, h, t Cegh t¯
