Capacity expansion of stochastic power generation under two-stage electricity markets
Introduction
The share of low-carbon power production technologies, such as wind and solar, is continuously increasing in current power systems to comply with the ambitious targets to reduce greenhouse gas emissions. Unlike conventional generating units, the production of these energy sources relies on weather conditions and therefore, they suffer from their uncontrollable and intermittent nature. Furthermore, the production of renewable energy sources can only be partially predicted in advance, which increases the uncertainty involved in the operation of a power system.
This shift of the generation mix creates unique challenges for operating power systems in an efficient and reliable manner and influences the performance of current electricity markets. One key issue on the successful integration of renewable generation into power systems is the appropriate use of system flexibility to efficiently deal with the variable and uncertain character of these energy sources. In fact, the utilization of system flexibility highly conditions the impact of power forecast errors on the total operation cost of a power system and the profitability of stochastic generating units [1], [2], [3], [4], [5]. Consequently, modeling the short-term operation of a power system accurately is crucial when evaluating the long-term capacity expansion of stochastic generating units [6]. Furthermore, the design of the electricity market may also have, by extension, an important effect on stochastic capacity investments, inasmuch as the market organization may determine the availability of flexible resources for the balancing operation of the power system [7], [8], [9].
The impact of renewable power generation on long-term capacity expansion models was first addressed three decades ago in [10], [11]. Since then, more recent works propose various models to investigate the impact of specific influential factors on the optimal capacity expansion of stochastic generating units. Neuhoff et al. [12] provide a methodology to capture the spatial variation and correlation of wind availability in investment models. The introduction of security constraints to optimally allocate wind power capacity is discussed in [13]. The impact of a high level of wind power penetration on the optimal future power generation mix is analyzed in [14]. Likewise, the models proposed in [15], [16] investigate the effects of unit commitment constraints on the generation expansion planning of power systems with renewable energy sources.
Although existing generation expansion models have focused on capturing the variability of renewable power production, they systematically disregard the uncertainty pertaining to short-term forecast errors of stochastic power generation. In this paper, we propose a modeling framework that allows us to investigate the effects of short-term forecast errors of renewable power production on generation expansion planning. While existing models approximate the functioning of the market by a single-stage clearing procedure, our contribution lies in enriching the current modeling of the short-term system operation by considering a two-stage electricity market that includes a forward market, which is cleared one day in advance based on forecast values, and a balancing market, which copes with real-time power imbalances. We propose a family of models formulated from a central planner perspective to determine the optimal generation expansion plan that minimizes the sum of operating and investment costs. The operating costs include both the day-ahead dispatch cost based on forecasts and the expected balancing costs according to the probability distribution of the forecast errors.
This paper also contributes to existing literature on investigating the impact of short-term market design on generation expansion planning. To this end, we model two market-design paradigms that differ on their ability to cope with forecast errors, namely:
- 1.
Inefficient market clearing: according to which the clearing of the day-ahead market does not anticipate the need for flexible power capacity to balance the system in real time. This clearing mechanism is representative of market designs where there is no provision of operating reserve capacity prior to or in concurrence with the determination of the day-ahead energy dispatch.
- 2.
Efficient market clearing: in which the clearing of the day-ahead market does account for the need for balancing power capacity in real time by jointly optimizing the day-ahead dispatch and the subsequent balancing operation of the power system. Therefore, this market-clearing mechanism endogenously determines the optimal amount of reserve capacity to be procured in concurrence with the day-ahead energy dispatch [17], [18], [19], [20], [21].
Under a deregulated market, the proposed cost-minimization approach provides the foreseen generation expansion decisions under a perfectly competitive market [22]. Alternatively, some works formulate generation expansion problems to determine the investment decisions that maximize the profit of strategic power producers owning stochastic power generating units [23], [24], [25], [26], [27].
In order to investigate the effects of competition at the investment level, we also propose in this paper a set of generation expansion models assuming that stochastic power producers cooperate to make investment decisions under a collusion agreement. The proposed models are first formulated as bi-level optimization problems (to capture the subordination of operation outcomes to generation expansion decisions) and then recast as mathematical programs with equilibrium constraints (MPECs) as in [28], [29], [30], [31], [32], [33], [34]. The upper-level problem determines the investment decisions to maximize the profit of the collusion of producers, while the lower-level problem represents the clearing of the electricity market. Unlike existing ones, the proposed models account for the projected revenues obtained by the stochastic power producers in both the day-ahead and the balancing markets under the two different market designs described above.
The resulting optimization problems consider the variability of the expected stochastic power production and demand level throughout the planning horizon by approximating their associated historic duration curves using time segments of different durations, as it is customary in expansion planning problems [25], [29], [32], [33], [35], [36], [37], [38]. Likewise, the uncertainty pertaining to forecast errors of power production and demand is characterized through scenario sets generated by randomly sampling their probability distribution functions.
The contributions of this paper are thus threefold. First, we develop a modeling framework to explicitly account for the impact of power forecast errors on the optimal generation expansion plan of stochastic power production. To do so, the short-term operation of the system is modeled through a two-stage electricity market that includes a day-ahead and a balancing market. To the best of our knowledge, this feature makes the resulting generation expansion models the first of its kind. Secondly, we analyze the effect of market design on generation expansion by considering two paradigmatic market-clearing mechanisms that differ in whether or not future balancing energy needs are taken into account when clearing the day-ahead electricity market. Thirdly, we investigate the impact of competition at the investment level by comparing the generation expansion plan that maximizes the social welfare with that obtained by a strategic collusion of power producers.
The remainder of this paper is organized as follows. Section 2 introduces generic generation expansion models from the standpoints of a social planner and a collusion of stochastic power producers. Section 3 presents the simplifying assumptions and procedures to recast the bilevel and three-level models proposed in Section 2 as single-level mixed-integer linear or non-linear optimization problems. Section 4 describes the procedure to characterize the variability and uncertainty of the stochastic power production. Section 5 presents the numerical results for a small power system to illustrate the differences among the proposed generation expansion models. Besides, the results of a more realistic case study based on the Danish power system are discussed in Section 6. Finally, Section 7 concludes the paper.
Section snippets
Investment models
This section introduces the family of optimization models used to investigate the impact of forecast errors of renewable power production on the generation expansion of stochastic power generating units. Our modeling framework covers two different market designs, which essentially differ in how proficient the electricity market is in coping with these forecast errors, and two different degrees of competition at the investment level, namely, perfect competition and collusion. The generation
Assumptions and solution strategy
The different generation expansion models proposed in this paper are formulated in Section 2 as single-, bi- and three-level optimization problems in a generic manner. Without any further assumptions, determining their global optimal solutions is very hard or impracticable using current solution algorithms, especially in the case of bi- and three-level optimization problems [50], [52]. In this section we present the set of simplifying assumptions and the detailed procedures required to recast
Uncertainty characterization
In order to capture the effect of forecast errors from renewables on investments in renewable power capacity, we propose in Section 2 a set of optimization problems in which the operation of the power system is modeled in two stages, namely, a day-ahead market and a balancing market. In this section, we describe the procedure to model the variability and uncertainty characterizing the production of stochastic power generating units within this two-stage market-clearing process.
First, we assume
Illustrative example
The main objective of this section is to report numerical results from a simple power system in order to illustrate the main features of the different generation expansion models as well as to quantify the impact of forecast errors and market design on optimal capacity expansion decisions. For clarity, we start by presenting the results for the perfect competition case, and continue with the imperfect competition approach.
Case study
We now use the models presented in Section 3 on the Danish power system. For the sake of illustration, we assume that the existing wind power generation is zero and determine the optimal capacity and location of wind power production for the year 2012.
The Danish power system is electrically separated into two areas, the western part (DK1), connected to the continental Europe system, and the eastern part (DK2), connected to the Nordic synchronous area. These two areas are interconnected through
Conclusions
The limited predictability of the power produced by stochastic power generating units implies that a significant share of both the operating cost of a power system and the profit of stochastic power producers comes from the real-time balancing of their power forecast errors. In this paper, we present a mathematical framework to model the impact of these errors on the generation expansion decisions of stochastic power generating units. To properly capture this impact, we characterize the
Acknowledgments
S. Pineda and J.M. Morales thank the Danish Council for Strategic Research for support through 5s – Future Electricity Markets project, no. 12-132636/DSF.
The work of J.M. Morales is also partly funded by DSF (Det Strategiske Forskningsråd) through CITIES project, no. 1035-00027B.
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2021, EnergyCitation Excerpt :However, the complexity of capacity expansion models is so significant that with today’s computational capabilities, the problem is tractable only with a limited number of scenarios. For instance, Pineda and Morales [2] solve this issue by considering yearly duration curves of the intermittent power production and the demand level, whereas Domínguez et al. [3] select representative days of the year. Moreover, it is hard to define a stochastic process that can reproduce the relevant specificities, especially because it is difficult to identify a unique pattern in their historical series.