Intertemporal Considerations for Supply Offer Development in Deregulated Electricity Markets Paul Stewart*,1, E. Grant Read and Ross James Energy Modelling Research Group, Department of Management, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
Abstract The literature refers to several methods for developing supply offers for generators in wholesale electricity markets, all of which consider much more simplified environments than those that occur in reality. In particular, we consider the approach suggested by members of the Electric Power Optimisation Centre, which involves forecasting and updating a “market distribution function”. The market distribution function describes the probability that a section of an offer curve at any point within the likely ranges of (price, quantity) offering space will be accepted by the market. This function is then used to produce an optimal offer. The focus of this paper is the construction of offers and offering strategies that consider intertemporal linkages, particularly those that arise in hydroelectric systems or fuel-constrained thermal units. The main linkages considered here are limited water availability over time, hydro inflows, and hydro reservoir storage bounds, while start-up and shut-down costs, ramp-rate restrictions, and the behaviours of other participants in the offering process over time are considered in ongoing research. To date, the work of Philpott and Anderson produces offers for single periods only. None of the intertemporal characteristics and restrictions of the generation units are considered, and thus the offers may not be optimal when viewed over several periods. It can be shown, for example, that when stochastic intertemporal effects are taken into account, the “optimal” offer from a hydro system will not necessarily be monotonically non-decreasing, as generally is required by market rules. It is also apparent that the optimal form of offers will change as real-time is approached and various uncertainties are resolved. Clearly, these issues require modification of the market distribution function approach, which doesn’t account for these intertemporal characteristics. We therefore examine practical methods for applying market distribution functions to these problems. Keywords: Hydro, intertemporal, reservoir modelling, marginal costs, electricity supply
* Corresponding author: Telephone: +64-3-366-7001 Fax: +64-3-364-2020 Email address:
[email protected] 1 Supported by the Foundation for Research, Science and Technology of New Zealand
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INTRODUCTION
Since the mid 1980’s, centralised electricity markets around the world have gradually converted to deregulated structures, whereby multiple generation companies are formed to compete for the right to supply electricity to meet the demands of the market. The idea behind such deregulation is that the introduction of competition will increase incentives to improve operational costs of these facilities and is also likely to produce competitive results. However, under a deregulated framework, there is no requirement for the generators to reveal their true cost structures, and hence there is significant potential to “game” the market by inflating their offers above costs, thus forcing the market price above the desired competitive levels. Traditionally, electricity market optimisation modelling work focused on the cost minimisation of the entire market as a whole. Clearly this goal is not appropriate in the new structure, and as such, the focus of much research work has changed to the optimisation of behaviour by various players in the market. This paper investigates the optimisation of offers made by an electricity generation company into a deregulated, pool electricity market. In particular, this paper considers the behaviour of a hydro electricity generator in the New Zealand electricity market. The New Zealand market is predominantly hydro-based, with hydro generation accounting for 60% of the total generation capacity and the remainder made up through gas, coal, and geothermal resources. Although there are many market nodes, there are two distinct market regions, corresponding to the two main islands of the country (North and South). These regions are joined by a single inter-connector, known as the high voltage direct current, or HVDC, link. Resulting from the deregulation of the New Zealand market, there are eight generation companies (supply-side) along with seven electricity retailers and several large industrial companies (demand-side) competing in the market, with many companies falling into both the demand and supply categories. In every half-hour period of the day, each of the generators provides a set of offer stacks and each of the demand-side participants provides a set of bid stacks to the central market coordinator, for each of many coming periods1. From a generator’s point of view, for each of these periods they are allowed to provide a five-stepped offer stack for every generation unit that they operate, or alternatively, they can aggregate their offers to a station–level, or even a block-level2. The market coordinator, or New Zealand Electricity Market (NZEM), then takes all this information for each single period individually and produces a dispatch (or pre-dispatch) schedule to minimise the overall cost of meeting demand for the country as a whole, while accounting for transmission constraints and losses. If we ignore these spatial, transmission aspects of the market, and thus consider the entire market to be located at a single market node, then this can easily be explained as the intersection of the aggregated market demand and supply stacks, as demonstrated in Figure 1. Note that because the market coordinator optimises the market for a single period at a time, any internal constraints faced by the generators must be accounted for within their own offers, contrary to what is the case in some other markets. The particular aspect of deregulated markets that this paper focuses on is the issue faced by a hydro generator, of producing supply offers in real-time. When we consider the marginal costs (λ) of generation for a hydro station, the approach is very different from that for considering marginal costs for many other types of plant. This is because some plant (eg wind) has no storable fuel, while thermal fuels, for example, while storable, are readily replenishable at the 1
To be precise, offers and bids are made for all periods from 2 hours away from the current time (offers can only be changed within two hours of real-time if there is a good reason behind the change), to the end of the current day if it is before 1pm, or to the end of the following day if it is after 1pm. The only binding offers or bids are those that are made on the nearest period, as there will be opportunities for all the others to be changed. 2 A block is a set of units or stations that may be dispatched together. In other words, dispatch that is allocated by the central coordinator to a unit within the block may in fact be generated by any of the stations or units within that block. This concept is known as “block dispatch”.
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going price, and can thus be considered to have a well defined marginal cost in a short-term planning horizon3. For a hydro generator, there is generally no direct cost for the water required to fuel the turbines, however there is likely to be a restriction on the amount of water available for use over the horizon. The cost associated with use of this fuel in a particular period is therefore an opportunity cost, related to the foregone opportunity to use this water later in the horizon4. Hence, effective fuel costs for a hydro generator are likely to vary significantly over time, and are likely to be changing right up until the final offers must be provided to the market.
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Figure 1: Single-Node Electricity Market Clearing Process Considering this aspect will enable us to produce an approach that takes into account intertemporal characteristics of a generator’s units and reservoirs, in addition to other intertemporal linkages, when providing pre-dispatch and final offers to the market coordinator. The main intertemporal characteristics considered by the approach presented in this paper are limited water availability over time, uncertain hydro inflows, and hydro reservoir storage bounds5. The particular results presented here explain how the optimal offer for any given reservoir level at any time could be constructed in real-time, given a marginal water value curve that accounts for these characteristics. This paper takes the viewpoint of a generator operating a block of units in a reservoir system which may be complex, with marginal generation costs thus defined by marginal water values at several reservoirs, but supplying a single node of the market. We consider that the market consists only of this node, while the effects of behaviour at other nodes of the actual network are accounted for implicitly, so that they are reflected in prices at this single node. The remainder of this paper is broken up as follows. Section 2 briefly describes the approach taken to this research area in the literature, with a focus on one particular research group, the Electric Power Optimisation Centre (EPOC), on whose work this paper expands. Section 3 motivates the interest and relevance of research into dealing with the proposed intertemporal issues. Section 4 describes the application and usefulness of work presented in this paper. Finally, Section 5 presents a new corollary to an idea presented in the literature, and explains a related development that is currently in progress. ,.
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Although fuels such as gas, bought under ‘take-or-pay’ contracts, may well have limited availability, over time, to the purchaser, and hence raise much the same “opportunity cost’ considerations as hydro. 4 Or the cost of pumping in the case of pumped-storage stations. 5 Other intertemporal characteristics which will be considered in future work include start-up and shutdown costs, ramp-rate restrictions, and the behaviour of other participants in the offering process as time progresses.
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CURRENT ELECTRICITY SUPPLY OFFER RESEARCH
There is much literature available on the subject of offering electricity into pool markets, but surprisingly little that focuses on how best to account for intertemporal linkages and other realistic market characteristics that are met by firms facing this task in markets of the type found in New Zealand. Here, we present a very brief summary of some approaches that have been taken to dealing with intertemporal issues and considerations, followed by a somewhat more thorough explanation of the work performed by EPOC. Villar and Rudnick (2003) use a dynamic programming simulation model under the assumption that fixed-quantity offers are to be made to the market, while accounting for the limited hydro resource over a 24 hour horizon. The main disadvantage with this approach is that market uncertainty does not affect the resulting dispatch level, contrary to what is likely to be the situation faced by real generators. Some authors have considered the dynamic aspects of a pool market, with respect to the generator’s ability to iteratively adjust offers as real-time is approached. Sakk et al (1997) develops a sequential offering scheme with information feedback, whereby generators can modify their offers after pre-dispatch notification. Li et al (1999) use parametric dynamic programming to produce hourly offer stacks in order to meet certain revenue-adequacy conditions in a iterative offering market (but with quite restrictive re-offering rules that limit gaming considerations). Gajjar et al (2003) present a system using Markov Decision Processes to select from a range of actions from a generator’s policy (set of possible actions). Reinforcement learning is used to simulate intelligent agents that can learn how to make good decisions by observing the results of their own behaviour (allowing Markov transition probabilities to evolve as a function of temporal difference error). Petrov (2002) develops the Roth-Erev reinforcement learning algorithm (Roth and Erev (1995) and Erev and Roth (1995)) such that agents learn how to offer to the auction and develop a winning strategy. However, this is restricted by its assumption that at any given time, a trader knows only their last offer and the amount of profit resulting, thus cannot take advantage of information from prior rounds of offering. The interested reader can also refer to Stothert and Macleod (2000) and Visudhipan and Ilic (1999) for examples of other feedback-type offer updating systems. The remainder of this section focuses on work by members of the Electric Power Optimisation Centre (EPOC) in Auckland, New Zealand, who have created a concept known as the Market Distribution Function (MDF). The MDF, denoted by ψ, is defined over the valid quantity and price offer combinations and takes a value of between 0 and 1, corresponding to the probability that an offer at each (q, p) point will not be dispatched6. In other words, ψ(q, p) is the probability that if the supply curve passes through (q, p), the point of dispatch will lie below (q, p). Because the probability of being dispatched a given quantity, q, at an offer price, p, cannot be greater than the probability of being dispatched the same quantity at a higher price, p + δ (and similarly for increasing dispatch quantities at a given price), this function clearly must be monotonically non-decreasing in both its dimensions. The market distribution function is essentially a stochastic representation of a residual demand curve for a single period of time. For example, if we knew the residual demand function for certain, then the market distribution function would have values of 0 below the curve, and 1 above the curve. Figure 2 and Figure 3 demonstrate sample non-linear MDF’s, in 3-D and 2-D form respectively. In Figure 3, the MDF is represented by contours corresponding to values of 0.1, 0.2, … , 0.8, 0.9, rather than explicitly showing the entire function.
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The foundation of this approach is in the work of Friedman, L. (1956), "A Competitive-Bidding Strategy", Operations Research, 4 (1), 104-112., which estimates the probability of winning a bid against competitors based on market data, then uses this probability to determine the optimal bid.
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Figure 2: Example 3D MDF
Figure 3: Example 2D MDF, Philpott (2002) In taking this market distribution function approach, there are two issues that need to be dealt with. The first is how to create the market distribution function from historical data and how to update it using new data as it becomes available. The second is, given the market distribution function, how to find the optimal offer of the required form for the market. The following subsection of this paper describes briefly the approaches that EPOC have proposed to the latter issue. The interested reader can refer to Anderson and Philpott (2001) and Pritchard et al (2002) for explanations of Bayesian and maximum likelihood approaches respectively, to the construction and updating issue. 2.1
Applying the Market Distribution Function
In recent papers, the EPOC group have proposed methods for constructing “optimal” offers for single periods, with a given market distribution function. Neame et al (2003) consider the case of a perfectly competitive market. It is well known in the literature that if the form of the offer is unrestricted in this market situation, then the optimal choice of offer will be to exactly reflect your marginal cost of generation. However, if stepped offers are required, then this may not be feasible, and so an optimal approximation must be determined. Under the assumption of piecewise linear marginal costs, this paper demonstrates that the breakpoints of the offer steps will occur only at breakpoints of the piecewise marginal cost function, and in particular, develops optimality conditions on the location of the steps. To then limit the number of steps to match market requirements, they use dynamic programming to determine where the steps should best be placed. Their results show, as would be expected, that the stack will follow the marginal cost curve more closely in regions where the price probability
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density function has high values. Unfortunately, using this approach, these results cannot easily be extended to the case of a price-maker. Taking another approach, Anderson and Philpott (2002a) create a set of optimality conditions for offer stacks, and then seek to maximise the expected return over the line integral of the offer curve. The focus of the paper is the derivation of the necessary conditions for optimality and, as such, it does not fully address the important practical question of efficient computation of the optimal offer. Anderson and Philpott (2002b) expand on these optimality conditions and demonstrate that, in the case of a convex cost function and inverse log concave rest-of-market supply curve, the optimal unrestricted offer will be monotonically non-decreasing in both the price and quantity dimensions, where the uncertainty in the market is in relation to the level of inelastic demand. This implies that, given these particular conditions, the optimal offer under uncertainty is simply the path, or locus, traced out by identifying the optimal dispatch point under each possible resulting position of the residual demand curve. However, it is likely that in a real market situation, these conditions will not be met. Thus, in Philpott et al (2002), a practical approach to more realistic scenarios of rest-of-market supply and uncertainty is suggested. As with Anderson and Philpott (2002a), this paper seeks to maximise expected return over the line integral of the supply function, given the payoff at each point and the probability of that payoff actually occurring. In this paper, it is assumed that for realistically sized problems, an explicit ψ may be impractical. As such, they break the (q, p) plane into a grid, with known payoffs on the edges of the grid. The offer curve is then constructed under the restriction that it must follow the discrete edges of this grid. For each vertex in the grid, the maximum expected additional payoff of any curve passing through the vertex is calculated dynamically by selecting the maximum of: a) The expected payoff on the ‘up’ edge + maximum expected payoff from the ‘up’ vertex b) The expected payoff on the ‘right’ edge + maximum expected payoff from the ‘right’ vertex Once these payoffs are all known, then it is a simple process to determine the offer curve, by using dynamic programming to determine the optimal path through the MDF surface. 2.2
Limitations of the Market Distribution Function Approach
This market distribution function approach, as described above, successfully considers the ability of a unit to affect the price, and the fact that price levels will differ at different times7. However, it has not yet been extended to deal optimally with the various intertemporal linkages that occur in the marketplace, as discussed above. 3
IMPORTANCE OF INTERTEMPORAL CONSIDERATIONS
For both hydro and thermal generation units, the importance and significance of intertemporal characteristics and considerations can easily be demonstrated. This section of the paper focuses on a couple of simple demonstrations of how ignoring such considerations can be detrimental to the performance of a generating company. These examples consider deterministic cases for ease of comprehension, but the results apply equally (or possibly even more so) to stochastic scenarios. Also, they make the assumption that beyond the monotonically non-decreasing requirement, there are no restrictions on the form of the offers. 3.1
Non-Monotonicity of the Optimal Offer Curve
It has been shown in Drayton-Bright (1997) that for a hydro operator it is not possible to construct a monotonically non-decreasing supply function that will achieve an efficient outcome over multiple periods. This is due to the linked nature of the periods, and therefore means that
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Although it does not consider the impact which one party’s offers may have on offers from other parties, as in a “gaming” model.
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intertemporal linkages make the electricity offering problem non-trivial even before gaming or risk are considered. The following example demonstrates why this is the case. Figure 4 illustrates two potential price paths over a 48 period horizon, while Figure 5 provides the associated optimal generation paths for a single hydro generation unit, taking into account many types of intertemporal constraints. Note that the price paths account for the fact that the unit’s generating level will affect the price, using fairly simple linear residual demand curves for each period. Specifically, in Figure 4, the line marked with diamonds shows a long-run expected daily price path. Consider that we observe a price in the current period (period 1) near the start of the day that is above the long-run average. If we assume that this means we expect it to be a high price day (a reasonable assumption), then our estimates of price will increase for all periods in the horizon, up to the line marked with squares. This particular example assumes that if the price is x% above the long-run average in this period, then it will be x% above the long-run average for all periods remaining in the horizon, and thus the magnitude of the increases will be more extreme for peak periods. Prices Expected Prices
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Figure 4: Two Potential Price Paths over a 48 Period Horizon Generation over Time
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Figure 5: Corresponding Optimal Generation Schedules If the unit being considered is hydro, then there may be a limited total generation capacity over the day associated with water restrictions. As a result of the increases in expected prices over the horizon, this unit’s optimal generation schedule will change. It will want to generate more at the peak periods (which now have even more extreme prices), and so must trade off on generation at the off-peak periods in order to still meet the limited total generation constraints. In particular, Figure 5 shows that in period 1, the optimal generation level has fallen from its previous level. In other words, despite the fact that price has increased for this period, the optimal generation level has fallen. Or alternatively, the higher the price is in this current off-peak period, the less that the unit will desire to generate. This implies a backward sloping optimal supply function rather than a traditional forward sloping one, as required by market conditions.
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If we take this example further and consider multiple potential price paths ranging from all prices being 50% above the long-run average down to 40% below the long-run average, we can form an “offer curve” for each period that will produce an optimal dispatch schedule if any one of these potential scenarios were to actually occur. These offers can hence be considered to account for the stochasticity in price under very simple circumstances. Take the “offer curve” for period 30 for example, as shown in Figure 6. We observe that as the price for this period increases from 30% below, up to 50% above the long-run average, the optimal quantity to generate in that period reduces. Thus, a backward-sloping offer curve is produced. Note that moving from 40% to 30% below the long-run average, the total water restriction is not binding, and so the offer appears as forwards sloping. Examining all of these “offers” together enables an interesting, and logical observation of a “fan effect”. The lower the overall vertical position of the offer curve (and thus the more off-peak the period is), the more backward sloping the optimal offer curve is for that period. As we move through to periods with mid-range prices, the offers become close to vertical, and then for the peak periods, the offers are very forwardsloping. Hence producing a set of offers that resemble a fan. Hydro offers under uncertain prices
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Figure 6: Fan of Offers for All Periods in the Horizon The problem with this though, is that backward sloping offers are not generally permissible under market restrictions, and so the closest that can be achieved in these off-peak or shoulder off-peak periods is to make the offers vertical at some generation quantity (a Cournot-style offer), and move these quantities as time progresses, depending on what has been learnt about the state of the market in that day. This confirms the findings of Drayton-Bright, that one cannot construct a monotonically increasing supply function that will be efficient over multiple periods. 3.2
Joint Effect of Many Intertemporal Constraints on a Hydro Unit
This second demonstration (Figure 7) of the importance of considering intertemporal characteristics of the generating units and of the marketplace compares three generation paths over a given price series: 1. The optimal generation path, accounting for intertemporal linkages (marked with squares) 2. The naïve desired generation path produced by optimising each period individually (marked with triangles) 3. The actual generation path that would result if one tried to follow the desired generation path but ended up hitting many intertemporal constraints (marked with crosses). The intertemporal constraints considered in this example include ramp-rates, minimum up and down times of the unit, minimum and maximum generation levels, limited total water (and hence generation) capability over the horizon, and intra-horizon reservoir bounds.
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Figure 7: Optimal, Desired, and Actual Generation Paths There are many points of interest in this figure: 1. The naïve generation path is above the optimal path for most of the horizon because it does not consider the overall water restrictions. 2. Observing periods 4 and 5, the naïve schedule implicitly assumes that the unit can ramp up very quickly, and as a result, the ramp up is left too late and the generators are unable to take full advantage of the high prices in these periods. This can be seen in that the line marked with crosses (“actual”) is below the line marked with squares (“optimal”) in these periods. 3. Observe the low prices periods 18 through to 26. The naïve schedule again ignores the restrictions of ramp rates and as such leaves it too late to shut down and thus gets caught generating at substantial losses in period 19. 4. From period 40 onwards, the actual schedule that results from the naïve desired schedule starts to run out of water, in order to meet the end of horizon reservoir targets. Therefore it must ramp down and shut off for the last five periods, missing out on some potentially profitable generation. To illustrate the sub-optimality of ignoring these intertemporal linkages by attempting to follow the naïve desired path, we can compare the profits that would result over the horizon under this deterministic scenario. The profit from the optimal schedule was around $151,000. The profit that would have resulted from the naïve schedule, had it been feasible was almost $158,000. However, the actual profit that resulted from trying to follow the naïve schedule was just over $132,000. In other words, 12.5% of the possible revenue was foregone by not explicitly considering intertemporal constraints when planning the generation path over the horizon, in advance. Therefore, this example again reiterates the importance of considering the whole planning horizon when planning generation schedules and therefore offers to the market, even under deterministic assumptions. 4
OFFER STRATEGY DEVELOPMENT
The remainder of this paper focuses on a proposed approach for forming an offering strategy over a horizon to account for inter-temporal linkages, in the context of a market employing supply function style offers, and considering uncertainty with respect to the residual demand curve that will be faced. In other words, the residual demand curve can be represented using a market distribution function, as described in Section 2 of this paper. Given the fact that the cost of fuel for a hydro station is effectively an opportunity cost related to the foregone opportunity to use the water later in the horizon (as discussed earlier), it seems that the best way to consider the intertemporal elements of the reservoir management problem is to approach the problem from a dual perspective. In a deterministic sense, this dual perspective says that we want to trade off the marginal value of releasing water with the marginal value of
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holding water in storage. Hence, the standard optimality condition for the multi-period reservoir problem is that, in each period: marginal value of release = marginal value of holding in storage Let us consider how this will affect the offers from a generator. Observe Figure 8, which uses standard economic theory of determining output decisions by setting marginal cost equal to marginal revenue8. If the marginal value of holding water in storage moves up9, this will increase the desired marginal value of release (equivalently, raising the effective marginal cost of using water in this period from MC to MC’), thus leading to a lower release level (Q to Q’)10.
MC’ Marginal Water Value Curve (MC)
Expected Residual Demand Curve
MR Q’ Q Figure 8: Effect of changing marginal water values In the case of stochastic rest-of-market behaviour, this process can be considered to be somewhat self-equilibrating. As an illustration, if, in the previous period, higher than expected release resulted (as a result of the uncertainty in the market), this would mean that water would become more valuable in the future (due to the lower than expected resulting reservoir level), and hence the offer in that following period would be more restrictive. This effect would tend to bring the final storage position back towards the original end-of-horizon reservoir target. A somewhat similar approach to this is considered in Scott (1998), under the scenario of Cournot (fixed-quantity) offers, considering hydrological uncertainty over the horizon. However, this and other related approaches, such as Villar and Rudnick (2003), have not considered the effects of market uncertainty under a supply curve offering structure. For a stochastic formulation, it is clear that market uncertainty in the current period is going to drive the circumstances (for example, profitability) both in that period and in future periods. For example, if prices in the current period are high, then this would lead to a higher level of release now and thus higher return in the current period, but then a lower storage level, reducing profitability for the remainder of the horizon. Due to this effect, we know that our offer curve in the current period should reflect the expected marginal water value curve defined in terms of storage at the end of the period. 8
The marginal revenue curve has twice the slope of the expected residual demand curve for this scenario, where the generator is effectively a monopolist over the residual market. 9 This could occur as the result of an increased reservoir target, higher anticipated future prices, lowerthan-expected current reservoir levels, etc. 10 Note that in the stochastic offering situation, the result would be a more restrictive offer.
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In real-life, offers made to the pool market by generators need to be constructed on the go, in real-time, so ideally the operators need an approach to offering that does not require any complicated calculations or mathematics on the spot. And, even if an inter-temporal optimisation is employed to determine the marginal water value, it must be quick and efficient. Hence, having a pre-computed table of optimal responses to various circumstances, as in Scott’s algorithm, would be very helpful. Here we propose a marginal cost based “offer curve patching” approach. For a given form of the market distribution function (or expected residual demand curve), a family of offer curves corresponding to a range of constant marginal costs (or marginal water values) can be constructed. This process could be repeated off-line for many alternative residual demand forms (constructed from historical data). In a continuous sense, this family of curves would effectively form an offer surface. Our basic proposition is that (as will be shown in the following section), if the marginal water value curve for the current period and reservoir storage level can be found, these families of offer curves can be optimally patched together,11 so as to produce the optimal offer for the coming period. Specifically, the motivation for the development of this patching approach lies in the production of a dual dynamic programming model (as in Scott) for the generation of a set of optimal offers (and hence an offering strategy) in real-time, for a short-term planning horizon. This is currently part of our ongoing research program. The idea is that an iterative method will be developed to produce the above-mentioned marginal water value surface for the planning horizon, and this surface will be updated at the end of every market period to reflect new information gained. This new information includes updated likely rest-of-market behaviour for the remainder of the horizon and updated information about the given generator’s circumstances, such as the current reservoir level(s) that have resulted from market uncertainty acting on the supply function-style offer that the firm had provided to the market. Once this marginal water value surface has been constructed, the marginal cost curve for the hydro generator can be found for any period and any reservoir level, by considering that an increased unit of generation means that they forego the opportunity to use that fuel in a later period. Under certain conditions, this marginal cost curve will be monotonically increasing, and quite possibly stepped, depending on the shape of the marginal water value surface. For example, there will be steps at the points, where extra generation implies using water out of a reservoir at which the water is valued more highly, or spilling water past a bottleneck generation station. But we note that the position of these steps will change dynamically as water values are revised over time. Using the offer surface corresponding to the expected market distribution function of the period in question in conjunction with the proposed marginal cost patching approach, the optimal offer for the marginal cost curve could then be constructed. As mentioned above, Scott (1998) successfully demonstrated a similar algorithm under the assumption of hydrological uncertainty and Cournot (fixed quantity) offers in an attempt to find equilibrium results. What we propose here goes further, to consider the way market uncertainty affects dispatch (and hence reservoir) levels, when the offers are provided to the market as supply curves, and to optimise those offer curves. In the following section, we explain and provide examples for a corollary on a result provided in Anderson and Philpott (2002b), which demonstrates that the proposed patching approach is appropriate under certain conditions. We then discuss whether this result can be extended to more general and realistic conditions, and present examples to support the proposal. 5
MARGINAL COST PATCHING
Anderson and Philpott (2002b) report the following theorem, under the assumption of known rest-of-market supply curve and uncertain inelastic demand level: 11
Given that a water value surface has already been constructed for the horizon, and considering the distribution of possible end-of-period reservoir levels from that point.
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“If C(.) is a nondecreasing convex function and S(.) is a differentiable inverse log concave function, then there is a supply-function response that is optimal for any h ∈ H.” Defining the notation used, C(.) is the generator’s cost function, S(.) is the rest-of-market supply curve, and h is a possible demand level from the set of all possible demand levels, H. This theorem can be taken to mean that when these conditions on C(.) and S(.) are met, the set of optimal dispatch points for all of the possible demand levels in the set H, can be connected to form a curve (or offer) that is monotonically increasing in both the quantity and price dimensions. I.e. the higher the demand level in the given period, the higher the optimal dispatch point will be with respect to both price and quantity. Our corollary that follows from this is that the optimal offer price at any particular quantity level will depend only on the local marginal cost level, thus being independent of the marginal cost levels that have come before that quantity and those that will occur beyond it. In other words, the optimal offer for a firm with a marginal cost that steps from MC1 to MC2 at BP will be the combination of: -
The section of the offer from 0 to BP under the assumption that marginal cost is equal to MC1 over the entire range, and
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The section of the offer from BP to Qmax under the assumption that marginal cost is equal to MC2 over the entire range.
To illustrate, Figure 9 shows a single residual demand curve scenario. We can see that the optimal dispatch point is at the quantity where marginal revenue (MR) is equal to marginal cost (MC). With a convex cost function and concave residual demand curve, there must be a unique point of intersection between these curves. Further, observe that this point of intersection is dependent only on the local marginal cost (i.e. the optimal point would not change if marginal costs at other generation levels were different, as indicated by the arrows).
Price P*
X
MC
Expected Residual Demand Curve
MR Quantity
Q*
Figure 9: Optimal Dispatch Point for a Single Residual Demand Scenario Note that to produce numerical examples of this corollary, we have applied a dynamic program (DP) that breaks (q, p) space into cells, with constant values for the market distribution function within each. The payoff on each edge of each cell is then calculated using a discrete equivalent of a line integral, before the DP is applied to establish the offer with the maximum expected payoff. This DP is similar to that discussed earlier, as used in Anderson and Philpott (2002a).
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Philpott et al (2002) presents these results in another way, which can be used to analytically derive expressions for the optimal offer curves without restriction to the above-mentioned conditions on C(.) and S(.), but does not guarantee monotonicity of the resulting offer. We will use this approach to illustrate our corollary. Again the assumption in this paper is that the restof-market supply curve for the particular period is known, while the position of the perfectly inelastic demand is unknown. Hence, uncertainty exists only in a horizontal sense, or along the quantity axis. We continue with this assumption of horizontal uncertainty throughout the rest of the paper. Philpott et al (2002) show that the expected payoff (profit) from any particular offer made to the market, V(s) is the line integral of the payoff function, R(q,p) along the offer path with respect to ψ (the market distribution function). I.e.
V ( s ) = ∫ R(q, p ).dψ (q, p ) s
Formulating an optimal offer curve problem that maximises V(s) over all possible choices of offer curve s, subject to monotonicity constraints, can then be viewed as a problem in the calculus of variations. It is shown that, for s to be stationary with respect to feasible variations, then it must at all points be vertical, horizontal or satisfy the first-order condition:
∂R ∂ψ ∂R ∂ψ − =0 ∂q ∂p ∂p ∂q
(1)
We now apply this formula to a case where the marginal cost is a stepped function, taking the value c1 before the breakpoint (BP) in the quantity direction, and c2 beyond. Therefore, we have two distinct payoff functions:
R(q, p ) = qp − c1q
- before BP
R (q, p ) = qp − c1BP − c2 (q − BP )
- after BP
Taking the partial derivatives of this payoff function with respect to price and quantity as before, produces:
∂R ∂R = p − ci , =q ∂q ∂p In other words, the derivative with respect to q has the same form as it would have if the marginal cost was constant across the entire range, but it uses the local marginal cost. Therefore, if we take the two derivatives established above, in combination with the partial derivatives of ψ provided by Philpott et al, and apply the optimality condition, we can establish the optimal form of the offer curve. In this case we consider a linear rest-of-market supply curve, and denote the slope of this curve σ . The optimal form of the offer in this simple case is thus:
p=
1
σ
q + ci
This case is demonstrated in Figure 10.
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Stewart et al. (2004) Uncertain Demand
p
Uncertain Residual Demand
Optimal Offer
1/σ
Inverse Restof-market supply, S-1(p)
c2 c1
BP
q
Figure 10: Example of Optimal Offer under Uncertainty, with Stepped Marginal Costs Now let us consider a more complicated scenario, whereby there is a fixed level of contracts, and the rest-of-market supply curve is piecewise linear, in a convex to the horizontal axis fashion (thus, the expected residual demand curve is concave to the horizontal axis). In this scenario, the payoff function for a two-step marginal cost curve is defined as:
R(q, p ) = (q − k ) p − c1q
- before BP
R(q, p ) = (q − k ) p − c1BP − c2 (q − BP )
- after BP
where k is equal to the fixed contract quantity12. This gives partial derivatives:
∂R ∂R = p − ci , = q−k ∂q ∂p We note that σ is now dependent on the slope of the rest-of-market supply at that particular point, hereafter denoted by σi, which has breakpoints in the price dimension (denoted BPP). Applying the first-order condition gives the form of the optimal offer:
p=
k q + ci − σi σi 1
Now, consider the optimal offer forms that would result from this scenario for two different constant marginal costs, c1 and c2 (i.e. ci is unchanged over the entire quantity range). In Figure 11, the light grey lines indicate information from the rest of the market, thus producing the uncertain residual demand curve, or equivalently, the 0.5 contour of the market distribution function. The broken grey line and the black line show the optimal offers corresponding to fixed marginal costs of c1 and c2 respectively.
12
Note that the revenue associated with this contracted amount is a fixed term in this equation so need not be considered from an optimisation perspective.
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p
Uncertain Demand
Broken Grey Line Black Line
Inverse Restof-market supply, S-1(p)
1/σ2
BBP c2 – k/σ1 c1 – k/σ1
Uncertain Residual Demand
1/σ1 BP
q
Figure 11: Example of Optimal Offer under Uncertainty, with Piecewise Linear Expected Residual Demand, and Stepped Marginal Costs Now let us consider what the optimal offer would look like if the marginal cost jumped from c1 and c2 at the quantity indicated by BP in Figure 11. Hence, we can segment (q, p) space by regions associated with (σi, ci) pairings as shown in Figure 12.
p (σ2, c1)
(σ2, c2)
BBP (σ1, c2)
(σ1, c1) BP
q
Figure 12: Regions for (σi, ci) pairings Now, apply the optimal offer formula found from the first-order condition, to each of these regions, and connect the offer segments by horizontal and/or vertical lines as appropriate. We join these segments in this way as it is the only way to feasibly connect the segments in a monotone fashion, and noting that Philpott et al (2002) claim that the optimal monotone offer has this form. The resulting offer in this case is shown with a heavy dotted black line in Figure 8. As we can observe, this provides the same result as simply following the path of the offer associated with the lower marginal cost up to the quantity BP, and then jumping up to the path of the offer associated with the higher marginal cost for the remainder of the range. On examination, it can be shown that this “marginal cost patching” will work no matter where the breakpoint is located in the quantity dimension. In order to investigate more complicated numerical examples with potentially complex analytical forms for the rest-of-market supply curve, we have used a dynamic programming approach, which produces the optimal monotonic offer (as required by real markets), even where the analytical results suggest that a non-monotone offer is appropriate.
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Consider an example which uses an MDF that corresponds to an inverse log concave rest-ofmarket supply curve, in fitting with the Anderson and Philpott theorem. In particular, the MDF shown13 has its 0.5 contour (i.e. the expected residual demand) as:
30 − 0.5q p= 44 − q
p ≥ 16 p < 16
Figure 13: An example MDF for a rest-of-market supply curve that is inverse log concave Given this market distribution function, we can apply the dynamic program to produce optimal offers under any form of payoff function. To start with, we can produce the three thin lines shown in Figure 14, corresponding respectively to constant marginal costs of 2, 5, and 25 ($/MWh) as we look from the bottom, up14. To confirm the analytic patching theory result demonstrated in Section 4, we can now construct a payoff function corresponding to a stepped marginal cost curve, that begins at 2, jumps up to 5 at an output quantity of 8 (MWh), and then up further to 25 at an output quantity of 43. The resulting optimal offer produced by the dynamic program is that indicated by the heavy, marked line. We can observe that this result supports the patching corollary, as the new offer (with stepped marginal cost), exactly matches up with the constant marginal cost offer for $2/MWh up to the first breakpoint of 8MWh, then jumps up to match the constant marginal cost offer for $5/MWh up to the second breakpoint of 43MWh, and likewise for the third step in the marginal cost curve. 50
MC = 2, 5, 25, BP = 8, 43
Price 45
MC = 5
40
MC = 2 MC = 25
35 30 σ=2
25 20 15
σ=1
10 5 0 0
5
10
15
20
25
30
35
40
45
50
Quantity
Figure 14: Optimal offers with fixed and stepped marginal costs 13
Observe that we find it more convenient to describe the market distribution function value at any given point as the probability that an offer at any given point will be dispatched by the market, rather than the probability that it won’t be dispatched. 14 Note that the downward sloping piecewise linear curve is the position of the expected residual demand curve.
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So, Anderson and Philpott’s theorem, in conjunction with our corollary shows that under the assumption of an inverse log concave rest-of-market supply curve with demand uncertainty, that offers corresponding to constant marginal costs can be patched together to produce the optimal offer for the case of stepped marginal costs. However, it is highly likely that the actual rest-ofmarket supply curve faced by a generator will not meet these conditions15. Therefore, we are interested in showing that our corollary will hold under any form of rest-of-market supply curve. At present, a definitive proof of this hypothesis is part of our ongoing research. We use the remainder of this section to explain the process, and present numerical examples that support our claim, while noting that we are unable to produce any cases that do not support it. Consider now an MDF corresponding to a rest-of-market supply curve that is not inverse log concave, and thus does not fit within the Anderson and Philpott theorem. In particular, the MDF has its 0.5 contour (i.e. the expected residual demand) as:
30 − 0.5q p= 44 − q
p < 16 p ≥ 16
Figure 15: An example MDF for a rest-of-market supply curve that is not inverse log concave When we consider the optimal form of the offer for a rest-of-market supply curve of this sort, we find some interesting results. It can be shown that under the MDF that results from this scenario, the offer given by the optimality condition (1) would be as demonstrated by the heavy line in Figure 16. Clearly, this offer is not of a form allowed by market rules, as it is not monotonically non-decreasing in both the dimensions. Hence, the optimal feasible offer must be some variation on this, most likely a connection between the upper and lower curves in the manner of the dotted line shown. The exact location of this connection will depend on secondorder effects. In other words, as we continue along the lower curve with increasing quantity, there will come a point where we are better off to try and raise the price by applying the upper curve, at the risk of not being fully dispatched. Despite this issue of non-convexity, if we consider stepped marginal costs and thus segment (q, p) space as demonstrated in Figure 12, we can produce examples that show the patching approach to still hold.
15
For example, any concave piecewise linear rest-of-market supply curve does not satisfy this condition.
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Stewart et al. (2004) Inverse Restof-market supply, S-1(p)
Uncertain Demand
p
1/σ2
BBP c – k/σ1
Uncertain Residual Demand
1/σ1
BP
q
Figure 16: Optimal (unrestricted) offer with a rest-of-market supply curve that is not inverse log concave Consider a numerical example, restricted by monotonicity, as shown in Figure 17. As in Figure 14, the thin lines represent constant marginal costs, in this case, of 5, 10 and 25 ($/MWh) respectively. We can see that the optimal offers for the lower two marginal cost values (which must pass through the breakpoint of price = $16/MWh), indeed contain a vertical step as suggested in Figure 16. Again, considering a stepped marginal cost function that steps up between these marginal cost levels at the breakpoints of 12 and 20 MWh, we can observe that even in this case, the patching result holds. 50 Price 45 40
σ=1
35
MC = 0, 2, 5, BP = 15, 16
30
MC = 5
25
MC = 10
20
MC = 25
15 σ=2
10 5 0 0
5
10
15
20
25
30
35
40
45
50
Quantity
Figure 17: Optimal offers with fixed and stepped marginal costs Finally, let us consider a rest-of-market supply function that has a very general form, with concave and convex segments, as well as horizontal steps and vertical jumps, implying an MDF as demonstrated in Figure 18.
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Figure 18: An example general market distribution function In this example, we also consider a much more complicated form of payoff function, considering both fixed and option contracts issued by the generator. As in the earlier examples, Figure 19 demonstrates with thin lines the optimal offers under this payoff structure if the marginal cost was fixed at the levels of 0, 10 and 25 ($/MWh) respectively, starting from the lower-most curve. The heavy line shows the optimal offer if we knew in advance that the marginal cost was going to step between these three levels at the breakpoints of 12 and 31 MWh’s. MC = 0, 10, 25, BP(Q) = 12, 31 MC = 0
50
MC = 10
Price 45
MC = 25
40 35 30 25 20 15 10 5 0 0
5
10
15
20
25
30
35
40
45
50
Quantity
Figure 19: Optimal offers with fixed and stepped marginal costs This illustration demonstrates an example whereby even under complicated payoff and rest-ofmarket supply forms, the optimal offers for different constant marginal costs can be patched together to produce the optimal offer that should be submitted to the market by a generator with an underlying stepped marginal cost curve. 6
CONCLUSIONS AND FUTURE RESEARCH
In this paper, we have proven that under the assumption of market uncertainty in the quantity dimension and an inverse log concave rest-of-market supply curve, segments of optimal offers for fixed marginal costs can be patched together to produce the optimal offer corresponding to a stepped marginal cost curve. Additionally, we have demonstrated examples under which this
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Stewart et al. (2004)
patching approach works when the latter assumption is not satisfied. We have also discussed how this knowledge is useful in the context of a hydro electricity generator, where the marginal costs are actually opportunity costs, and hence a firm estimate cannot be established until close to real-time. Ongoing research involves a definitive proof of the marginal cost patching theory where the rest-of-market supply curve is not inverse log concave, in addition to producing a method of constructing and characterising the water value surface for use in this approach, and using it to produce the marginal water value curve for any system state within the horizon. There are many possible further extensions including adapting the approach for a limited number of offer curve tranches, considering thermal applications, and the actual generation of market distribution functions that will be suitable for this type of approach. 7
REFERENCES
Anderson, E. J. and Philpott, A. B. (2001), "Estimation of Electricity Market Distribution Functions", Australian Graduate School of Management. Anderson, E. J. and Philpott, A. B. (2002a), "Optimal Offer Construction in Electricity Markets", Mathematics of Operations Research, 27 (1), 82-100. Anderson, E. J. and Philpott, A. B. (2002b), "Using Supply Functions for Offering Market Generation into an Electricity Market", Operations Research, 50 (3), 477-489. Drayton-Bright, G. (1997), Coordinating energy and reserves in a wholesale electricity market, PhD dissertation, Department of Management, University of Canterbury. Erev, I. and Roth, A. E. (1995), "On the Need for Low Rationality, Cognitive Game Theory: Reinforcement Learning in Experimental Games with Unique, Mixed Strategy Equilibria", University of Pittsburgh. Friedman, L. (1956), "A Competitive-Bidding Strategy", Operations Research, 4 (1), 104-112. Gajjar, G. R., Khaparde, S. A., Nagaraju, P., and Soman, S. A. (2003), "Application of Actor-Critic Learning Algorithm for Optimal Bidding Problem of a Genco", IEEE Transactions on Power Systems, 18 (1), 11-18. Li, C., Svoboda, A. J., Guan, X., and Singh, H. (1999), "Revenue Adequate Bidding Strategies in Competitive Electricity Markets", IEEE Transactions on Power Systems, 14 (2), 492-497. Neame, P., Philpott, A. B., and Pritchard, G. (2003), "Offer Stack Optimisation in Electricity Pool Markets", Operations Research, 51 (3), 397-408. Petrov, V. (2002), Exploring Computational Power Markets with Evolutionary Algorithms, Masters, Computer Engineering, Iowa State University. Philpott, A. B., Pritchard, G., Neame, P., and Zakeri, G. (2002), "Market Distribution Functions in the Electric Power Industry", Electric Power Optimisation Centre, Working Paper. Pritchard, G., Zakeri, G., and Philpott, A. B. (2002), "Estimation of Market Distribution Functions in Electricity Pool Markets", Proceedings of the 37th Annual Conference of the Operational Research Society of New Zealand, Auckland, 2002. Roth, A. E. and Erev, I. (1995), "Learning in Extensive Form Games: Experimental Data and Simple Dynamic Models in the Intermediate Term", Games and Economic Behaviour, 8 (Special Issue: Nobel Symposium), 164-212. Sakk, E., Thomas, R. J., and Zimmerman, R. (1997), "Power System Bidding Tournaments for a Deregulated Environment", Proceedings of the Thirtieth Hawaiian International Conference on System Sciences, 5, 681-686. Scott, T. (1998), Hydro Reservoir Management for an Electricity Market with LongTerm Contracts, PhD dissertation, Department of Management, University of Canterbury. Stothert, A. and Macleod, I. (2000), "Competitive Bidding as a Control Problem", IEEE Transactions on Power Systems, 15 (1), 88-94. 20
Stewart et al. (2004)
Villar, J. and Rudnick, H. (2003), "Hydrothermal Market Simulator Using Game Theory: Assessment of Market Power", IEEE Transactions on Power Systems, 18 (1), 91-98. Visudhipan, P. and Ilic, M. D. (1999), "Dynamic Games-Based Modeling of Electricity Markets", Proceedings of IEEE PES 1999 Summer Meeting, 1, 274-281.
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