- Email: [email protected]
Modelling hydro reservoir operation in a deregulated electricity market
Pergamon
Int. Trans. Opl Res. Vol. 3, No. 3/4, pp. 243-253, 1996 Copyright © 1996 IFORS. Publishedby ElsevierScience Ltd Printed in Great Britain. All rights reserved S0969-6016(96)00019-6 0969-6016/96 $15.00 + 0.00
Modelling Hydro Reservoir Operation in a Deregulated Electricity Market T. J. SCOTT and E. G. READ University of Canterbury, New Zealand We describe a medium term market simulation model which was built to help analyse the effects of contracts, and of company structure, on the efficiency of a wholesale electricity market. Our approach employs a Dual Dynamic Programming methodology, with the sub-models at each stage being Cournot duopolies. The model played a part in the analysis which helped shape the recent New Zealand Government decisions which move the industry towards a market driven structure. Although similar moves have been made in other countries in recent years, and there have been several electricity market studies reported recently in the literature, gaming by mixed hydro-thermal firms does not seem to have been modelled previously. Copyright © 1996 IFORS. Published by Elsevier Science Ltd.
Key words: Electricity market, Cournot game, reservoir management, dual dynamic programming.
INTRODUCTION To date, hydro reservoirs in New Zealand have been managed by a single State Owned Enterprise (ECNZ) so as to minimise the combined cost of fuel and shortages. The New Zealand Government has for some time been considering splitting ECNZ into several firms. It has been recognised that the managers of these firms may have incentives to adjust output so as to manipulate prices, leading to concerns that such 'gaming' will cause so much distortion to traditional 'merit order' operation as to outweigh any gains from such reform. Our concern, here, has been to quantify the extent of such coordination losses, thus enabling them to be compared with gains estimated to arise in other areas. Similar concerns have been studied in other markets (Allaz and Vila, 1993; Green and Newberry, 1992; Bolle, 1992; Powell, 1993), but none of these studies considered the behaviour of hydro reservoir managers in a mixed hydro thermal system. In particular, concerns did arise in the NZ context that decentralisation of decision making might lead to excessive spill and/or greater probabilities of shortage. The major contribution of this paper is to present a way of modelling these issues, which are critically important in a system where 80% of production is from hydro. In the course of this study we have also developed a practical method to optimise hydro reservoir management in a deregulated environment. Managers of hydro reservoirs need to determine how much water to release given the stock of water, and the current and expected future market conditions. In New Zealand this is done by ECNZ using a centralised planning model based on Dual Dynamic Programming (DDP) (Read, 1985; Read and George, 1990). This model was developed from an analysis of optimality conditions for a traditional DP applied to problems for a continuous state space. It differs from the DDP approach developed for a Stochastic Bender's Decomposition framework by Pereira and Pinto (1991), in that it produces an analytical (piece-wise linear) representation of the entire marginal water value surface for a relatively small number of reservoirs (currently 2), rather than employing sampling techniques to produce an approximation for a much larger number. Here we generalise this analytical approach to DDP so as to optimise reservoir management in a competitive market over a medium term planning horizon (1 year), given a tool to calculate generation for any given sub-period. We interpret the water value curves as demand curves, and show how DDP is analogous to the economic concept of adding demand curves. This conceptual shift represents a major advance with respect to the understanding, and hence acceptance, by the method of potential users. Our model differs from conventional optimisation problems in that it has a Cournot game as a submodel, and hence cannot be directly optimised via standard mathematical programming techniques. The focus of our investigation differs slightly from that of some other studies. For example, Green
Correspondence: E. G. Read, Management Science Department, University of Canterbury, Christchurch, New Zealand ~A,'I
244
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation
and Newbery (1992) report that five firms would be required to make the U K market adequately competitive. But creating even that number of firms in the small NZ market would involve creating very small firms and they could only be given a balanced portfolio by dividing ownership of some key plant between firms. In any case, such a "radical' break-up option is widely held to be politically unattainable. Thus our investigation focused on contractual mechanisms, both to discipline gaming behaviour, and to balance portfolios, in a market involving only two major firms, and a fringe of smaller stations. Finally we note that, since this modelling system was intended for use in a 'one-off" policy study, rather than a routine application, it was implemented with the goals of minimising programming effort, improving the clarity of the code and minimising the learning curve for potential users, at the expense of some computational efficiency. Although computation times are reported, these times could well be substantially reduced by more efficient programming. The fact that our method can be successfully implemented in such an environment, though, argues well for the potential for application by both small and large scale utilities.
S T R U C T U R E OF" A W H O L E S A L E ELECTRICITY M A R K E T The Wholesale Electricity Market considered in this paper consists of several independent generating firms who each offer generation to the market. A market coordinator matches these offers against a market demand curve. Companies are paid for the power used at the spot price, as determined from the intersection of the supply and demand curves. In practice, demand may be exogenous, and independent of the spot price, or there may be a full set of demand side bids. Here, though, the demand for the whole market is aggregated, and it is assumed that the form of the market demand curve is known, at least locally. We have explicitly considered two types of demand curve, linear and constant elasticity, although other types would also fit with our analysis. Since the spot price is directly affected by the offers of the firms, the possibility exists that the firms will adjust their bids in order to extract higher profits. In particular, a station could reduce its output so as to force the spot price to rise. This will be a profitable strategy if the increased revenue on the remaining sales exceeds the lost profit on the production foregone. In practice, generator offers may be quite complex, and will certainly involve prices as well as quantities. Other studies [Drayton, 1996) suggest, though, that the bidding strategy followed by energy limited (hydro) generators can be reasonably represented as targeting a generation quantity. Thus, the formulation we have used for the market model is of the Cournot type, where the firms set the quantities, and the market price is determined by the demand. Other equally valid formulations are possible, and would work with our methodology, provided they produce (nearlyt monotone 'Demand Curves for Release', as discussed later.
CONTRACTS Each of the generating firms may have contracts with consumers to sell them a pre-arranged quantity of electricity, at a pre-arranged contract or strike price. These contracts are private, in the sense that they are arranged outside of the spot market and the contract prices are not directly affected by the spot price in any given period. One way of viewing this transaction is to say that the generating firm sells all the electricity it generates to the spot market at the spot price, then buys back the contracted amount from the market at the spot price, and on-sells it to the consumer at the contract price. Equivalently the generator and consumer each deal independently with the spot market, with the consumer being compensated by the generating firm (or vice versa) for the difference between the spot price and the contract price on the contract quantity. Technically such a contract is known as a two way option [see, e.g., Brealey and Myers (1984), pp. 432-440]. As well as two way contracts, the generating companies may buy and sell one way contracts, known as call options which are effective if the spot price rises above a specified strike price. Typically we would expect these to be sold by thermal station owners (with a strike price related to their
I n t e r n a t i o n a l Transactions in Operational Research Vol. 3, No. 3/4
245
marginal production cost) and bought by hydro station owners, who need to 'buy in' back-up power during low flow periods. Thus we refer to these as back-up contracts. The same effect can be achieved by having the 'back-up' generators offer their services directly to the consumers, who can then buy 'non-firm' energy from hydro generators. Note that these contracts substantially alter generator objectives. Normally we would expect generators to have incentives to force prices up by restricting supply. But this incentive is greatly reduced if most sales are covered by contracts. Indeed a firm which has contracted for more than it can economically produce in a give period will be a nett buyer, and will try to drive the market price down (see Fig. 3 for examples of this). HYDRO T H E R M A L C O O R D I N A T I O N Generating firms are faced with the problem of coordinating their use of hydro and thermal stations throughout the time horizon. They must trade offthe use of water now with saving it for use in later periods when it may be more valuable. Limited water storage capacity must be taken into account, as must restrictions on maximum (and possibly minimum) generation levels for each station. In practice, the inflow is not known in advance, and must be forecast, with some uncertainty. Thus the problem is actually stochastic in nature, a complication which we must deal with in the multi-period model. In other words, the objective of each firm could be described as: To maximise nett revenue minus fuel costs over all time periods, accounting for contract obligations and subject to meeting the restrictions on generation levels and water storage in each period. The optimisation problem for a firm with several thermal stations and one hydro station could be stated as: max~ O
P'(G t
t=lk
91-' k t LieJ'
_
, t
i "CXgi)
(1)
such that si, = si, - 1 + f , i - 1
- 9i, - 1Vt = 1. . . . . T, V i E ~
(2)
g~ <~ 9~ <~ glVt = 1..... T, V i ~ J
(3)
s_i' <~ si' ~<~iiVt = 1. . . . . T, V i e ~ .
(4)
where p is the spot price, a function of the total market generation, G, ~¢ is the set of all power stations, ~¢' is the set of power stations belonging to the firm, each of which produces #i, k is the contracted quantity, c is the production cost, s is the amount of water stored in the reservoir, f is the inflow into the reservoir, ~ is the set of hydro stations and _x and ~ represent lower and upper bounds on x, respectively, Unfortunately (1)-(4) is not a simple optimisation problem, although it does decompose naturally into time periods, assuming that the only link from one stage to the next is the water held in storage. The difficulty lies in the fact that the market price, p(G), is a function of the firm's own generation, g, and also a function of the other firms' generation. Also, each firm influences the market price, and this will influence the decision of the other firms, which in turn will affect this firm's decision. Hence we do not attempt to solve (1)-(4) directly as a mathematical programming problem, but instead treat it as a (continuous state) dynamic programming problem with a market model at each stage of the DP. This allows treatment of the non-linear (and possibly non-convex) objective function. The state transformation function is simply (2), above, and at each stage the optimality conditions for each firm are, by differentiation, (Scott, 1996): dP---~ t ff rrt dg' [9' - k'] + =
(5)
dc'i where ~rt represents this firm's marginal cost of generation, which may be the marginal fuel cost, ~ of a thermal station i, or the multiplier on the constraint (2) for a hydro station. The necessary condition,
246
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation water value curve
4
3.5
3 > £ 2.5 I c
2
1.5
0.5
0
5' 0
' 100
L 150
~ , 2 0 250 Hydro Generation
, 300
350
400
Fig. 1. Short term d e m a n d curve for release (DCR).
(5), is that the firm should generate at a level which will equate marginal cost with marginal profit. This result is identical in form to that for the case of a Cournot oligopolist, except that we have the
nett generation, g' - k', instead of g'. If
= O, as is the case when the firms are all too small to affect
the market price, the 'marginal profit' is just the (shadow) price offered by the rest of the system, and (5) produces the same results as for a reservoir coordinated by centralised optimisation of a national system. Otherwise, the effect of (5) is to 'distort" the output towards the contract quantities.
D E M A N D CURVES FOR RELEASE Suppose we have a method for calculating the generation levels of each firm given a set of market conditions for a particular period, including generation constraints, fuel costs, contracts and the energy demand curve. Now, if we allow the marginal water value (i.e. the 'fuel cost' of the hydro station) to vary over a range of values, we can build up a curve showing optimal hydro generation for a given water value, which we term a Demand Curve for Release (DCR) for a single period. An example is shown in Fig. 1. To apply D D P we require that the DCR be monotone non-increasing over the range of possible releases. Although this cannot be guaranteed in all circumstances, experimental results to date have shown that a small amount of smoothing will produce the desired non-increasing property for the gaming models we have used. Such smoothing does not seem unreasonable to us, given that we are trying to model realistic operational strategies, and it seems unlikely that real managers would concern themselves with small non-monotonicities given the uncertainty in the market many weeks into the future. We assume that 'the game' is played independently in each period, and we aggregate DCRs corresponding to each sub-period in an assumed (five point) Load Duration Curve to obtain the aggregate DCR for each planning period. This implies that generators are not required to maintain any kind of consistency between offers in successive periods. This simplification is justified on the grounds that an earlier study (Culy et al., 1990) showed that such restrictions tend to be counter-productive.
T H E DUAL DYNAMIC P R O G R A M M I N G APPROACH The D D P approach follows the same recursive structure as conventional DP, but concentrates on the dual space of the problem, in this case the marginal value of water, as opposed to the primal
International Transactions in Operational Research Vol. 3, No. 3/4
247
4
3.5
3
~ 2.5
¥
1.5
il :1
:1
1 ZR
i
:
iEor,t:~S ::
0.5
I I I
dw500
I
1000
I
i
1500 de 2000 Storage Level
dt
2500
Fig. 2. H o w to s t e p b a c k o n e p e r i o d .
variable, the generation. Assuming we have an end of horizon demand curve for water in storage (DCS), we recursively add the DCR within each period to the DCS at the end of that period, producing the DCS for the end of the previous period. Details may be found in (Scott, 1996), but reference to Fig. 2 may help in understanding this process. At the beginning of the period, each additional unit of water we hold in storage could either be released during the period or carried over to the next period for later use. At a particular marginal water value, @ we would want to release d w units within the period, and use d e units to meet the demand for storage. Adding these together gives a total demand for water at the beginning of the period at that price, d r We proceed similarly for all values of~, adding the demand curves for the two individual 'markets' to form an aggregate demand for the 'combined market'. If we assume that all the inflow, F, arrives at the beginning of the period, the marginal water value when holding s units of water in storage at the end of the previous period is equal to the marginal water value when holding s + F units in storage at the beginning of the current period, and so we simply shift the beginning of period curve back along the storage axis by the inflow, F, to get the end of period storage demand curve for the previous period. In practice we do not known exactly what the inflow will be in a given period, especially many months ahead, but we do have historical data which is used to estimate a range of likely inflows, and their likelihood of occurring. This gives rise to a set of possible end of previous period marginal value curves, each with a given likelihood, and the expected marginal value of water for a particular storage level is simply a weighted sum of the marginal values from the set of curves we have. Thus we now average these curves along the Marginal Water Value axis to give us the expected marginal value of water in storage at the end of the previous period. Those parts of the curve lying above or below the storage bounds for the reservoir are simply truncated. We have now completed one step in our D D P analysis, giving us an end of period storage demand curve based on the demand curve for release within the next period and the demand curve for storage at the end of the next period. This process can be repeated for the rest of the periods until we have a storage demand curve for each week of the year. These can be through of as determining a water value surface for the year. Given the marginal water value for a given storage level in a given week of the year, we may use our short term market model to determine the appropriate generation levels, which is the goal of the model. The level contours of such a surface are equivalent to the reservoir operating 'guidelines' currently used by ECNZ, below which particular thermal plant are used to conserve storage. Perhaps surprisingly, our results indicate that these guidelines are relatively unaffected by the presence of gaming opportunities. This probably reflects the fact that while the traditional interpretation is no longer strictly valid for thermal plant outside the firm, it remains so for plant
248
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation
within the firm. In particular, radically different operating strategies, such as deliberately spilling water, do not seem to arise.
MARKET EQUILIBRIUM We have outlined a reservoir optimisation approach which will work in a competitive environment, provided we can produce a monotone DCR. A Cournot duopoly is one model of gaming in the short run market which is capable of producing such curves. That is, we suppose that each of the firms has an objective function of the form defined earlier, and assumes that the output of the other firms will not change, even if its own output changes. This approach is rather extreme. In reality, for example, a smaller, lower cost firm might simply expand production in the expectation that the larger firm will reduce its own production rather than accept a lower price. This would lead to higher output and lower prices than those produced here. In the limit the result would approach the other bound discussed below, in which the smaller firm acts as a perfect competitor. Several studies in the literature on gaming in electricity markets have employed this model, though, including (Green, 1993; Allaz and Vila, 1993). Our approach differs from theirs in that they assume both firms to be identical, whereas we have modelled an asymmetric duopoly, as in (Tirole, 1988, Chap. 5). Under these assumptions at a market equilibrium (5) must be satisfied for each of the firms simultaneously. This means that each firm will have reached a local maximum in its profits, and that it can do no better given that the others do not change their output. Averaging (5) over all J firms gives:
dp G - K piG) = ~ . . . . de
J
(6)
where J j=l J
K = ~ ks
(8)
i=l J
j-1
In other words, (6) simply states that the market price will be equal to the average marginal cost minus the change in price due to the average amount of surplus generation. (This assumes that the marginal costs are constant over the relevant range of generation, a point which we will shortly discuss further.) If we know the marginal cost and contract levels for each firm, and we have an expression for market price as a function of total generation, then we may solve (6) for G, the total generation level. With this we can evaluate p, the market price, and substitute this back into (5) for each player, and hence calculate their individual generation levels, gj. In practice each firm's marginal cost will not be a constant, but will instead be a function of their own generation. Typically this will be a stepped function, with the steps representing the fuel costs of the different stations. (Within a single time period water is no different rom other fuels in that it has a constant marginal cost, and so we treat it in the same way as the thermal stations.) Thus we are presented with an integer decision of which stations will be generating and which will not. For the small systems we have studied, with less than 10 stations in total, we have found that an exhaustive search for equilibria of all combinations is not prohibitive. If our method were to be applied to larger systems then a more intelligent search strategy would be appropriate. For the demand curves studied, and with all firms acting to influence the market price there will be only one admissible solution. [See Scott (1996) for a proof of this, based on the idea that moving to a region of higher (or lower) marginal cost can only increase the concavity of the objective function.] In the special case where there is only one firm acting to manipulate the price, and all others are perfect competitors, there may be several alternative equilibria, but the gaming firm can be assumed to choose the admissible solution which maximises its profit.
International Transactions in Operational Research Vol. 3, No. 3/4
249
The key unknown in our model is the way in which price varies with generation, that is, the demand curve. Here we consider two approximations to reality, the constant elasticity demand curve, and the linear demand curve. Define the price elasticity of demand to be OG p
-- - - ' - -
(10)
Op G
where p is the market price, and G is the quantity demanded. If we assume constant elasticity of demand then the market price may be described by
[~Gl~
,11)
p(G) = Po Lgod
where (Po,9o) is a reference point somewhere on the demand curve. Using this in (6) yields the following optimality condition for the optimal price, ~: [1 + aJ]~f - a J ~ f - '
(12)
KP°~ - 0
go Equation (12) is simply a polynomial in ~, and can be solved numerically to give the equilibrium market price for the given marginal costs and contract amounts, from which we can easily determine individual generation levels. For a linear demand curve, (13)
p(G) = Po + p i G - go],
the optimality condition is *
p=
Po + ffJ + p K
--
Pgo
(14)
J+l
This is a closed form expression for ~ which we may solve directly, and then as before use the price to give us the individual generation levels. See also (Green, 1993; Allaz and Vil,a 1993; Scott, 1996).
C O M P U T A T I O N A L E X P E R I E N C E AND RESULTS We chose to write this model using MATLAB because it is easy to understand and to adapt, and produces easy to interpret graphical results, and for the reasons noted earlier. A single period response curve such as those shown in Fig. 3, or the DCR's actually used by the model, each take around 2 min to calculate on a Sun SPARC 10, and it may be necessary to compute, say, 50 of these to give an adequate representation of the situations likely to be found in a year. In the cases illustrated here, there are only two generating companies, each with three power stations, as described in Table 1. We only present results using constant elasticity demand curves in this paper, but see Scott (1996) for linear demand. The elasticity of demand is set at -½, a value which lies somewhere between the short-run and long-run elasticities suggested in other studies, e.g. Green and Newbery (1992). This may be thought of as corresponding to a situation in which generators realise that, even if the market responds to gaming by accepting higher prices without much demand reduction on a particular day, it will react much more strongly to such behaviour over the ensuing weeks and months. Equation (5) implies that if the contracts match the perfect competition output levels exactly then Table 1. Station characteristics Company 1
Station 1 Station 2 Station 3
Company 2
Capacity
Marginal cost
Capacity
Marginal cost
300 400 300
1.0 2.0 3.0
300 500 500
0.8 1.9 2.4
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation
250
Key: + 10% over, * 10% under
10 8 i.
6 4
E
2
J
oI 0
I
i
I
I
500
1000
1500
2000
2500
generation Fig. 3. Suppliers contracted for 10% under and 10% over the perfect competition demand. Both firms are game players. 10
Key: + 50% over, * 50% under
6
4
• + +
0
0
+
+
+
+
+
~d- + +
+ + ++
++
+++
+
~
3~" I
i
I
I
500
1000
1500
2000
2500
generation Fig. 4. Suppliers contracted for 50% under and 50% over the perfect competition demand. Both firms are game players. One of the series of constant elasticity demand curves is shown as an example. the market spot price will exactly equal the perfect competition market price. If a supplier is over-contracted, making it a nett buyer, then it will try to force the market price down to reduce its losses. If a supplier is undercontracted, making it a nett supplier, then it will try to force the market price up to increase its profits. Figures 3 and 4 show the effects of a range of contract levels from 50% under to 50% over the a m o u n t which the firms would have generated if they had all behaved as perfect competitors. Both firms are game players. The resulting response curves follow the solid line of the true supply curve closely for the range of + 10%, but with + 50% contracting there are significant discrepancies, and hence inefficiencies. The curves are no longer strictly m o n o t o n e when the contract levels are not close to the perfect competition levels, although the n o n - m o n o t o n i c i t y is minor and does not necessarily imply a n o n - m o n o t o n e DCR. Figures 5 and 6 show the corresponding response curves for the case where one of the companies behaves as a perfect competitor. It is interesting to note that although the behaviour is generally less extreme, there are now multiple solutions for a given d e m a n d curve. However it is the game player alone who gets to choose which o u t c o m e will occur, giving a unique solution, as shown in the figures. Note that this response curve is not only n o n - m o n o t o n e , but is now discontinuous at some points. Where necessary, we have s m o o t h e d curves to produce m o n o t o n e DCRs, as noted earlier.
ILLUSTRATIVE RESULTS FROM THE SIMULATION MODEL The full market simulation produces operating guidelines for a one year horizon with weekly periods, each having three sub-periods and five possible inflow levels. This simulation takes a r o u n d 6 min of C P U time, in addition to time spent c o m p u t i n g the DCRs. For this example we calculated a total of six D C R s representing low, medium and high load for both winter and summer, an overhead of a r o u n d 12 min. We have completed tests on a model representing the New Zealand system, with one firm
International Transactions in Operational Research Vol. 3, No. 3/4 10
251
Key: +, 10% under contracted !
46-81
".
E 2
:
.............
I
nl 0
.=, ~ .=, ~ ~ =:7:':.~'-4-1--I1--11- + + + - ÷ 4 +
i 500
v
. . . .
"
"i i i
J
+++++ + t t t t-l~t-li-li~..~ : .-. : : : :: : : .I.+ + + + + ...............................
4++4+
+ + + + + + +
, 1000
, 1500
2000
2500
generation 10
Key: + 10% over contracted
8 6
i
4 E 2 oI
0
i¸ I
I
500
1000
iil
1500
2000
2500
generation Fig. 5. Suppliers contracted for 10% under and 10oVoover the perfect competition demand. Firm 1 is a game player, Firm 2 is behaving as a perfect competitor. The preferred solutions are marked with circles.
"'-
10
Key: + 50% under contracted
6
'
'
2 0
500
I000
1500
2000
2500
generation Key: + 50% over contracted
10 o
8-
•r~
6
~
. .....
!
... ~
~. . . . . .
" "
.....
4
0
t
i
i
)
500
1000
1500
2000
2500
generation Fig. 6. Suppliers contracted for 50% under and 50% over the perfect competition demand. Company 1 is a game player, Company 2 is a perfect competitor. The preferred solutions are marked with circles. One of the series of constant elasticity demand curves is shown as an example.
252
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation
Table 2. Summary statistics of distortions for simulation of four different scenarios Scenario
Contracts firm one Contracts firm two Back-up contracts Price Total profit Consumer surplus Total benefit
A
B
C
D
80% 80% 0% +21.2 + 17.8 2I .0 3.1
80% 80°/; 100% + 19.1 + 17.5 19.1 1.6
100% 100% 0% +5.8 + 3.2 - 5.8 - 2.2
100% 100% 100% +0.0 +0.0 - 0.0 -0.0
controlling the major storage reservoirs and a single thermal, the second controlling the remainder of the thermal, and fringe firms owing several (non-storage) hydro systems. The results shown here assume the Cournot duopoly model, and should be taken as upper bounds on the distortion possible. The simulation was run for a large number of hydrological sequences, and the summary results are presented in Table 2 for four scenarios, each with different contract levels. The distortion is a measure of the deviation from perfect competition of a variety of key indicators for each scenario. (Apart from price, all changes are expressed in absolute terms with the unit of measurement being 1% of the total value of the market.) Scenario A which represents 80% contracting with no back-up between firms shows quite high distortions, with a large transfer of wealth from the consumers to the producers and the spot price is up by more than 20%. This only directly affects the 20% of the energy which is traded at the spot price. It may also affect the remaining 80% sold via the contracts as the spot price level feeds through into prices for new contracts, although this level of distortion is clearly not sustainable in the long term as the threat of entry will put a price cap on the spot market. The most important distortion measure is the dead-weight loss to society, in this case 3. t %. Scenario B again has 80% contracting, but this time with full back-up between the firms which nearly halves the deadweight loss. For Scenario C, 100% contracting with no backup, the spot price is raised by less than 6%, but the deadweight loss to society, indicated by the drop in total benefit is still more than 2%. However for Scenario D, which is a close approximation to 100% contracting with full back-up, the distortions are eliminated within the accuracy of the model. A key message to take from these results is that contracts do have a significant impact on market behaviour, and that the market can be made to closely mimic perfect competition if contracts are set appropriately. It is also worth noting that the model is very sensitive to changes in the elasticity of demand. We will consider this in Scott (1996), but preliminary results show that the distortions in a spot market with an elasticity of demand of -0.1 and 90% contracting are as great as those in a market with an elasticity of demand of - 0 . 3 3 and 50% contracting.
CONCLUSIONS AND FUTURE WORK We have shown how the existing D D P techniques, as currently used by ECNZ, may be adapted to model a competitive wholesale electricity market. As long as we have some way of deriving monotone single period water demand curves, the procedure is relatively simple, only requiring the ability to add demand curves together. Our results suggest that, assuming a fairly extreme model of gaming, there is relatively little loss in coordination efficiency if, but only if, there is a high level of contracting and/or a high effective elasticity. In that case the efficiency losses may well be kept within acceptable limits, and be offset by gains in other areas. In particular, some of the more bizarre outcomes in which companies might find it beneficial to withhold generation from the market, whilst simultaneously spilling water, for example, or deliberately push storage towards either spill or shortage, did not occur, contrary to predictions made by some critics of reform. In fact this type of contracting gives generators, if anything, excessive incentives to avoid generation falling below contract levels. Of course this whole analysis ignores a crucial question: 'What level of contracts would occur in
International Transactions in Operational Research Vol. 3, No. 3/4
253
equilibrium?'. In theory, the threat of gaming by generators should give customers strong incentives to buy contracts, both to protect themselves and, by taking up more contracts than their load, to take advantage of any extra profits generated by such gaming activities. In equilibrium, we might expect a high level of contracting, and, therefore, little gaming activity, with prices set by the cost of building new capacity, since any party who is not satisfied with the price being offered in this kind of market is free to build their own. Indeed, it seems unlikely that much new plant will be built in such markets unless consumers are prepared to sign long term contracts for the output. In practice the level of contracting may be substantially lower, for a variety of reasons, including the fact that end customers are represented in the market by retailers who are reluctant to commit themselves to long term contracts without having similar arrangements with their customers, and the fact that such retailers may see little need to contract with existing plant if they think Government will exert pressure to limit gaming even if they do not enter into contracts. It can be argued that, provided reasonably priced contracts are available, consumers have all the protection they need. As it happens in this case, the NZ Government has decided to restructure the industry in a form very similar to that studied here. One generator controls the two major reservoir storage systems, and one thermal plant which most commonly sets the spot price, and hence the water value, partly in order to ensure that these three key elements are closely correlated. This gives the generator considerable market power, but this has been restricted by making a high proportion of its capacity available on long term contract at 'reasonable' prices. The market is being left to correct the imbalance between the firms, in terms of plant portfolios, either with direct 'back-up' arrangements between them, or indirectly, via generator to consumer contracts. Our studies have also underlined the importance of consumers holding (call) options which cover their core requirements when prices are high, both to protect their own position in times of shortage, and to give generators strong incentives to cover their requirements. Further, while 'gaming' has not been prohibited, the two major firms are being retained in public ownership, and this seems likely to restrain tendencies in that direction. It is hoped that, as load grows and contracts expire, competitive entry will be sufficient to restrain prices and limit gaming to levels which do not unduly distort merit order operation. This "solution' is clearly a compromise which does not entirely satisfy any party. The actual outcome remains to be seen. Acknowledgements - - This work is one of the projects of the energy Modelling Research Group at the University of Canterbury, funded by ECNZ and Trans Power. The authors would like to thank ECNZ and Trans Power for their support. and acknowledge the helpful comments of the anonymous referee, but of course the views expressed in this paper and the responsibility for any errors or omissions are entirely our own.
REFERENCES Allaz, B. & Vila, J.-L. (1993). Cournot competition, forward markets and efficiency. Journal of Economic Theory 59, 1-16. Bolle, F. (1992). Supply function equilibria and the danger of tacit collusion: the case of spot markets for electricit). Energy Economics 14(2), 94-102. Brealey, R.A. & Myers, S. C. (1984). Principles of Corporate Finance. McGraw-Hill Series in Finance, McGraw-Hill Book Company, Singapore. Culy, J. G., Mayes, K. & Read, E.G. (1990). Short term gaming in the power pool, theory and results. Report for corporate development. ECNZ, New Zealand Institute of Economic Research (Inc.) and EYMSL. Drayton-Bright, G.R. (1996). Energy and Reserve Coordination in a Decentralised Electricity Market. Ph.D. thesis, University of Canterbury, Department of Management. Green, R. J. (1993). Long-term contracts and the electricity spot market. In Proceedingsof the Royal Economic Society Annual Conference, York. Green, R. J. & Newberry, D. M. (1992). Competition in the British electricity spot market. Journal of Political Economy 100(5), 929-953. Pereira, M. V. F. & Pinto, M. V.G. (1991). Multi-stage stochastic optimization applied to energy planning. Mathematical Proyrammin9 52, 359-375. Powell, A. (1993). Trading forward in an imperfect market: the case of electricity in Britain. The Economic Journal 103, 444~.53. Read, E, G. (1985). A new variant of stochastic DP for multi-reservoir release scheduling. In Proceedings twenty-first ORSNZ Conference, pp. 4-7. Read, E.G. & George, J.A. (1990). Dual dynamic programming for linear production/inventory systems. Computers aml Mathematics With Applications 19(I1), 29-42. Scott, T. J. (1996). Hydro reservoir management for an electricity market with long-term contracts. Ph.D. thesis, Uni,,ersity of Canterbury, Department of Management, Tirole, J. (1988). The Theory of Industrial Organization. Massachusetts Institute of Technology Press, Cambridge, Massachusetts.
Int. Trans. Opl Res. Vol. 3, No. 3/4, pp. 243-253, 1996 Copyright © 1996 IFORS. Publishedby ElsevierScience Ltd Printed in Great Britain. All rights reserved S0969-6016(96)00019-6 0969-6016/96 $15.00 + 0.00
Modelling Hydro Reservoir Operation in a Deregulated Electricity Market T. J. SCOTT and E. G. READ University of Canterbury, New Zealand We describe a medium term market simulation model which was built to help analyse the effects of contracts, and of company structure, on the efficiency of a wholesale electricity market. Our approach employs a Dual Dynamic Programming methodology, with the sub-models at each stage being Cournot duopolies. The model played a part in the analysis which helped shape the recent New Zealand Government decisions which move the industry towards a market driven structure. Although similar moves have been made in other countries in recent years, and there have been several electricity market studies reported recently in the literature, gaming by mixed hydro-thermal firms does not seem to have been modelled previously. Copyright © 1996 IFORS. Published by Elsevier Science Ltd.
Key words: Electricity market, Cournot game, reservoir management, dual dynamic programming.
INTRODUCTION To date, hydro reservoirs in New Zealand have been managed by a single State Owned Enterprise (ECNZ) so as to minimise the combined cost of fuel and shortages. The New Zealand Government has for some time been considering splitting ECNZ into several firms. It has been recognised that the managers of these firms may have incentives to adjust output so as to manipulate prices, leading to concerns that such 'gaming' will cause so much distortion to traditional 'merit order' operation as to outweigh any gains from such reform. Our concern, here, has been to quantify the extent of such coordination losses, thus enabling them to be compared with gains estimated to arise in other areas. Similar concerns have been studied in other markets (Allaz and Vila, 1993; Green and Newberry, 1992; Bolle, 1992; Powell, 1993), but none of these studies considered the behaviour of hydro reservoir managers in a mixed hydro thermal system. In particular, concerns did arise in the NZ context that decentralisation of decision making might lead to excessive spill and/or greater probabilities of shortage. The major contribution of this paper is to present a way of modelling these issues, which are critically important in a system where 80% of production is from hydro. In the course of this study we have also developed a practical method to optimise hydro reservoir management in a deregulated environment. Managers of hydro reservoirs need to determine how much water to release given the stock of water, and the current and expected future market conditions. In New Zealand this is done by ECNZ using a centralised planning model based on Dual Dynamic Programming (DDP) (Read, 1985; Read and George, 1990). This model was developed from an analysis of optimality conditions for a traditional DP applied to problems for a continuous state space. It differs from the DDP approach developed for a Stochastic Bender's Decomposition framework by Pereira and Pinto (1991), in that it produces an analytical (piece-wise linear) representation of the entire marginal water value surface for a relatively small number of reservoirs (currently 2), rather than employing sampling techniques to produce an approximation for a much larger number. Here we generalise this analytical approach to DDP so as to optimise reservoir management in a competitive market over a medium term planning horizon (1 year), given a tool to calculate generation for any given sub-period. We interpret the water value curves as demand curves, and show how DDP is analogous to the economic concept of adding demand curves. This conceptual shift represents a major advance with respect to the understanding, and hence acceptance, by the method of potential users. Our model differs from conventional optimisation problems in that it has a Cournot game as a submodel, and hence cannot be directly optimised via standard mathematical programming techniques. The focus of our investigation differs slightly from that of some other studies. For example, Green
Correspondence: E. G. Read, Management Science Department, University of Canterbury, Christchurch, New Zealand ~A,'I
244
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation
and Newbery (1992) report that five firms would be required to make the U K market adequately competitive. But creating even that number of firms in the small NZ market would involve creating very small firms and they could only be given a balanced portfolio by dividing ownership of some key plant between firms. In any case, such a "radical' break-up option is widely held to be politically unattainable. Thus our investigation focused on contractual mechanisms, both to discipline gaming behaviour, and to balance portfolios, in a market involving only two major firms, and a fringe of smaller stations. Finally we note that, since this modelling system was intended for use in a 'one-off" policy study, rather than a routine application, it was implemented with the goals of minimising programming effort, improving the clarity of the code and minimising the learning curve for potential users, at the expense of some computational efficiency. Although computation times are reported, these times could well be substantially reduced by more efficient programming. The fact that our method can be successfully implemented in such an environment, though, argues well for the potential for application by both small and large scale utilities.
S T R U C T U R E OF" A W H O L E S A L E ELECTRICITY M A R K E T The Wholesale Electricity Market considered in this paper consists of several independent generating firms who each offer generation to the market. A market coordinator matches these offers against a market demand curve. Companies are paid for the power used at the spot price, as determined from the intersection of the supply and demand curves. In practice, demand may be exogenous, and independent of the spot price, or there may be a full set of demand side bids. Here, though, the demand for the whole market is aggregated, and it is assumed that the form of the market demand curve is known, at least locally. We have explicitly considered two types of demand curve, linear and constant elasticity, although other types would also fit with our analysis. Since the spot price is directly affected by the offers of the firms, the possibility exists that the firms will adjust their bids in order to extract higher profits. In particular, a station could reduce its output so as to force the spot price to rise. This will be a profitable strategy if the increased revenue on the remaining sales exceeds the lost profit on the production foregone. In practice, generator offers may be quite complex, and will certainly involve prices as well as quantities. Other studies [Drayton, 1996) suggest, though, that the bidding strategy followed by energy limited (hydro) generators can be reasonably represented as targeting a generation quantity. Thus, the formulation we have used for the market model is of the Cournot type, where the firms set the quantities, and the market price is determined by the demand. Other equally valid formulations are possible, and would work with our methodology, provided they produce (nearlyt monotone 'Demand Curves for Release', as discussed later.
CONTRACTS Each of the generating firms may have contracts with consumers to sell them a pre-arranged quantity of electricity, at a pre-arranged contract or strike price. These contracts are private, in the sense that they are arranged outside of the spot market and the contract prices are not directly affected by the spot price in any given period. One way of viewing this transaction is to say that the generating firm sells all the electricity it generates to the spot market at the spot price, then buys back the contracted amount from the market at the spot price, and on-sells it to the consumer at the contract price. Equivalently the generator and consumer each deal independently with the spot market, with the consumer being compensated by the generating firm (or vice versa) for the difference between the spot price and the contract price on the contract quantity. Technically such a contract is known as a two way option [see, e.g., Brealey and Myers (1984), pp. 432-440]. As well as two way contracts, the generating companies may buy and sell one way contracts, known as call options which are effective if the spot price rises above a specified strike price. Typically we would expect these to be sold by thermal station owners (with a strike price related to their
I n t e r n a t i o n a l Transactions in Operational Research Vol. 3, No. 3/4
245
marginal production cost) and bought by hydro station owners, who need to 'buy in' back-up power during low flow periods. Thus we refer to these as back-up contracts. The same effect can be achieved by having the 'back-up' generators offer their services directly to the consumers, who can then buy 'non-firm' energy from hydro generators. Note that these contracts substantially alter generator objectives. Normally we would expect generators to have incentives to force prices up by restricting supply. But this incentive is greatly reduced if most sales are covered by contracts. Indeed a firm which has contracted for more than it can economically produce in a give period will be a nett buyer, and will try to drive the market price down (see Fig. 3 for examples of this). HYDRO T H E R M A L C O O R D I N A T I O N Generating firms are faced with the problem of coordinating their use of hydro and thermal stations throughout the time horizon. They must trade offthe use of water now with saving it for use in later periods when it may be more valuable. Limited water storage capacity must be taken into account, as must restrictions on maximum (and possibly minimum) generation levels for each station. In practice, the inflow is not known in advance, and must be forecast, with some uncertainty. Thus the problem is actually stochastic in nature, a complication which we must deal with in the multi-period model. In other words, the objective of each firm could be described as: To maximise nett revenue minus fuel costs over all time periods, accounting for contract obligations and subject to meeting the restrictions on generation levels and water storage in each period. The optimisation problem for a firm with several thermal stations and one hydro station could be stated as: max~ O
P'(G t
t=lk
91-' k t LieJ'
_
, t
i "CXgi)
(1)
such that si, = si, - 1 + f , i - 1
- 9i, - 1Vt = 1. . . . . T, V i E ~
(2)
g~ <~ 9~ <~ glVt = 1..... T, V i ~ J
(3)
s_i' <~ si' ~<~iiVt = 1. . . . . T, V i e ~ .
(4)
where p is the spot price, a function of the total market generation, G, ~¢ is the set of all power stations, ~¢' is the set of power stations belonging to the firm, each of which produces #i, k is the contracted quantity, c is the production cost, s is the amount of water stored in the reservoir, f is the inflow into the reservoir, ~ is the set of hydro stations and _x and ~ represent lower and upper bounds on x, respectively, Unfortunately (1)-(4) is not a simple optimisation problem, although it does decompose naturally into time periods, assuming that the only link from one stage to the next is the water held in storage. The difficulty lies in the fact that the market price, p(G), is a function of the firm's own generation, g, and also a function of the other firms' generation. Also, each firm influences the market price, and this will influence the decision of the other firms, which in turn will affect this firm's decision. Hence we do not attempt to solve (1)-(4) directly as a mathematical programming problem, but instead treat it as a (continuous state) dynamic programming problem with a market model at each stage of the DP. This allows treatment of the non-linear (and possibly non-convex) objective function. The state transformation function is simply (2), above, and at each stage the optimality conditions for each firm are, by differentiation, (Scott, 1996): dP---~ t ff rrt dg' [9' - k'] + =
(5)
dc'i where ~rt represents this firm's marginal cost of generation, which may be the marginal fuel cost, ~ of a thermal station i, or the multiplier on the constraint (2) for a hydro station. The necessary condition,
246
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation water value curve
4
3.5
3 > £ 2.5 I c
2
1.5
0.5
0
5' 0
' 100
L 150
~ , 2 0 250 Hydro Generation
, 300
350
400
Fig. 1. Short term d e m a n d curve for release (DCR).
(5), is that the firm should generate at a level which will equate marginal cost with marginal profit. This result is identical in form to that for the case of a Cournot oligopolist, except that we have the
nett generation, g' - k', instead of g'. If
= O, as is the case when the firms are all too small to affect
the market price, the 'marginal profit' is just the (shadow) price offered by the rest of the system, and (5) produces the same results as for a reservoir coordinated by centralised optimisation of a national system. Otherwise, the effect of (5) is to 'distort" the output towards the contract quantities.
D E M A N D CURVES FOR RELEASE Suppose we have a method for calculating the generation levels of each firm given a set of market conditions for a particular period, including generation constraints, fuel costs, contracts and the energy demand curve. Now, if we allow the marginal water value (i.e. the 'fuel cost' of the hydro station) to vary over a range of values, we can build up a curve showing optimal hydro generation for a given water value, which we term a Demand Curve for Release (DCR) for a single period. An example is shown in Fig. 1. To apply D D P we require that the DCR be monotone non-increasing over the range of possible releases. Although this cannot be guaranteed in all circumstances, experimental results to date have shown that a small amount of smoothing will produce the desired non-increasing property for the gaming models we have used. Such smoothing does not seem unreasonable to us, given that we are trying to model realistic operational strategies, and it seems unlikely that real managers would concern themselves with small non-monotonicities given the uncertainty in the market many weeks into the future. We assume that 'the game' is played independently in each period, and we aggregate DCRs corresponding to each sub-period in an assumed (five point) Load Duration Curve to obtain the aggregate DCR for each planning period. This implies that generators are not required to maintain any kind of consistency between offers in successive periods. This simplification is justified on the grounds that an earlier study (Culy et al., 1990) showed that such restrictions tend to be counter-productive.
T H E DUAL DYNAMIC P R O G R A M M I N G APPROACH The D D P approach follows the same recursive structure as conventional DP, but concentrates on the dual space of the problem, in this case the marginal value of water, as opposed to the primal
International Transactions in Operational Research Vol. 3, No. 3/4
247
4
3.5
3
~ 2.5
¥
1.5
il :1
:1
1 ZR
i
:
iEor,t:~S ::
0.5
I I I
dw500
I
1000
I
i
1500 de 2000 Storage Level
dt
2500
Fig. 2. H o w to s t e p b a c k o n e p e r i o d .
variable, the generation. Assuming we have an end of horizon demand curve for water in storage (DCS), we recursively add the DCR within each period to the DCS at the end of that period, producing the DCS for the end of the previous period. Details may be found in (Scott, 1996), but reference to Fig. 2 may help in understanding this process. At the beginning of the period, each additional unit of water we hold in storage could either be released during the period or carried over to the next period for later use. At a particular marginal water value, @ we would want to release d w units within the period, and use d e units to meet the demand for storage. Adding these together gives a total demand for water at the beginning of the period at that price, d r We proceed similarly for all values of~, adding the demand curves for the two individual 'markets' to form an aggregate demand for the 'combined market'. If we assume that all the inflow, F, arrives at the beginning of the period, the marginal water value when holding s units of water in storage at the end of the previous period is equal to the marginal water value when holding s + F units in storage at the beginning of the current period, and so we simply shift the beginning of period curve back along the storage axis by the inflow, F, to get the end of period storage demand curve for the previous period. In practice we do not known exactly what the inflow will be in a given period, especially many months ahead, but we do have historical data which is used to estimate a range of likely inflows, and their likelihood of occurring. This gives rise to a set of possible end of previous period marginal value curves, each with a given likelihood, and the expected marginal value of water for a particular storage level is simply a weighted sum of the marginal values from the set of curves we have. Thus we now average these curves along the Marginal Water Value axis to give us the expected marginal value of water in storage at the end of the previous period. Those parts of the curve lying above or below the storage bounds for the reservoir are simply truncated. We have now completed one step in our D D P analysis, giving us an end of period storage demand curve based on the demand curve for release within the next period and the demand curve for storage at the end of the next period. This process can be repeated for the rest of the periods until we have a storage demand curve for each week of the year. These can be through of as determining a water value surface for the year. Given the marginal water value for a given storage level in a given week of the year, we may use our short term market model to determine the appropriate generation levels, which is the goal of the model. The level contours of such a surface are equivalent to the reservoir operating 'guidelines' currently used by ECNZ, below which particular thermal plant are used to conserve storage. Perhaps surprisingly, our results indicate that these guidelines are relatively unaffected by the presence of gaming opportunities. This probably reflects the fact that while the traditional interpretation is no longer strictly valid for thermal plant outside the firm, it remains so for plant
248
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation
within the firm. In particular, radically different operating strategies, such as deliberately spilling water, do not seem to arise.
MARKET EQUILIBRIUM We have outlined a reservoir optimisation approach which will work in a competitive environment, provided we can produce a monotone DCR. A Cournot duopoly is one model of gaming in the short run market which is capable of producing such curves. That is, we suppose that each of the firms has an objective function of the form defined earlier, and assumes that the output of the other firms will not change, even if its own output changes. This approach is rather extreme. In reality, for example, a smaller, lower cost firm might simply expand production in the expectation that the larger firm will reduce its own production rather than accept a lower price. This would lead to higher output and lower prices than those produced here. In the limit the result would approach the other bound discussed below, in which the smaller firm acts as a perfect competitor. Several studies in the literature on gaming in electricity markets have employed this model, though, including (Green, 1993; Allaz and Vila, 1993). Our approach differs from theirs in that they assume both firms to be identical, whereas we have modelled an asymmetric duopoly, as in (Tirole, 1988, Chap. 5). Under these assumptions at a market equilibrium (5) must be satisfied for each of the firms simultaneously. This means that each firm will have reached a local maximum in its profits, and that it can do no better given that the others do not change their output. Averaging (5) over all J firms gives:
dp G - K piG) = ~ . . . . de
J
(6)
where J j=l J
K = ~ ks
(8)
i=l J
j-1
In other words, (6) simply states that the market price will be equal to the average marginal cost minus the change in price due to the average amount of surplus generation. (This assumes that the marginal costs are constant over the relevant range of generation, a point which we will shortly discuss further.) If we know the marginal cost and contract levels for each firm, and we have an expression for market price as a function of total generation, then we may solve (6) for G, the total generation level. With this we can evaluate p, the market price, and substitute this back into (5) for each player, and hence calculate their individual generation levels, gj. In practice each firm's marginal cost will not be a constant, but will instead be a function of their own generation. Typically this will be a stepped function, with the steps representing the fuel costs of the different stations. (Within a single time period water is no different rom other fuels in that it has a constant marginal cost, and so we treat it in the same way as the thermal stations.) Thus we are presented with an integer decision of which stations will be generating and which will not. For the small systems we have studied, with less than 10 stations in total, we have found that an exhaustive search for equilibria of all combinations is not prohibitive. If our method were to be applied to larger systems then a more intelligent search strategy would be appropriate. For the demand curves studied, and with all firms acting to influence the market price there will be only one admissible solution. [See Scott (1996) for a proof of this, based on the idea that moving to a region of higher (or lower) marginal cost can only increase the concavity of the objective function.] In the special case where there is only one firm acting to manipulate the price, and all others are perfect competitors, there may be several alternative equilibria, but the gaming firm can be assumed to choose the admissible solution which maximises its profit.
International Transactions in Operational Research Vol. 3, No. 3/4
249
The key unknown in our model is the way in which price varies with generation, that is, the demand curve. Here we consider two approximations to reality, the constant elasticity demand curve, and the linear demand curve. Define the price elasticity of demand to be OG p
-- - - ' - -
(10)
Op G
where p is the market price, and G is the quantity demanded. If we assume constant elasticity of demand then the market price may be described by
[~Gl~
,11)
p(G) = Po Lgod
where (Po,9o) is a reference point somewhere on the demand curve. Using this in (6) yields the following optimality condition for the optimal price, ~: [1 + aJ]~f - a J ~ f - '
(12)
KP°~ - 0
go Equation (12) is simply a polynomial in ~, and can be solved numerically to give the equilibrium market price for the given marginal costs and contract amounts, from which we can easily determine individual generation levels. For a linear demand curve, (13)
p(G) = Po + p i G - go],
the optimality condition is *
p=
Po + ffJ + p K
--
Pgo
(14)
J+l
This is a closed form expression for ~ which we may solve directly, and then as before use the price to give us the individual generation levels. See also (Green, 1993; Allaz and Vil,a 1993; Scott, 1996).
C O M P U T A T I O N A L E X P E R I E N C E AND RESULTS We chose to write this model using MATLAB because it is easy to understand and to adapt, and produces easy to interpret graphical results, and for the reasons noted earlier. A single period response curve such as those shown in Fig. 3, or the DCR's actually used by the model, each take around 2 min to calculate on a Sun SPARC 10, and it may be necessary to compute, say, 50 of these to give an adequate representation of the situations likely to be found in a year. In the cases illustrated here, there are only two generating companies, each with three power stations, as described in Table 1. We only present results using constant elasticity demand curves in this paper, but see Scott (1996) for linear demand. The elasticity of demand is set at -½, a value which lies somewhere between the short-run and long-run elasticities suggested in other studies, e.g. Green and Newbery (1992). This may be thought of as corresponding to a situation in which generators realise that, even if the market responds to gaming by accepting higher prices without much demand reduction on a particular day, it will react much more strongly to such behaviour over the ensuing weeks and months. Equation (5) implies that if the contracts match the perfect competition output levels exactly then Table 1. Station characteristics Company 1
Station 1 Station 2 Station 3
Company 2
Capacity
Marginal cost
Capacity
Marginal cost
300 400 300
1.0 2.0 3.0
300 500 500
0.8 1.9 2.4
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation
250
Key: + 10% over, * 10% under
10 8 i.
6 4
E
2
J
oI 0
I
i
I
I
500
1000
1500
2000
2500
generation Fig. 3. Suppliers contracted for 10% under and 10% over the perfect competition demand. Both firms are game players. 10
Key: + 50% over, * 50% under
6
4
• + +
0
0
+
+
+
+
+
~d- + +
+ + ++
++
+++
+
~
3~" I
i
I
I
500
1000
1500
2000
2500
generation Fig. 4. Suppliers contracted for 50% under and 50% over the perfect competition demand. Both firms are game players. One of the series of constant elasticity demand curves is shown as an example. the market spot price will exactly equal the perfect competition market price. If a supplier is over-contracted, making it a nett buyer, then it will try to force the market price down to reduce its losses. If a supplier is undercontracted, making it a nett supplier, then it will try to force the market price up to increase its profits. Figures 3 and 4 show the effects of a range of contract levels from 50% under to 50% over the a m o u n t which the firms would have generated if they had all behaved as perfect competitors. Both firms are game players. The resulting response curves follow the solid line of the true supply curve closely for the range of + 10%, but with + 50% contracting there are significant discrepancies, and hence inefficiencies. The curves are no longer strictly m o n o t o n e when the contract levels are not close to the perfect competition levels, although the n o n - m o n o t o n i c i t y is minor and does not necessarily imply a n o n - m o n o t o n e DCR. Figures 5 and 6 show the corresponding response curves for the case where one of the companies behaves as a perfect competitor. It is interesting to note that although the behaviour is generally less extreme, there are now multiple solutions for a given d e m a n d curve. However it is the game player alone who gets to choose which o u t c o m e will occur, giving a unique solution, as shown in the figures. Note that this response curve is not only n o n - m o n o t o n e , but is now discontinuous at some points. Where necessary, we have s m o o t h e d curves to produce m o n o t o n e DCRs, as noted earlier.
ILLUSTRATIVE RESULTS FROM THE SIMULATION MODEL The full market simulation produces operating guidelines for a one year horizon with weekly periods, each having three sub-periods and five possible inflow levels. This simulation takes a r o u n d 6 min of C P U time, in addition to time spent c o m p u t i n g the DCRs. For this example we calculated a total of six D C R s representing low, medium and high load for both winter and summer, an overhead of a r o u n d 12 min. We have completed tests on a model representing the New Zealand system, with one firm
International Transactions in Operational Research Vol. 3, No. 3/4 10
251
Key: +, 10% under contracted !
46-81
".
E 2
:
.............
I
nl 0
.=, ~ .=, ~ ~ =:7:':.~'-4-1--I1--11- + + + - ÷ 4 +
i 500
v
. . . .
"
"i i i
J
+++++ + t t t t-l~t-li-li~..~ : .-. : : : :: : : .I.+ + + + + ...............................
4++4+
+ + + + + + +
, 1000
, 1500
2000
2500
generation 10
Key: + 10% over contracted
8 6
i
4 E 2 oI
0
i¸ I
I
500
1000
iil
1500
2000
2500
generation Fig. 5. Suppliers contracted for 10% under and 10oVoover the perfect competition demand. Firm 1 is a game player, Firm 2 is behaving as a perfect competitor. The preferred solutions are marked with circles.
"'-
10
Key: + 50% under contracted
6
'
'
2 0
500
I000
1500
2000
2500
generation Key: + 50% over contracted
10 o
8-
•r~
6
~
. .....
!
... ~
~. . . . . .
" "
.....
4
0
t
i
i
)
500
1000
1500
2000
2500
generation Fig. 6. Suppliers contracted for 50% under and 50% over the perfect competition demand. Company 1 is a game player, Company 2 is a perfect competitor. The preferred solutions are marked with circles. One of the series of constant elasticity demand curves is shown as an example.
252
T. J. Scott and E. G. Read--Modelling Hydro Reservoir Operation
Table 2. Summary statistics of distortions for simulation of four different scenarios Scenario
Contracts firm one Contracts firm two Back-up contracts Price Total profit Consumer surplus Total benefit
A
B
C
D
80% 80% 0% +21.2 + 17.8 2I .0 3.1
80% 80°/; 100% + 19.1 + 17.5 19.1 1.6
100% 100% 0% +5.8 + 3.2 - 5.8 - 2.2
100% 100% 100% +0.0 +0.0 - 0.0 -0.0
controlling the major storage reservoirs and a single thermal, the second controlling the remainder of the thermal, and fringe firms owing several (non-storage) hydro systems. The results shown here assume the Cournot duopoly model, and should be taken as upper bounds on the distortion possible. The simulation was run for a large number of hydrological sequences, and the summary results are presented in Table 2 for four scenarios, each with different contract levels. The distortion is a measure of the deviation from perfect competition of a variety of key indicators for each scenario. (Apart from price, all changes are expressed in absolute terms with the unit of measurement being 1% of the total value of the market.) Scenario A which represents 80% contracting with no back-up between firms shows quite high distortions, with a large transfer of wealth from the consumers to the producers and the spot price is up by more than 20%. This only directly affects the 20% of the energy which is traded at the spot price. It may also affect the remaining 80% sold via the contracts as the spot price level feeds through into prices for new contracts, although this level of distortion is clearly not sustainable in the long term as the threat of entry will put a price cap on the spot market. The most important distortion measure is the dead-weight loss to society, in this case 3. t %. Scenario B again has 80% contracting, but this time with full back-up between the firms which nearly halves the deadweight loss. For Scenario C, 100% contracting with no backup, the spot price is raised by less than 6%, but the deadweight loss to society, indicated by the drop in total benefit is still more than 2%. However for Scenario D, which is a close approximation to 100% contracting with full back-up, the distortions are eliminated within the accuracy of the model. A key message to take from these results is that contracts do have a significant impact on market behaviour, and that the market can be made to closely mimic perfect competition if contracts are set appropriately. It is also worth noting that the model is very sensitive to changes in the elasticity of demand. We will consider this in Scott (1996), but preliminary results show that the distortions in a spot market with an elasticity of demand of -0.1 and 90% contracting are as great as those in a market with an elasticity of demand of - 0 . 3 3 and 50% contracting.
CONCLUSIONS AND FUTURE WORK We have shown how the existing D D P techniques, as currently used by ECNZ, may be adapted to model a competitive wholesale electricity market. As long as we have some way of deriving monotone single period water demand curves, the procedure is relatively simple, only requiring the ability to add demand curves together. Our results suggest that, assuming a fairly extreme model of gaming, there is relatively little loss in coordination efficiency if, but only if, there is a high level of contracting and/or a high effective elasticity. In that case the efficiency losses may well be kept within acceptable limits, and be offset by gains in other areas. In particular, some of the more bizarre outcomes in which companies might find it beneficial to withhold generation from the market, whilst simultaneously spilling water, for example, or deliberately push storage towards either spill or shortage, did not occur, contrary to predictions made by some critics of reform. In fact this type of contracting gives generators, if anything, excessive incentives to avoid generation falling below contract levels. Of course this whole analysis ignores a crucial question: 'What level of contracts would occur in
International Transactions in Operational Research Vol. 3, No. 3/4
253
equilibrium?'. In theory, the threat of gaming by generators should give customers strong incentives to buy contracts, both to protect themselves and, by taking up more contracts than their load, to take advantage of any extra profits generated by such gaming activities. In equilibrium, we might expect a high level of contracting, and, therefore, little gaming activity, with prices set by the cost of building new capacity, since any party who is not satisfied with the price being offered in this kind of market is free to build their own. Indeed, it seems unlikely that much new plant will be built in such markets unless consumers are prepared to sign long term contracts for the output. In practice the level of contracting may be substantially lower, for a variety of reasons, including the fact that end customers are represented in the market by retailers who are reluctant to commit themselves to long term contracts without having similar arrangements with their customers, and the fact that such retailers may see little need to contract with existing plant if they think Government will exert pressure to limit gaming even if they do not enter into contracts. It can be argued that, provided reasonably priced contracts are available, consumers have all the protection they need. As it happens in this case, the NZ Government has decided to restructure the industry in a form very similar to that studied here. One generator controls the two major reservoir storage systems, and one thermal plant which most commonly sets the spot price, and hence the water value, partly in order to ensure that these three key elements are closely correlated. This gives the generator considerable market power, but this has been restricted by making a high proportion of its capacity available on long term contract at 'reasonable' prices. The market is being left to correct the imbalance between the firms, in terms of plant portfolios, either with direct 'back-up' arrangements between them, or indirectly, via generator to consumer contracts. Our studies have also underlined the importance of consumers holding (call) options which cover their core requirements when prices are high, both to protect their own position in times of shortage, and to give generators strong incentives to cover their requirements. Further, while 'gaming' has not been prohibited, the two major firms are being retained in public ownership, and this seems likely to restrain tendencies in that direction. It is hoped that, as load grows and contracts expire, competitive entry will be sufficient to restrain prices and limit gaming to levels which do not unduly distort merit order operation. This "solution' is clearly a compromise which does not entirely satisfy any party. The actual outcome remains to be seen. Acknowledgements - - This work is one of the projects of the energy Modelling Research Group at the University of Canterbury, funded by ECNZ and Trans Power. The authors would like to thank ECNZ and Trans Power for their support. and acknowledge the helpful comments of the anonymous referee, but of course the views expressed in this paper and the responsibility for any errors or omissions are entirely our own.
REFERENCES Allaz, B. & Vila, J.-L. (1993). Cournot competition, forward markets and efficiency. Journal of Economic Theory 59, 1-16. Bolle, F. (1992). Supply function equilibria and the danger of tacit collusion: the case of spot markets for electricit). Energy Economics 14(2), 94-102. Brealey, R.A. & Myers, S. C. (1984). Principles of Corporate Finance. McGraw-Hill Series in Finance, McGraw-Hill Book Company, Singapore. Culy, J. G., Mayes, K. & Read, E.G. (1990). Short term gaming in the power pool, theory and results. Report for corporate development. ECNZ, New Zealand Institute of Economic Research (Inc.) and EYMSL. Drayton-Bright, G.R. (1996). Energy and Reserve Coordination in a Decentralised Electricity Market. Ph.D. thesis, University of Canterbury, Department of Management. Green, R. J. (1993). Long-term contracts and the electricity spot market. In Proceedingsof the Royal Economic Society Annual Conference, York. Green, R. J. & Newberry, D. M. (1992). Competition in the British electricity spot market. Journal of Political Economy 100(5), 929-953. Pereira, M. V. F. & Pinto, M. V.G. (1991). Multi-stage stochastic optimization applied to energy planning. Mathematical Proyrammin9 52, 359-375. Powell, A. (1993). Trading forward in an imperfect market: the case of electricity in Britain. The Economic Journal 103, 444~.53. Read, E, G. (1985). A new variant of stochastic DP for multi-reservoir release scheduling. In Proceedings twenty-first ORSNZ Conference, pp. 4-7. Read, E.G. & George, J.A. (1990). Dual dynamic programming for linear production/inventory systems. Computers aml Mathematics With Applications 19(I1), 29-42. Scott, T. J. (1996). Hydro reservoir management for an electricity market with long-term contracts. Ph.D. thesis, Uni,,ersity of Canterbury, Department of Management, Tirole, J. (1988). The Theory of Industrial Organization. Massachusetts Institute of Technology Press, Cambridge, Massachusetts.