Power Scheduling with Forward Contracts - CiteSeerX

Power Scheduling with Forward Contracts Stein-Erik Fleten a

a;1 Stein W. Wallace a

Norwegian University of Science and Technology, N{7034 Trondheim, e-mail: [email protected]

Abstract Electricity producers participating in the Nordic wholesale-level market face signi cant uncertainty in in ow to reservoirs and prices on the spot and contract markets. Taking the view of a single risk-averse producer, we propose a stochastic programming model for the coordination of physical generation resources with hedging through the forward market. Numerical results are shown from a model with a two year horizon, and with the rst period being one week. Key words: Stochastic programming, hydro scheduling, portfolio management, deregulated electricity market.

1 Introduction We present a portfolio model for a hydropower producer operating in a competitive electricity market. By portfolio is meant one's own production and a set of power contracts for delivery or purchase, including contracts of nancial nature. We illustrate the dierence between such a model and current industry practice through an example. The model is based on earlier work by Fleten, Wallace & Ziemba (1997) and Grundt, Dahl, Fleten, Jenssen, Mo & Stness (1998). The model is general, but is specialized to t producers in Scandinavia. Following deregulation, the producers have had to change their focus from reliable and cost-ecient supply of electricity to more pro t oriented and competitive objectives. Many countries are in the process of deregulating the electricity Supported by the Norwegian Research Council and members of the Norwegian Electricity Federation through the project \Theory of Portfolio Management in the Electricity Industry". 1

Proceedings of the Nordic MPS, Molde, Norway, May 9{10, 1998. 31 August 1998

industry, at least at the wholesale level, and markets for electricity exists to various extents in the UK, Norway, Sweden, Finland, New Zealand, Australia, USA, Canada and some Latin American countries. We assume the producer has access to functioning electricity forward and futures markets, providing instruments for portfolio management. These markets exist today in some countries, but are, so far, not perfect. Even so, opportunities for diversi cation of risk, using futures markets, has made capital asset pricing and portfolio management techniques relevant for planning in the electric power industry. Employing nancial instruments has important operational implications with respect to power scheduling (Oren 1996). A producer who controls generation facilities and demand side resources now also controls trading and nancial settlements such as forward contracts and other types of derivatives. The existing physical resources will have to be coordinated in concert with the nancial resources, in order to mitigate risk associated with uctuating in ow and prices. Our numerical example is consistent with this view. An ideal electricity portfolio management model should incorporate the risks and also the dynamics of the problem. Norwegian electricity prices, as an example, have shown greater volatility than in most comparable markets (Amundsen & Singh 1992). A stochastic programming approach is therefore appropriate to support the managing of both the \power portfolio" and the nancial portfolio of hydropower producers. On the operations side of hydropower production, stochastic programming methods have been used for years (Gessford & Karlin 1958, Stage & Larsson 1961, Lindqvist 1962). We model the integrated portfolio selection and hydropower scheduling problem as a multistage stochastic program. Stochastic variables are electricity prices, prices on nancial instruments and in ow to reservoirs. In ow uncertainty and price uncertainty are of particular importance for explaining varying nancial performances for the producers. The industry is based on a mix of thermal power and hydropower, with the hydropower dependence making spot prices correlated with the amount of in ow to reservoirs. The last fact is due to the correlation between local and regional precipitation: water shortage or abundance is often national, not just local. Besides, much of the residential heating is done via electricity, so that if the temperature is very low, then not only is demand higher, but there is also likely to be less in ow. The number of realizations of the random variables, as well as the number of stages, must be restricted in order to achieve computational tractability. In our example we have ve stages and four sets outcomes for the random variables per period. This size is chosen so that the time it takes to solve one instance of the model is less than 30 seconds. The model can accommodate 2

multiple interconnected reservoirs, although the number of reservoirs has to be restricted also. It encompasses the decisions regarding buying and selling various classes of contracts, such as futures, forwards and options. In the example there is one reservoir, and two dierent forward contracts. We believe the model can provide the producer with a starting point in making decisions regarding power scheduling, contracting and coordination between those activities. It can provide important information regarding tradeo between risk and expected return on short and long term given the resources available. It should also be able to give useful boundary conditions for use in shorter term power scheduling. Using several types of electricity contracts and other ( nancial) instruments, the producers maximize expected pro ts subject to a risk constraint. First we discuss some aspects of the electricity markets.

2 Electricity markets We assume that transmission and generation services are unbundled, i.e. that there is a general third party access, and discuss here the generation side of the business. For a producer who sells (buys) electricity, the transmission costs do not depend on where the buyer (seller) is located. In Norway, this unbundling is accomplished by regulating the transmission side and having a \free" market on the generation side of the industry. There is a legal requirement for power utilities to have separate nancial reporting for transmission and generation. This is weaker than requiring splitting of companies, although this is the intention of the Norwegian Energy Act. In Norway and Sweden there are two markets for electricity contracts excluding the shorter term spot markets: the Nord Pool ASA organized markets and a bilateral market for over-the-counter (OTC) contracts. Today about 75% of the total turnover of derivative contracts is in the OTC market. In the OTC market the most common contract types are forward contracts with dierent ( xed) load pro les, forward contracts with exibility in the load pro le (load factor contracts) and options. An important type is the exible load factor contract. A typical load factor contract has a one year maturity, 5000 hours of maximum load, with the additional constraint that 2/3 of the contract energy volume must be utilized in the summer season, and 1/3 in the winter season. The markets organized by Nord Pool are classi ed according to the time scale of the contracts traded; there is a control market, a spot market and a futures market. The control market is operated by Statnett, who has the technical responsibility for the main grid. It is used for matching real time supply and 3

demand. Market participants with technical ability to rapidly control their power ow submit bids to Statnett on how they can ramp up or down at which price, and Statnett chooses the most economical way to control the system according to the merit order list. The prices in this market are settled ex post to be equal to the bid price of the producer that was picked by Statnett. What is termed the spot market is actually a forward market settled daily at noon for delivery in the next 12{36 hours. It is meant to re ect the marginal price under the prevailing conditions, and was based on the former power pool market established with restricted access in 1970. The individual supply and demand curves submitted by all participants are aggregated by Nord Pool, and the market is cleared each hour according to the competitive equilibrium model. The actual price and quantities for each hour are then submitted back to the participants. The Nord Pool futures market is organized as a futures market having the spot market price as the underlying reference price. The contracts have a time resolution of one week with no physical delivery. Contracts with delivery up to three years from now can be traded, and contracts that mature more than 5{8 weeks from now are stacked into blocks of 4 weeks and contracts that matures more than a year from now are stacked into seasonal contracts of 4{6 blocks. As the maturity of a block draws nearer, the block is dissolved and new ones created. The main dierence between futures (Nord Pool) and forwards (OTC) is the method of settlement. Futures are settled daily, marking to market. Forwards are settled during the delivery period of the contract. In our model we do not distinguish between the two types, and denote both types forward contracts.

3 Portfolio modeling 3.1 Hydropower scheduling model with spot prices

Power scheduling is often modeled with one aggregated equivalent reservoir and having the reservoir stock as the state variable, stochastic dynamic programming is employed to solve the problem (Gjelsvik, Rtting & Rynstrand 1992). Approximate methods for multireservoir systems can be found in for example (Turgeon 1980, Egeland, Hegge, Kylling & Nes 1982, Sherkat, Campo, Moslehi & Lo 1985). Operations scheduling in deregulated markets with simpli ed contracting is discussed in (Wangensteen, Mo & Haugstad 1995, Scott & Read 1996). The multireservoir problem can be solved exactly by using for example nested Benders decomposition (Pereira & Pinto 1985, Morton 1996) 4

Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 (= T ) Period 1 Period 2 Period 3 Period 4 (= T ? 1) Fig. 1. Example time scale

or variants (Pereira 1989). We formulate the problem here as having one equivalent reservoir, because that simpli cation allows for more focus on the interaction with the new aspects of the model, namely contracts and risk aversion. A production scheduling problem is formulated. There are T ? 1 time periods, or T stages, as illustrated in Figure 1. Periods are time intervals between stages, which are discrete points in time. The time periods of the model do not have to be of equal length. In our example, the rst time period is one week, the second period four weeks, the third period 16 weeks, and then a fourth period with time length 64 weeks. This structure could be changed to re ect the hydrological season. In such a model, we would have to nd an optimal plan for the second stage, dependent on what happened with the price and in ow in the rst period. That is, we would have to model the decisions as if no information is revealed during the periods. The producer is running an ongoing business with an inde nite future. We would like to avoid end eects, which are distortions in the model decisions due to the fact that the model has a nite horizon, whereas the real business problem has an inde nite horizon. For example, if in the model the value of the reservoir at the end of the model horizon is too low, say equal to zero, then the end eect would be that too much water is sold in the last stage. A satisfactory theoretical approach to end eects is found in Grinold (1980). The author proposes a \dual equilibrium" approach where the dual variables, i.e. resource prices, for periods T; : : : increase in value with an appropriate in ation rate. This yields an equivalent one-period constraint set for the in nite horizonperiods T; : : :. For an application of this in the stochastic case, see Cari~no, Myers & Ziemba (1997). We leave this issue for future work. The stochastic variables are price, PRICE , and in ow, INFLOW . They might be correlated; this is re ected in the scenarios. Consider Figure 2. Let t be a vector describing what happens to the uncertain parameters in period t?1 (i.e. a realization of ~t), where  is deterministic. Denote ( ; : : :; T ) a scenario. Scenarios are possible histories up to the end of the horizon, and the gure, called an event tree, shows how the uncertainty unfolds through the horizon. A scenario in the event tree is a path from the root node to a leave node. Let n be a node in the event tree for the problem, corresponding to a realization of ( ; : : :; t). The root node is n = 1, and scenarios are uniquely identi ed by nodes at the last stage, belonging to the set S . Each node has a probability 1

1

2

5

f

 P  , l PPP  l , lf PP  f, f

1

f

 BS  BS  BS  BS S S S     B B    B f f f f f f f f f f f f f f BfSf


 DA f
fDfAf


 DA
ff DfAf

...

...


 DA
 DfAf ff


 DA
 DfAf ff

2 3 4 5

Fig. 2. Example event tree and time scale when T = 5. The nodes represent decisions, while the arcs represent realizations of the uncertain variables.

PROBn, and every node except the root node has an ancestor (parent) node A(n). The stage of node n is STG (n). Assume that t is not known until the end of period t ? 1, i.e. at stage t. The decision variables are reservoir discharge, rELEASEn, and spill. There are two kinds of in ow; one going directly into the power station, ST INFLOWn, and one going into the reservoir, RSRV INFLOWn. Spill has therefore also two components, station spill, sT SPILLn and reservoir spill, rSRV SPILLn . Each node in the event tree represents a decision point, so each variable in the problem must be indexed by the node they belong to. Let rRSV n be the reservoir storage levels. Let G(rSRV;A n ; rSRV n ; rELEASEn) be power generation as a function of reservoir level and discharge for node n, period STG(n) ? 1. Power generation is generally a nonlinear funtion of the height of the water in the reservoir and the discharge, and could easily be non-convex (Tshidimba 1995). However, in our example we disregard head variation eects, and assume that generation gENn is proportional to ow through the station sT FLOWn . ( )

Let V (rSRV n) be the value of the reservoirs at the end of the horizon as a function of the reservoir level. This function has to be speci ed in order to avoid end eects. If a long term scheduling model is available, V may be extracted from this, e.g. in the form of incremental value of stored water in reservoirs. In our example we have used the spot prices so that V (rSRV n) = PRICEn
rSRV n for all leaf nodes n 2 S . This is a crude approximation, and in most runs of our example we x the end-of-horizon reservoir level instead of using that function. The revenue in period STG(n) ? 1 is PRICEn
nETSSPn , where nETSSPn is net sale of power in the spot market in period STG(n) ? 1. Assume no cost of production, and initially that all power generated is valued at the spot price. This will be modi ed later. 6

The problem is max

NX MAX n=1

+

X

s2S

PROBn(1 + )?NSTG n PRICEn nETSSPn ( )

PROBs(1 + )?NT PRICEsrSRV s

(1)

subject to

rSRV n ? rSRV;A n + rELEASEn + rES SPILLn = RSRV ( )

INFLOWn ;

(2)

gENn ? nETSSPn = 0;

(3)

gENn ? 
sT FLOWn = 0;

(4)

sT FLOWn ? rELEASEn + sT SPILLn = ST INFLOWn;

(5)

MINU;STG n  rELEASEn  MAXU;STG n ;

(6)

( )

( )

MIN FLOW;STG n  sT FLOWn  MAX FLOW;STG n ;

(7)

MIN RSRV;STG n  rSRV n  MAX RSRV;STG n ;

(8)

sT SPILLn ; rES SPILLn ; nETSSPn  0

(9)

( )

( )

( )

( )

for n = 1; : : : ; NMAX and with initial reservoir level given. Here  0 is a discount interest rate , (2) is the reservoir balance, and (6), (7) and (8) imposes upper and lower limits on release, station ow and reservoir. The discount factor is adjusted for time periods having unequal length; Nt is the number of years to stage t. Due to concession limits, there might exist minimum stream ow constraints such as 2

rSRV

SPILLn

 MIN RSRV

SPILL;STG (n)

(10)

for n = 1; : : : ; NMAX . When there is a con ict between reservoir constraints and stream ow constraints, the stream ow constraints gets priority. Alternatively, violations of the stream ow constraints could be penalized in the This interest rate should ideally be stochastic and re ect the price of systematic risk, for example as can be inferred from the term structure of futures prices (Carr & Jarrow 1995).

2

7

objective function . Equation (10) would then become 3

rSRV

SPILLn + sTRFL SHORTn = MIN RSRV SPILL;STG (n)

(11)

and the variable sTRFL SHORTn would be penalized in the objective function. In our example, however, we have not included such constraints. 3.2 Extending the model to include electricity contracts

Two contract types are introduced into the model, namely electricity contracts with and without exibility in the release pro le. For the forward contract type there are few special diculties. Of course, we must make sure that the prices of these contracts are statistically consistent with the spot price movements. The delivery pro le of forwards has a constant power level during the whole delivery period. For a survey of multistage portfolio models, speci cally asset and liability management models, see (Mulvey & Ziemba 1995). We draw upon that literature when modeling contracts and risk aversion. The most important contract type is the forward contract. The position in a forward contract at node n having delivery in period k is denoted fWDPOSkn . Let negative fWDPOSkn represent a short position. The prices of these contracts are denoted FPRICEkn{for node n and delivery in period k. We assume that the prices are not in uenced by the trading decisions, i.e. in nitely liquid markets. The contract level, or position, accumulated at node n is

fWDPOSkn = fWDPOSk;A n + fWDBUY kn ? fWDSALEkn ; ( )

(12)

for n = 1; : : : ; NMAX and all k, with the initial forward position given. Rebalancing decisions are made at each stage t, after the realizations of the random variables for the period t ? 1 are known. Purchases and sales of forwards are represented by the variables fWDBUY kn and fWDSALEkn de ned for k > STG(n). Transaction costs are de ned via the coecients b and s . The introduction of electricity contracts means that we need to consider the power balance of the producer. The net ow of power must be zero, so Equation (3) becomes:

gENn + fWDPOS;STG n ? ;n ? nETSSPn = 0; ( ) 1

(13)

This would have the desirable political eect of having environmental aspects count in the objective function

3

8

for n = 1; : : : ; NMAX . The fWDPS;STG n ? ;n variable represents net energy supply from forwards. If the producer has any xed commitments for power delivery it should be added on the right hand side. The accumulated pro t for period t at node n is ( ) 1

8 > > rEV t;A(n) > > > < +F f (1 ? s ) rEV tn = > PRICEn WDSALEtn > ?FPRICEnfWDBUY tn(1 + b ) if STG(n) < t > > > : rEV t;A(n) + PRICEn nETSSPn if STG (n) = t;

(14)

for all n 2 fn : STG(n)  tg. Company policy may restrict investment in contract categories. Limits on short sale, liquidity considerations and risk policies can often be expressed in the form of linear constraints. Illiquidity can also be incorporated as higher transaction costs. Many of the contracts are nancial in nature, which entitle or obligates the holder (the power purchaser) to receive or pay the dierence between the spot price and the strike price, which is the agreed contract price. So far we have modeled forwards as if all of them were contracts for physical delivery. Since some forwards are settled nancially, however, in Equation (13) the variables for selling and buying are not necessarily equal to the physical exchange of power on the power pool spot market. For example, if a producer has hedged against a price decrease by short selling ( nancial) forwards or futures, the producer may end up having to physically sell power cheaply, but is compensated through the hedge contracts. Thus in terms of risk and expected return, the producer may have ended up nancially buying power in a favorable price situation. If the producer is worried about the physical exchange of electricity also, for example due to transmission constraints, it is necessary to model the nancial forwards dierently, so that engaging in such contracts do not aect the power balance. Denoting the position in purely nancial forwards as fIN FWDPOSkn , the accumulated pro t is changed to 9

8 > > rEV t;A(n) > > > +FPRICEn(1 ? s ) [fWDSALEtn > > > > +f ] > < IN FWDSALEtn rEV tn = > ?FPRICEn(1 + b ) [fWDBUY tn > > > +fIN FWDBUY tn] if STG(n) < t > > > > rEV t;A(n) + PRICEn [nETSSPn + > > > : fIN FWDPOStn ] if STG(n) = t;

(15)

for all n 2 fn : STG(n)  tg. In our example there are no constraints relating to the physical power exchange, so this separation of nancial and physical contracts is not used. A critical issue in a portfolio modeling eort is to determine the choice of an objective function and an underlying preference structure. The rst proposed objective is to maximize the rm's wealth at the end of period T |or the present value of all cash ows. This corresponds to a risk-neutral approach with the linear objective function: max +

X k

NX MAX n=1

PROBn

(1 + )?NSTG (n)



PRICEn nETSSPn )

FPRICEkn [(1 ? s)fWDSALEkn ? (1 + b)fWDBUY kn] +

X

s2S

(16)

PROBs(1 + )?NT V (rSRV s )

However, a risk neutral decision maker would never enter into forward contracts unless there is some expected gain|in which case the desired volume would be in nite. In our example we assume that the price on a forward in node n to be the conditional-on- n expected spot price for the delivery period of the forward. Thus we avoid here aspects that relate to speculative motivations of trading. A basic feature of the model is risk aversion; we mainly support the hedging decisions of the producer. Modeling of risk is dependent on the views of the decision maker. The study by Grundt et al. (1998) indicates that decision makers want to maximize expected pro t, subject to achieving certain pro t targets at dierent points in time. A way of accomodating this in a model is to have target levels for nancial performance at dierent stages. The extent to which these targets are not met is called target shortfall, and one would progressively penalize target shortfalls in the objective, e.g. in the form of a piecewise linear cost function as shown in Figure 3. This way of penalizing operational risk has 10

Shortfall costs 6D D

D BB B

B B

B B

B l l l

lh hhhh

hhh Pro t target Achieved pro t at a given stage

Fig. 3. Shortfall cost function

been successful in asset and liability models (Kusy & Ziemba 1986, Cari~no, Kent, Myers, Stacy, Sylvanus, Turner, Watanabe & Ziemba 1994). Let j be an index for the linear segments of the target shortfall variable (assume all targets have the same number of linear segments), and let SCOSTj;STG n be the marginal shortfall cost in segment j , and denote sHORTFjn the shortfall. The following equation de nes the shortfall variables: ( )

rEV tn +

X j

sHORTFjn  TARGt;

(17)

for all n 2 fn : STG(n) = tg and all t for which there is a pro t target TARGt. The proposed objective function now becomes NX MAX

(1 + )?NSTG(n)

(

PRICEnnETSSPn PROBn n =1 X + FPRICEkn [(1 ? s)fWDSALEkn ? (1 + b)fWDBUY kn] k 9 max

?

+

X

X

s2S

j

=

SCOSTj;STG n sHORTFjn; ( )

(18)

PROBs(1 + )?NT V (rSRV s ):

It should be noted that an approach where risk is penalized in the objective function through shortfall costs, leads to a piecewise linear concave objective function in wealth, and the objective function could thus be interpreted as a utility function that re ects risk aversion. See Figure 4. With this approach risk is incorporated in the objective, and the discount interest rate should not 11

Objective function 6

? ? % %

? ?

? ?

%

% %



-



 



Pro t

Fig. 4. Piecewise linear concave utility function

be adjusted for risk . 4

A special type of OTC contract is the so-called load factor contract. With these contracts the holder must continuously decide how much of the contract's energy shall be released. For example, the 5000 hour contract mentioned above can be employed at the maximum power level (which is given in the contract) for no more than 5000 hours of the total 8760 hours in the contract delivery period. The holder can choose not to release the total contract volume, but should always do that because the electricity price is positive. Let rEL LFCn be the energy released from the contract at node n, lFCPOSn the volume of the contract, MAX LFC RELt the maximum release in period t, U the set of summer periods in the contract's delivery period and W , the set of winter periods. The node at which the trading period ends and delivery period begins is denoted D(n). This is for a given contract, i.e. with a speci ed delivery period and load factor (for example, 5000/8760). The release in period t becomes part of the power balance and is thus valued in the objective function at the current spot price. The following constraints, valid for the delivery period of the contract, ensure that the release is according to the contract terms: X

U[W

rEL LFCn  MAX LFC REL;STG n

( )

rEL LFCn  MAX LFC REL;STG n

( )

(19) (20)

It can be argued that this objective function is too simple to be interpreted as an intertemporal utility function, and thus discounting should in some way re ect risk. However, our approach must be considered in the light of our criterion for choosing an objective function, namely, what is computationally feasible and what is acceptable for the decision maker.

4

12

X U

X W

rEL LFCn = lFCPOS;D n FRACS

(21)

rEL LFCn = lFCPOS;D n FRACW

(22)

( )

( )

where FRACS and FRACW is the fraction of the volume to be released in summer and winter respectively, and FRACS + FRACW = 1. In the trading period of the contract the ordinary rebalancing apply and is similar to Equation (12). With short sale of load factor contracts, we must make assumptions about how the holders of contracts that we have sold will behave regarding release from the contract over time. One such assumption could be to assume that these holders will in aggregate behave as if they were risk neutral. Option contracts are simply modeled as another set of contract categories. In some scenarios the option is worthless at maturity. In other scenarios the option has value, in which case it is sold at the moment it expires. The price of the option has to be consistent with the price of the underlying instrument. Like all prices in this model, this is taken care of in the description of the scenarios. In our example there are neither options nor load factor contracts.

4 Scenario Generation The generation of scenarios takes a lot of eort in large-scale stochastic programming models. In Norway there is more than 60 years of observed data on in ow, and a spot market has been in operation for more than 25 years. However, there was restricted access to this market before the deregulation eective from January 1, 1991. Forecasting spot prices and in ow is not a new activity, but forecasting the prices of forwards and futures is more recent. Observed market prices for futures on electricity are available only from the last few years. The Scandinavian futures market have unfortunately had periods of very limited liquidity, which not only makes the historical data less appropriate for forecasting purposes, but also creates a need in portfolio management to limit the sizes of purchases and sales of these contract types. However, the liquidity problems are becoming so small that they are not worth modelling. Since a forward contract essentially is a bet on the average spot price for the delivery period, it is natural to assume that the forward prices are equal to the conditional expected spot price for the delivery period. In order to maintain consistency with the market, the decision maker should then set up the spot price scenarios so that the expected spot price for a period is equal to the current market price of a forward with delivery in that period. Posing 13

this constraint on scenario generation means that the model supports solely the hedging aspect of trading in contracts. If the decision maker expects the average spot price to be dierent from the forward price in any period, she has a speculative motivation for entering into a position in that contract. This aspect of contracting is important to some producers, but the procedure indicated above does not give a model that supports that aspect. The advantages of doing this lies in the importance for risk control and reporting to separate between hedging and speculation. An alternative approach would be to allow for a gap between the forward price and the expected future spot price that gradually is diminished as the time to maturity approaches. In a practical application, such a feature would be valued, because most producers also make speculative trades.

4.1 Price Forecasting

A market equilibrium model frequently used for price forecasting in Scandinavia is the Multiarea Power Scheduling (MPS) model developed by the Norwegian Electric Power Research Institute (EFI) (now a part of SINTEF Energy Research), and is described in (Egeland et al. 1982) and (Botnen, Johannesen, Haugstad, Kroken & Frystein 1996). In the MPS, process models describe production, transmission and consumption activities within the Nordic and adjacent areas. The various demand/supply regions are connected through the electrical transmission network. A solution of the model results in a set of equilibrium prices and production quantities, for each week over the time horizon considered (usually 3 years) and for each historical in ow year. The demand side of the model consists of price dependent and price independent load for each region. Important input for the model is demand, thermal generation costs, and initial reservoir levels. The model is short term in the sense that there are no mechanisms for endogenously increasing production capacity. The MPS recognizes that hydro scheduling decisions are made under the uncertainty of reservoir in ows; to determine the opportunity hydro generation costs production in each region, stochastic dynamic programming is employed on the scheduling problem where production in the region is aggregated into an equivalent reservoir/power station pair. The MPS model generates independent scenarios for price and in ow. However, this structure is not appropriate for multistage programming. What is needed is a scenario tree where information is revealed gradually, and not only after the rst stage. 14

4.2 Generating Scenario Trees

Generation of scenarios can be based on simulation or construction. The applied scenario generation method is a combination of the two. We construct small scenario trees with a method based on nonlinear optimization, see (Hyland & Wallace 1996), and aggregate several small trees into one large tree which is input to the optimization model. We let the decision maker specify the market expectations by any statistical properties that are considered relevant for the problem to be solved, and construct the tree so that these statistical properties are preserved. Generally, expectations will depend on the state of the economy. Some statistical properties are clearly state dependent, while others might be speci ed independently of the state. As an example of state dependency, consider autocorrelation of spot market prices. If prices have been high in the previous period, then it is likely that prices in the following period are high also. We model this eect by letting the price in period t + 1 be a function of the outcome in period t.

5 Numerical Example We consider a producer having one reservoir with a storage capacity of 16473 GWh. The reservoir is presently half full, and there is no initial contracts in the portfolio. The generating capacity per week is 847 GWh. The decision maker wants to nd the optimal release from the reservoir, and the optimal buying and selling of contracts which have delivery in some critical future periods. 5

There is uncertainty in spot market prices, in ow going directly into the reservoir, and in ow going directly to the power station. We employ a four period ( ve stage) model with a total of 256 scenarios. The rst period has a length of one week, the second has a length of four weeks, then 16 weeks and nally 64 weeks. Both prices and in ows are nonstationary time series, for example, in ows are likely to be low in the winter and high in early summer. In order to keep the example simple, we do not consider these eects here. The basis for generating the scenarios is

 user supplied statistical moments for the rst period marginal distributions This reservoir is an aggregate of many smaller reservoirs located in the south of Norway. Because of this, the statistical properties of the in ow will not conform with intuition for one reservoir. 5

15

Table 1 Speci cations of market expectations. Uncertain variable Distribution property Period 1 Period 2-4 Spot market price expected value NOK/MWh 180.0 State dep standard deviation 70.0 State dep skewness 1.23 1.23 kurtosis 4.04 4.04 Reservoir in ow expected value GWh 270.0 State dep standard deviation 200.0 State dep skewness 1.43 1.43 kurtosis 1.71 1.71 Station in ow expected value GWh 90.0 State dep standard deviation 70.0 State dep skewness 2.76 2.76 kurtosis 8.17 8.17 Table 2 Speci cation of correlation.

Period 2 3 4 -0.62 -0.78 -0.87 -0.63 -0.79 -0.88 0.62 0.78 0.87

Correlation 1 Price{Reservoir in ow -0.34 Price{Station in ow -0.36 Reservoir in ow{Station in ow 0.35

of all random variables,  correlation between the variables,  de nition of the state dependent statistical properties and  bounds on outcomes and probabilities Price forecasting and analysis of historical in ow data was done using the MPS model. We assume that the rst four moments and and correlation are the relevant statistical properties, and the speci cations are given in Tables 1 and 2. In this example we have modeled state dependent expected values and standard deviations for all uncertain variables. The other statistical properties are assumed state independent, meaning that they are the same in all states of the world at a certain point in time. 16

Table 3 Adjustment in mean and standard deviation due to nonuniform period length. The numbers for standard deviation are based on analyzing autocorrelation in the time series. Period Uncertain variable Parameter 2 3 4 Spot market price RPL E 1.0 1.0 1.0 RPL SD 0.96 0.74 0.54 Reservoir in ow RPL E 4.0 4.0 4.0 RPL SD 3.69 2.88 2.16 Station in ow RPL E 4.0 4.0 4.0 RPL SD 3.77 3.34 2.32

The state dependent mean in period t > 1 is

E (xit) = [MRiE (xi;t? ) + (1 ? MRi )xi;t? ] RPL Eit 1

1

(23)

where E (xit) is the expected outcome of random variable i in period t, xit is the outcome of random variable i in period t, MRi 2 [0; 1] is the mean reversion factor (a high MRi leads to a large degree of mean reversion), and RPL Eit is a scaling factor depending on the period length relative to the previous period and on the type of random variable i. We let the mean reversion factor, MRi = 0:2 for price and = 0:3 for in ow. For standard deviation we assume that the state dependency is given by

SD(xit) = [V Cijxi;t? ? E (xi;t? )j + (1 ? V Ci)SDBAS (xit)] RPL SDit (24) 1

1

where V Ci 2 [0; 1] is the variance clumping parameter for random variable i (a large V Ci means that variance in the next period is highly in uenced by variance in the previous period), and SDBASit is the average standard deviation of the outcome of random variable i in period t. We let the volatility clumping parameter, V Ci = 0:15 for all variables. The scaling factor RPL SDit is needed due to the nonuniform period lengths. In Table 3 these scaling factors are listed. There are two types of bounds implemented here; relative and absolute. Relative bounds means that the bounds are expressed in terms of minimum or maximum number of (stated dependent) standard deviations from the (state dependent) mean. An absolute bound is expressed in terms of the outcome or the probability. Table 4 lists these bounds. 17

Table 4 Speci cation of bounds.They are assumed constant for all periods. Uncertain Relative Relative Absolute variable upper bound lower bound lower bound Price 9.00 4.50 25.00 NOK/MWh Reservoir in ow 5.00 3.25 29.45 GWh Station in ow 8.00 3.25 2.00 GWh Probability (cond.) 0.01

The rst four stages of the tree is shown in Figure 5. Careful examination of the numbers in the gure will reveal that the speci cations in the tables above is not met 100%. This is probably due to overspeci cation; too many statistical properties are to be satis ed at once. Pro t target shortfall is measured and penalized at stages 4 and 5. There are two contracts, with delivery in the last and next to last period, respectively. There is a s = b = 1% transaction cost on both buying and selling of contracts. The objective (Equation (19)) is maximized for dierent weights on the shortfall costs. In order to mitigate the eect of the possibly wrong speci cation of the value of the water in the reservoir at the model horizon, V (rSRV s), we set target levels for the end-of-horizon reservoir levels, one for each scenario. We found this target by solving rst with no weight on the shortfall costs, i.e. a risk-neutral run, with the value of the reservoir set at spot market prices. In all subsequent runs in this paper, these target reservoir levels are used. Figure 6 shows points on the ecient frontier. We see that the risk neutral point, found at the high right end of the graph, has a risk that is 2.8 times higher than at the minimum risk point at the low left end of the graph. The expected pro t is only 0:7% higher. We conclude that for a hydropower producer, employing a dynamic stochastic model with risk aversion and forward contracts, it is possible to reduce risk signi cantly compared to a risk neutral approach without contracts, and only losing marginally in terms of expected pro t. 5.1 The performance of static portfolio approaches

The current industry practice is to schedule production without contracts rst, and decision support for contract trading is based on static portfolio models. The two approaches should ideally have been compared using rolling horizon 18

0.60 434 1060 307

0.04 277 1700 1460 0.06 402 463 123

0.01 819 4830 655

0.36 277 2430 597

0.08 25 2450 2600

0.13 25 1530 1110 0.04 113 29.4 376 0.74 128 29.4 1170

0.49 440 2680 938 0.06 317 5400 2020 0.02 621 3330 367 0.43 368 3670 1040 0.06 0.44 0.42 0.08

140 265 252 422

6640 7280 4720 4750 5380 4460 4390 3740

0.06 561 19300 3210 0.05 1060 12800 1630 0.36 689 15100 2120 0.53 782 12400 2070 0.36 0.06 0.06 0.52

235 440 151 281

7690 1930 6410 1530 9580 2920 6530 1900

0.82 25 7830 8650 0.06 25 4360 5460 0.11 177 4180 7550 0.01 232 4430 5460 0.06 25 0.01 25 0.11 190 0.81 25

2690 2690 2700 2690 2470 3330 5060 3690

0.67 143 29.4 3710 0.06 43.8 2260 4640 0.22 93.5 1090 3760 0.05 280 29.4 3350

0.06 336 29.4 989

0.06 184 2270 4400 0.67 333 29.4 3120 0.05 542 29.4 2640 0.22 257 1090 3210

0.34 161 1470 441

0.06 64.9 7120 2520 0.02 380 4500 905 0.44 124 4920 1500 0.47 198 3640 1330

1.0 180 268 83.5

0.01 443 5350 2

0.52 467 16500 2 0.02 155 21400 3480 0.02 763 9640 2 0.44 346 13900 2

0.06 25 4880 1620

0.12 173 10900 4570 0.81 25 14700 5470 0.06 25 11100 2910 0.01 25 8980 2910

0.13 62.9 762 174

0.60 25 2710 629

0.06 57.3 2830 1140

0.06 25 0.12 174 0.81 25 0.01 25

6770 1390 6610 1870 8200 2140 6770 1390

0.01 94.8 0.81 44.8 0.06 98.2 0.11 227

4460 1580 8870 3920 5530 1580 5530 3150

0.51 185 388 269

0.19 189 586 911 0.66 176 1560 878 0.07 86.7 3380 1650 0.08 311 378 659

0.35 168 1330 130

0.01 158 1810 1920 0.15 85.9 5570 982 0.04 322 4540 312 0.79 175 3760 377

0.77 186 197 59

0.08 352 382 27

0.05 535 898 2 0.43 357 781 2 0.07 202 3360 1260 0.45 305 1710 142

Fig. 5. The generated event tree. The last period is not shown. The numbers in the boxes represent conditional probability, average spot market price for the period in NOK/MWh, reservoir in ow in GWh and station in ow in GWh. The rst box on the left is deterministic and represent the outcome in the period before the rst stage. Thus the numbers in this box was not generated in the procedure.

19

6560

E(profit)

6550 6540 6530 6520 6510 6500 6490 6480 0

200

400

600

800

1000

E(shortfall costs)

Fig. 6. The ecient frontier shows the tradeo between expected portfolio pro t and risk, and is obtained by solving the model for dierent weights on the shortfall costs in the objective function.

simulations like in (Fleten, Hyland & Wallace 1998), in order to re ect that both with the dynamic and static approaches, the decision maker will use only the rst stage decision, and then rerun the model based on any new information. However, a simpler approach to comparison is adequate for this example. The performance of the static approach in terms of expected pro t and risk at the end of the horizon can be found approximately by rst nding all decisions in the following way:

 First the model is run without contracts, and a risk averse production strat-

egy is obtained.  This schedule is kept xed. For each stage beginning with the rst, the model is rerun with buying and selling allowed only for the current stage. This is repeated until stage 3, where the contract with delivery in the last period is last traded. 6

This means that at any stage, the model only sees a now-or-never opportunity for trading. Ideally, the production should also have been rerun at each stage, but this eect is small, as Section 5.2 shows. The resulting point in the mean-risk diagram is shown as the square o the frontier in Figure 6. The reduction in total objective function value is 1:5%, and one can obtain a 20% reduction in risk with the same level of expected pro t when employing a dynamic approach instead of a static one. We conclude The production scheduling could also have been done with a risk neutral objective. However, the dierence is small.

6

20

that a dynamic stochastic model can add value to portfolio management in this industry. The rst stage decision for the dynamic approach regarding contracts, was to sell (2670; 2174)GWh of the two contract types. For the static approach, the corresponding recommended sale was (4225; 5878)GWh, i.e. 2{3 times as much. This is due to the fact that in the static approach, the model does not see the value of waiting for more information, so that unneccessary transaction costs can be avoided. All risk that can be dealt with through the two forwards, must be mitigated in the rst (current) stage. 5.2 How production is in uenced by contracting

When comparing the dynamic approach with a model run with no contracting at all, we nd that production decisions are marginally in uenced by the possibility of trading. There is slightly more release from the reservoir in the rst stage when there are no contracts, but the dierence is less than 0.2% of release capacity. In the second stage the dierence in expected release is 8.4% of capacity.

6 Further Development As in any model, many aspects of the real system under study have been omitted in order to focus on particularly interesting aspects. We wanted to highlight the coordination of physical generation resources and nancial instruments such as forwards, i.e. portfolio management. Several issues should still be examined before the model can be fully speci ed and then implemented and solved. For example, some producers may control large parts of the total power supply, or there can also exist dominating buyers. We have assumed that the scheduling decisions made by the producer under study does not aect the uncertainty in prices or other random variables. Dominating producers, as in the UK and Scandinavia, may be able to distort spot prices and thereby inhibit the ecient operation of futures markets. The issue is discussed in (Newbery 1984, Amundsen & Singh 1992). How this aects electricity portfolio management needs to be sorted out. For a study with xed contracts and game theory see (Scott & Read 1996). Bushnell (1998) study hydro scheduling in an imperfect market/mixed generation technology system setting. In Norway, there are tax issues causing distortions in the production decisions. These tax rules should ideally be incorporated in some way in a portfolio 21

model. Also, the issue of maintenance and forced outages, and existence of pumped storage units, has been ignored. The scenario generation also needs further development.

7 Solution issues The problem has a considerable (generalized) network structure, but there are some non-network constraints from the load factor contract constraints, and more are likely to follow. Hence it does not seem likely that the network structure is going to be used. However, it is possible that some decomposition scheme will produce non-linear generalized network sub-problems. For a discussion of such problems, see for example (Mulvey & Vladimirou 1991a, Mulvey & Vladimirou 1991b). The model size explodes in the number of time periods, so with deterministic solution methods at hand, the number of stages should be kept below 10. Experiences from industry applications of asset/liability models by stochastic programming also indicate this (Cari~no et al. 1994, Cari~no et al. 1997, Cari~no & Ziemba 1997). The number of scenarios, or realizations of the random variables, must also be restricted in order to achieve acceptable computing times. These modeling constraints may cause disturbances in the recommended decisions, but it focuses on the decisions today, and is likely to give good recommended decisions for the rst stage. We expect that among algorithms that can be used to solve the model, Benders' decomposition (Benders 1962), also referred to as the L-shaped decomposition method (van Slyke & Wets 1969), stochastic dual dynamic programming (SDDP), or a combination of dierent decomposition schemes, will be best suited. SDDP was developed in the context of multireservoir hydroelectric power scheduling by Pereira (1989). 7

Large-scale linear programs can also be solved by commercial optimization packages like CPLEX and IBM's OSL. The recent advances in interior point methods (Lustig, Marsten & Shanno 1994) and the simplex method actually makes this approach an alternative to decomposition. We have used MINOS 5.4 for our example. The model has been formulated having each node in the scenario tree as an index. An earlier formulation used a split-variable form where the problem is This algorithm is used in a commercial prototype of the model outlined in this paper and in (Grundt et al. 1998). The prototype is being developed for Norsk Hydro by SINTEF Energy Research.

7

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separable in each scenario except for explicit informational constraints. This approach is called scenario aggregation (Wets 1989, Rockafellar & Wets 1991). The stochasticity in the problem must be described by a tree, like we have done in this paper, and strong correlation is welcome, because this means that the tree is smaller. The state space is hardly a concern, meaning for example that reservoir aggregation is not necessary. For a solution procedure see for example Mulvey & Ruszczynski (1995) or Kall & Wallace (1994), where an augmented Lagrangian procedure is recommended with the informational constraints removed from the problem and put into the objective function. However, some authors report that these solution methods will probably not prove competitive (Consigli & Dempster 1996). Scenario aggregation models have been developed for hydro scheduling problems in (Escudero, de la Fuente, Garcia & Prieto 1996).

8 Conclusion This note presents a model for portfolio management in a deregulated hydropower based electricity market. A general framework has been set up, and major issues discussed. Many aspects remains to be developed. For example, for the Scandinavian and UK markets, some producers are so large that an assumption of perfect competition in production is not realistic. The presence of markets for electricity makes it necessary for power producers to coordinate physical generation resources with the trading and nancial settlements of \paper" resources such as forward contracts or other types of derivatives that can replace physical deliveries and mitigate the risk associated with uctuating prices on electricity. The industry practice is to use dynamic stochastic models for production sheduling, and static models for contracts, running these sequentially. Using the integrated dynamic model on an example portfolio shows that risk can be reduced by about 20% (for the same level of expected pro t) compared to the industry practice.

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