**3.3. Trading electricity in forward markets**

**Figure 3.** Profit function of selling power in the spot market

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immediately stop production if *pS* <*MC*.

under these conditions is then:

¥

*MC*

*MC*

the optimal operating policy is described by the following equation:

The expected value of the profit in the spot market per unit of generating capacity *bS* (*t*) under

0 () ( ) ¥

It is noteworthy to observe that the first term in equation is the probability of obtaining a zero profit in the spot market. Note also that E *pS* =*MC* only if *MC* =0. By selling the production in the spot market, the generator never incurs in operating losses, i.e. Pr(*bS* <0)=0 as it can

We consider now the more general case where generating units are unavailable, either for planned or unplanned reasons, during a fraction of the time. Let *p* the failure probability and *q* =1− *p* the probability of the unit being available, provided the failure and operating states are the only two mutually exclusive states in which the generator resides. We further assume that the price level and the state of the generator are statistically independent. Under these considerations, the generator cannot always capture de spread *pS* −*MC* and thus the proba‐ bility of obtaining a positive profit will decrease accordingly. The expected operating profit

( ) ( )

E *b q p MC f p dp* (46)

é ù = - é ù ë û *S S SS* ò ë û

¥

*MC*

The probability of having zero profit *β*<sup>0</sup> =Pr(*bS* =0) is given by:

*S SS S S S S MC*

( ) () E

é ù = - ³- é ù é ù ë û *S S SS S* ò ë û ë û

éù é ù =× + - é ù ëû ë û ò ò ë û

( )

E *b p f p dp p MC f p dp* (44)

E *b p MC f p dp p MC* (45)

Given the dramatic volatility of real time electricity prices, a major activity of power trading is structuring hedging strategies by means of tradable derivative instruments like future and option contracts [8]. A power company owning a set of generating units may decide either to sell electricity in advance at a fixed price in a forward market, or wait to the time of delivery and receive the spot price. Deciding on committing production forward or being exposed to volatility of real-time power prices has however a drastic impact on risk.

By selling forward its production, the generator may hedge against a sudden decline of electricity spot prices during the delivery horizon, thereby securing an operating margin. This hedging strategy isolates the generator from the price risk. However, the generator in exchange resigns the opportunity of selling electricity in the spot market if high prices happen.

Electricity markets are typically arranged under a two-settlement system. This approach preserves the economy and efficiency of the physical operation of the power system from any financial commitment the market players have entered into in the past. Under the twosettlement scheme, only deviations from contractual obligations are negotiated in the spot market.

The revenue from the forward contracting is given by the volume sold *PF* times the price *pF* agreed in the forward contract, i.e. *RF* = *pF PF* . On the other hand, the revenue captured by selling in the spot market is given by *RS* = *pSΔP* = *pS* (*PS* −*PF* ). So, the total revenue *RF* from forward contracting and delivering power in the spot market is given as:

$$R\_T = R\_F + R\_s = p\_F P\_F + p\_s (P\_S - P\_F) = P\_F (p\_F - p\_s) + p\_s P\_S \tag{48}$$

the condition E(*bF* )=E(*bS* ) holds [1]. This means that for a risk-neutral generator both policies, either selling in the spot market or hedging in forward markets, are entirely equivalent. Nevertheless, for risk-averse players (which is the rule in real market settings), the hedged strategy is clearly preferred as profit expectation remain unaltered while price risk is elimi‐

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If we consider again unplanned outages of generating units, hedging price risk in the forward markets exposes generating companies to other class of risk, i.e. delivery risk, which also referred as quantity or volume risk. We further examine this important issue. When a generator under a contractual obligation is unable to deliver in the spot market the contracted amount, i.e. *PS* ≠*PF* , the generator is forced to buy replacement power in the spot market at the prevailing price at that time. This may configure a very significant loss if while the generator is down the spot price is considerable higher than its own marginal costs, i.e. *pS* ≫*MC*. Under this situation, the generator may be compelled to buy very expensive replacement power to honor the obligation, incurring in a potentially high financial loss. It is interesting to note that if when the unit is unavailable spot prices are lower or equal than the marginal cost, the generator can even make an extra profit *bF* = *pF* − *pS* > *pF* −*MC*. The probability density function of the forward position under consideration of positive failure probability and the associated

The expected profit can be computed as the expected value of the contract under the hypothesis

é ù ¥ ê ú =- + ë û

As for modern units *q* ≅1, the change in the expected profit due to unit unavailability is

Assuming statistical independence between the unit´s failure and the level of spot prices, the

( ) ( ) <sup>0</sup> ¥ <sup>=</sup> - < ò *<sup>F</sup> F S SS F*

0

*F F SS F S SS* E *b q p MC f p dp p p f p dp* (51)

Pr( 0) Pr( ) *b pp p F SF* <= > (52)

*L p p f p dp b* (53)

( ) ( )

ò ò*MC*

[ ] ( ) ( )

*MC*

typically negligible. However, downside risk increases substantially.

and the conditional expectation on the value of losses can be written as:

*p*

nated.

**3.4. Delivery risk in forward contracting**

delivery risk is illustrated in Figure 5C.

probability of incurring in losses is given by:

of fully reliable unit times the probability of being available:

We can observe the utility of forward contracts by inspecting this equation. If generator delivers in the spot market an amount equal to its contractual obligation, i.e. *PS* =*PF* , the total revenue is set equal to *pF PF* irrespective of the fluctuations of the spot price *pS* .

At the time of delivery, and assuming the generator is price-taker, the term *PF* (*pF* − *pS* ) is fixed and represents the profit of the forward contract against the spot market. Therefore, the profitmaximizing production policy is the same and given by the spot price, irrespective of the contractual obligations. The profit the generator can make by selling electricity in forward markets is given by the expression:

$$B\_F = P\_F \left( p\_F - p\_S(t) \right) + \max \left\lfloor P\_S \left( p\_S(t) - MC \right), 0 \right\rfloor \tag{49}$$

In Figure 5A, a probability density function of the spot price is depicted. In the following, it is assumed that the forward market price is an unbiased estimator of the spot price at the time of delivery. Therefore, the condition *pF* =E(*pS* ) holds. In the forward contract, the generator makes a profit for unit of capacity *bF* = *pF* −*MC*, assuming *pF* >*MC*. Otherwise, the generator is better by avoiding entering into a forward obligation with negative profit. For realizing this profit, the generator must be able to deliver in the spot market the contracted volume in the exact amount. This profit level is achieved as long as the spot price exceeds the marginal cost, i.e. the probability of making this profit is Pr(*bF* = *pF* −*MC*) =Pr(*pS* >*MC*).

Graphically, this probability is represented by the dark grey area under the pdf of the spot price (cf. Figure 5A). The generator can make additional profits in the forward contract, *bF* = *pF* − *pS* > *pF* −*MC*, each time the spot price drops below the marginal cost, i.e. *pS* <*MC*. In fact, the generator is better buying replacement power in the spot market than incurring in fuel costs generating with its own facilities. Figure 5B illustrates the pdf of the profit of a forward contract. When compared with the profit distribution in the spot market (cf. Figure 5), it is easily noticeable the drastic reduction of the profit variance under forward contracting. The forward obligation sets a floor for profits, reducing dramatically dispersion of results and thereby the price risk. In exchange, the generator also foregoes the chance of profiting at times of high power prices in the spot market. The expected profit of a forward contract in terms of the pdf of the spot price *f* (*pS* ) can be expressed as:

$$\begin{aligned} \mathrm{E}\left[\boldsymbol{b}\_{F}\right] &= \left(\boldsymbol{p}\_{F} - \mathrm{MC}\right) \int\_{\mathrm{MC}}^{\mathrm{v}} f(\boldsymbol{p}\_{S}) \, d\boldsymbol{p}\_{S} + \int\_{0}^{\mathrm{MC}} \left[\left(\boldsymbol{p}\_{F} - \boldsymbol{p}\_{S}\right) f(\boldsymbol{p}\_{S})\right] \, d\boldsymbol{p}\_{S} = \\ &= \boldsymbol{p}\_{F} - \mathrm{MC} \int\_{\mathrm{MC}}^{\mathrm{v}} f(\boldsymbol{p}\_{S}) \, d\boldsymbol{p}\_{S} - \int\_{0}^{\mathrm{MC}} \left[\boldsymbol{p}\_{S} f(\boldsymbol{p}\_{S})\right] \, d\boldsymbol{p}\_{S} \end{aligned} \tag{50}$$

It can be mathematically demonstrated that under rational expectations and efficiency of forward markets, i.e. the forward price is an unbiased estimator of the spot price *pF* =E(*pS* ), the condition E(*bF* )=E(*bS* ) holds [1]. This means that for a risk-neutral generator both policies, either selling in the spot market or hedging in forward markets, are entirely equivalent. Nevertheless, for risk-averse players (which is the rule in real market settings), the hedged strategy is clearly preferred as profit expectation remain unaltered while price risk is elimi‐ nated.
