Building the Mathematical Model 53out at a given electricity generation. The shadow price can therefore be used for indicatingconstrained heat- and electricity production.In the case of electricity generation being constrained by the district heating demand,electricity prices are often lower that the price it would take to cover the marginal powergeneration costs. Therefore, the contribution for covering the loss from generating constrainedpower (at a lower price) will additionally come from an increasment of the marginalheat cost.As the electricity prices are expected to drop due to an increased wind capacity ofin thefuture, the heat production costs of the marginal utility will often be influenced by theunpleasant amount of “waste electricity”. Figure 5.4 describes the two markets are connected,by showing heat price as a function of el. price for a system with an extractionunit (being the marginal heat producer)El.c vHeat pricec mHeatEl. priceP h maxFigure 5.4, In a system with extraction unit as marginal producer, the relation betweenheat and el. price is given by an isoquant-like function, formed by the upper boilerlimit as well as lower backpressure limit, respectively (Source: Balmorel brunch).Economic system equivalenceIn many ways, the characteristics of the unit commitment model differ from the realpower and market system. First of all, performing an economic optimization of the wholeeconomic system like in the model would in the real system correspond to the existenceof just one producer with a complete monopoly 19 . At the real market, producers each decidewhich price is attractive producing under (Nord pool spot) and they add biddingconditions such as block offers. Nonetheless, the mathematical model in this project canbe assumed equivalent to a system where, first of all, the market players at all timesseek to maximize their profit, and second of all, that the market players are pricetakers20 .Unfortunately, a consequence of this is that the model, in its basic form, cannot includemarket power, and thus, that the total costs of the system will be underestimated tosome extent (Bregnbæk 2008)19In the monopoly days, technical considerations of the different power plants were often taking intoaccount with higher when scheduling operation hours.20By price-takers are meant, that they can alter their rate of production and sales without significantlyaffecting the market price of their product, which would be the case when all players are infinitelysmall (Investopedia 2009).

52 Building the Mathematical ModelShadow pricesAs mentioned earlier, the overall objective of optimization modeling is to minimize asystem’s total costs. However, the optimal value of the function in itself may containvery little information, while changes in this optimum can be given important interpretations.To evaluate the costs of these changes, GAMS/Cplex automatically calculates what isknown as the shadow prices. Shadow prices can in this connection be interpreted as thechange in the objective value resulting from a one-unit increase in the constant of thespecific constraint function. Most important is properly the shadow price of the electricityand heat demand satisfaction constraints, respectively. They indicate the specificchange in the total-cost function C(e t i,h t i) (known as the objective function) when raisingthe consumption levels by just one MWh, which can be interpreted as a heat- and electricityprice, respectively. Shadow prices on constraints such as interconnectivity capacities,and upper and lower boundaries of productions units, indicates the value of changingthese restrictions, and are therefore often used in this project for evaluating potentialinvestments. Figure 5.3 shows a principle sketch of a two-dimensional situationwhere tree equilibrium points, one that is free (a) and two that are restricted (b and c),are generating shadow prices. The shadow price is zero at a and non-zero at b and c.byacxFigure 5.3, Three types of optimal solutions (equilibrium) found within a convexregion “shaped” by linear constraints, can assume a shadow price that is eitherzero (a), positive or negative.When the shadow price is positive (b), it is an indication of extra costs connected withraising the upper bound constraint by one unit – and the opposite, if the restriction is alower bound constraint (c). In the case of equilibrium a, it would make no (economic)sense to modify the constants of the constraining functions, as the shadow price at eachlinear constraint is calculated to zero.As the optimal solution found is the equilibrium of a number of linear constraints, theshadow price represents the marginal loss or gain, of deviating one unit from this point.In the case ofCHP units, every optimal operation point is a state of equilibrium betweenheat and el. production, respectively. In the case of extraction units, the operation pointsare often found positioned along one of the limits, representing either the upper or lowerbound of electricity generation or the lower back pressure limit, forcing maximum heat

Building the Mathematical Model 53out at a given electricity generation. The shadow price can therefore be used for indicatingconstrained heat- and electricity production.In the case of electricity generation being constrained by the district heating demand,electricity prices are often lower that the price it would take to cover the marginal powergeneration costs. Therefore, the contribution for covering the loss from generating constrainedpower (at a lower price) will additionally come from an increasment of the marginalheat cost.As the electricity prices are expected to drop due to an increased wind capacity ofin thefuture, the heat production costs of the marginal utility will often be influenced by theunpleasant amount of “waste electricity”. Figure 5.4 describes the two markets are connected,by showing heat price as a function of el. price for a system with an extractionunit (being the marginal heat producer)El.c vHeat pricec mHeatEl. priceP h maxFigure 5.4, In a system with extraction unit as marginal producer, the relation betweenheat and el. price is given by an isoquant-like function, formed by the upper boilerlimit as well as lower backpressure limit, respectively (Source: Balmorel brunch).Economic system equivalenceIn many ways, the characteristics of the unit commitment model differ from the realpower and market system. First of all, performing an economic optimization of the wholeeconomic system like in the model would in the real system correspond to the existenceof just one producer with a complete monopoly 19 . At the real market, producers each decidewhich price is attractive producing under (Nord pool spot) and they add biddingconditions such as block offers. Nonetheless, the mathematical model in this project canbe assumed equivalent to a system where, first of all, the market players at all timesseek to maximize their profit, and second of all, that the market players are pricetakers20 .Unfortunately, a consequence of this is that the model, in its basic form, cannot includemarket power, and thus, that the total costs of the system will be underestimated tosome extent (Bregnbæk 2008)19In the monopoly days, technical considerations of the different power plants were often taking intoaccount with higher when scheduling operation hours.20By price-takers are meant, that they can alter their rate of production and sales without significantlyaffecting the market price of their product, which would be the case when all players are infinitelysmall (Investopedia 2009).