Building the Mathematical Model 55It is in this connection relevant to remind that the scope was to build a model with someof the same characteristics as the Danish energy system, which can be used to help clarifyhow increased wind penetration will affect the system, as well as how different toolsfor increased production flexibility might reduce some of the documented problems – notto create a perfect approximation of the West Danish heat- and power system. However,a great part of the system characteristics, when it comes to dimensioning the productioncapacities as well as the profiles for wind production and the heat- and electricity demands,are either heavily inspired by, or have directly been taken from, data of theWestern energy system.5.3.1. Power system simplificationsOne of the key problems of building an optimization model is to find the right balancebetween, on the one side, creating a simple and efficient model that still meets the objectiveof the model and, on the other side, building a compute-heavy, high detailed modelwith great similarities to a real system. Finding the right balance often takes an experiencedmodel programmer, but can also be achieved by starting out with a simple constellationand then, step by step, adding more elements until the final result – althoughthis can be a rather time-consuming process. Figure 5.5 shows the final geographicalsystem.MAIN REGIONEXCHANGE REGION(Hydro)Central CHP-areas (x 7)Decentralized CHP-area (x 1)Extraction units (CHP/coal)Condensing units (El/coal)Backpressure units (CHP/gas)REGIONS (el. transmission)AREAS (CHP distribution)El. peak load facilities (El/oil)Boilers (Heat/gas)Heat pumps (Heat/el.)*Transmission (lossless)Distribution (lossless)* Not a part of reference scenarioFigure 5.5, An illustrative, graphical overview of the fictional heat and power system used forapproximation of the impacts from increasing windpower as well as inclusion of heat pumps andbypass.Transmission regions and distribution areasAs seen above the geographical system has been limited to consist of just two regionswith the exchange region approximated as mainly the Norwegian hydro system. Bydoing this, intention is to imitate the storage mechanism of today’s wind-hydro interplay

54 Building the Mathematical ModelThe time factor is another main difference between the model and the real system.Where market players in the real system mostly plan operations one day ahead, themodel plans and optimizes operations for the entire modeling period, which, as latershown, varies from a week to a month. Only a perfect foresight of future events would bethe equivalent to this (Ravn 2001). For example, having knowledge of the exact windproduction as far into the future as the modeled periods is a good example of a deviationbetween the model and the real system. Nevertheless, the model results can still be regardedas a reasonable approximation to the impacts of the 50 % windpower scenario.As long as the system characteristics are realistic, the deviations in price formations justdescribed, will not be crucial for the approximation of these consequences.5.2.4. Summary, applied theory and methodsSo far, we have seen that the problem to be solved best can be solved by the unit commitmentmodel since there are boundary restrictions connected to the commitment ofsingle power plants, like minimum generation capacity and start-up considerations. Thefundamental problem of unit commitment is figuring out an optimal combination ofcommitted capacities; however, when optimizing this kind of problem, the feasible solutionarea of the root problem is no longer convex, which calls for Mixed Integer Programming(MIP). MIP is considered among the more complex problems to optimize, giventhe binary constraints. Luckily, today there are a number of available solvers to undertakethis rather heavy part, which means, that this project can focus more solely onthe modeling as an application study rather than a method study. The principles behindthe unit commitment problem have been shown, and the arguments for choosing theGAMS language and the Cplex solver have been stated.Furthermore it has been shown, that the formation of the heat- and electricity prices, isa result of the marginal costs of the different types of plants. In the case ofCHP plants,the marginal costs of heat and electricity are mutually dependent by the formation formationsin each market. Since a great part of the production comes from power unitsproducing both heat and electricity, the heat price will usually increase as the electricityprice decreases, and vice versa.It has also been shown that the prices are calculated from what can be interpreted asshadow prices, and finally it has been argued, that the model is equivalent to a marketsystem consisting of price-takers, which means that all player are seeking to maximizetheir profit at all times, by which market power is neglected, and that – despite the differentdeviations – the model can approximate the impacts ofincreasing wind capacityin a market area like West-DK.5.3. Model formulationThis section reviews the formulation of the mathematical model, from defining its “geographical”areas and boundaries to modeling the characteristics of the different productionunits. The basis of this is the proposals regarding the extensions of the wind capacitiesin West-DK stated by Energinet.dk (outlined in Chapter 3).

Building the Mathematical Model 55It is in this connection relevant to remind that the scope was to build a model with someof the same characteristics as the Danish energy system, which can be used to help clarifyhow increased wind penetration will affect the system, as well as how different toolsfor increased production flexibility might reduce some of the documented problems – notto create a perfect approximation of the West Danish heat- and power system. However,a great part of the system characteristics, when it comes to dimensioning the productioncapacities as well as the profiles for wind production and the heat- and electricity demands,are either heavily inspired by, or have directly been taken from, data of theWestern energy system.5.3.1. Power system simplificationsOne of the key problems of building an optimization model is to find the right balancebetween, on the one side, creating a simple and efficient model that still meets the objectiveof the model and, on the other side, building a compute-heavy, high detailed modelwith great similarities to a real system. Finding the right balance often takes an experiencedmodel programmer, but can also be achieved by starting out with a simple constellationand then, step by step, adding more elements until the final result – althoughthis can be a rather time-consuming process. Figure 5.5 shows the final geographicalsystem.MAIN REGIONEXCHANGE REGION(Hydro)Central CHP-areas (x 7)Decentralized CHP-area (x 1)Extraction units (CHP/coal)Condensing units (El/coal)Backpressure units (CHP/gas)REGIONS (el. transmission)AREAS (CHP distribution)El. peak load facilities (El/oil)Boilers (Heat/gas)Heat pumps (Heat/el.)*Transmission (lossless)Distribution (lossless)* Not a part of reference scenarioFigure 5.5, An illustrative, graphical overview of the fictional heat and power system used forapproximation of the impacts from increasing windpower as well as inclusion of heat pumps andbypass.Transmission regions and distribution areasAs seen above the geographical system has been limited to consist of just two regionswith the exchange region approximated as mainly the Norwegian hydro system. Bydoing this, intention is to imitate the storage mechanism of today’s wind-hydro interplay