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Your market model: the mathematics

In this article, Frank Krysiak explains the mathematical formulation of an optimization model to figure out how the supply mix was defined.
© University of Basel

Let us now have a somewhat more detailed look at the model that you have used two steps ago. On the surface, this model looks rather similar to the models that we have used and discussed before. However, behind the scenes, it has a very different structure.

Mathematical equatations

In the models before, you had to choose how much to invest and which technology should be deployed first. Using this power, you have tried to minimize costs or to reduce emissions.

In the new model, you do not have to do this basic work anymore. Rather, you only specify general objectives and the model searches by itself for the best possible solution, given these objectives. Thus, the model is the first real optimization model used in this course.

How does this work? In the first three models, the mathematical structure was always the same; focusing on checking the supply-demand balance of your supply mix. We just added more information on costs and emissions resulting from the supply mix.

However, for this model we first have to define an objective to obtain an optimization problem. As discussed before, we want to minimize the social costs of satisfying a given electricity demand.

These social costs stem from three major sources:

  1. The conventional costs of investment and operation (investment costs and variable costs). These are typically paid by the firms that own the technologies used to generate electricity.
  2. The costs of CO2-emissions. In many cases, these are external costs, that is, these costs are not directly paid by the firms causing the emissions, but rather by the society in general (ie by all persons that are harmed by climate change) not only by the firms.
  3. The potential costs of nuclear accidents. Again, these are external costs.

In our model the social costs (SC) are therefore defined as:

(1) (SC =sum_{k,l}(c_{k}^{inv}cdot q_{k}^{max}+ q_{k}^{var} cdot phi_{l}^{k} cdot q_{k}^{max}) + scc cdot sum_ke_{k}+ scn cdot q_{Nuclear}^{max}).

In this formulation, emissions are weighed by the term (scc), which describes the social costs of carbon. The costs of nuclear accidents are described by the term (scn), which describe the contribution of one more unit of installed nuclear capacity to the social costs of nuclear accidents. In the model, you have chosen these two values, which describe central parts of the total social costs.

Second, we need to define the constraints that we need to take into account while looking for the lowest social costs.

The social costs have to be minimized subject to the constraint that demand has to be met in all four hours. As discussed last week, this constraint can be written in the following way:

(2) (phi_1^{PV}cdot q_{PV}^{max}+phi_1^{Wind}cdot q_{Wind}^{max}+phi_{1}^{Coal}cdot q_{Coal}^{max}+phi_{1}^{Gas}cdot q_{Gas}^{max}+phi_{1}^{Nuclear}cdot q_{Nuclear}^{max} = d_{1},)

(3) (phi_2^{PV}cdot q_{PV}^{max}+phi_2^{Wind}cdot q_{Wind}^{max}+phi_{2}^{Coal}cdot q_{Coal}^{max}+phi_{2}^{Gas}cdot q_{Gas}^{max}+phi_{2}^{Nuclear}cdot q_{Nuclear}^{max} = d_{2},)

(4) (phi_3^{PV}cdot q_{PV}^{max}+phi_3^{Wind}cdot q_{Wind}^{max}+phi_{3}^{Coal}cdot q_{Coal}^{max}+phi_{3}^{Gas}cdot q_{Gas}^{max}+phi_{3}^{Nuclear}cdot q_{Nuclear}^{max} = d_{3},)

(5) (phi_4^{PV}cdot q_{PV}^{max}+phi_4^{Wind}cdot q_{Wind}^{max}+phi_{4}^{Coal}cdot q_{Coal}^{max}+phi_{4}^{Gas}cdot q_{Gas}^{max}+phi_{4}^{Nuclear}cdot q_{Nuclear}^{max} = d_{4}.)

Remember that the deployment of the controllable technologies (ie, the (phi_l^k)) are decision variables that are chosen during the optimization. Furthermore, the production capacities, (q_k^{max}), are also optimized.

To close the model, we have to link the emissions, (e_{k}), to the other variables. We can do this by assuming that there is some given emission intensity of production for each technology, which we denote by (epsilon_{k}). Thus emissions for technology (k) are given by

(6)  (e_{k} = epsilon_{k} cdot sum_{l=1}^4 phi_l^k cdot q_k^{max}).

Of course, for many technologies (PV, wind power, nuclear), we have (epsilon_{k} = 0).

Altogether, the model thus consists of minimizing our objective equation (1) over all (phi_k^l) and (q_k^{max}) and subject to the constraints given by equations (2-6).

This optimization is carried out behind the scenes for every value of (scc) (social costs of carbon) and (scn) (social costs of nuclear energy) that you chose.

© University of Basel
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Exploring Possible Futures: Modeling in Environmental and Energy Economics

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