From The Developing Economist VOL. 1 NO. 1
Efficient Pricing of Carbon in the EU and its Effect on Consumers
III. Computational Model
To evaluate the efficacy of the carbon tax on the utilization rates of multiple generation technologies, a computation model was created. The model uses a Monte Carlo process in conjunction with a statistical tool known as the Dirichlet distribution to randomly create an array of portfolios, each with a different share of the various generation technologies. These energy portfolios were then evaluated based on their cost and CO2 emissions. Certain thresholds were placed on all portfolios to ensure that they were plausible in the real world, namely that a percentage of power was generated from dispatchables (§2.3).
Monte Carlo Simulation
Monte Carlo simulations are often used in modeling in situations where a closed-form analytic solution is not readily available or exceedingly computationally intensive. A Monte Carlo simulation relies on repeated random sampling of input variables to obtain an optimal result. In the case of portfolio management (whether it be energy or equities), by randomly choosing the asset allocation percentages and calculating the costs repeatedly, the law of averages states that as the number of simulations approaches infinity, an optimal solution will be found.
In the proposed model, sampling was done according to the Dirichlet distribution, which ensures that some number of points in a set, n, are randomly sampled such that their sum is equal to some specified total. Thus, the Dirichlet distribution has the statistically appealing property of being the conjugate prior to the parameters of the multinomial distribution. In the computational model, the distribution generates N values, bounded by [0,1] that sum to unity.
Dispatchable Reserve Rate
As mentioned in §2.2, grid operators maintain a reserve of dispatchable capactiy based on the fraction of total capacity generated from renewables. Since this is an a posteriori calculation after the desired supply from renewables has been decided, it is calculated in the same manner in the model.
The Dirichlet distribution first creates a portfolio consisting of the generation technologies listed in Figure 76 where the sum of the weights equals 1. Afterwards, an algorithm adds the requisite percentage of dispatchable reserves to the portfolio.7 For reserve requirement λ:
Over 20,000 portfolios were generated and stored in a matrix. However, many of these did not meet certain the base load requirement, and were removed. With the remaining portfolios, matrix operations were performed to calculate the cost and emissions. The portfolios that had the highest and lowest cost and CO2 emissions were then kept for each tax rate.
*Base Load If energy portfolios did not meet a predefined percentage of power generation from dispatchables (ρ), they were culled from the dataset.