Review of the Literature

^{Awerbuch and Berger (2003)} utilized the ^{Markowitz Model (1952)} to evaluate electricity generation projects in the European Union, including both conventional (coal, oil, gas, and nuclear) and renewable (in this case, wind) technologies.

For his part, ^{Awerbuch (2006)} analyzes the case of the European Union and adds two more case studies, one in the United States and one in Mexico. He concludes that the Markowitz Model is a better alternative for planning a mix of power generation sources than a lower cost analysis, considering each option separately. The Markowitz Model helps identify the least risky mix.

DeLaquil, ^{Awerbuch, and Stroup (2005)} used the Markowitz Model in the planning of electricity in the State of Virginia in the United States. At the time of the analysis, the mix of generation sources was dominated by hydrocarbons, mainly natural gas, which at that time had high volatility and high cost.

^{Awerbuch, Jansen, Bursken, and Drennen (2005)} utilize the Markowitz Model to show that including geothermal and renewable generation in the mix of electricity generation in the western region of the United States-the generation tradition of which is of fossil origin-favors risk reduction in the cost of electricity.

^{Woo, Horowitz, Horii, and Karimov (2004)} propose the use of the Markowitz Model for the purchase of energy coverages, for an electricity trader who wants to keep costs and risks low.

^{Lesser, Lowengrub, and Yang (2007)} submitted an analysis of electric power generation mixes for the year 2020 to the California State Energy Commission. The analysis used an optimization study based on the Markowitz Model. This work concludes that the involvement of non-fuel technologies can help reduce the cost and risk of the generation portfolio in California. The use of the Markowitz Model makes it possible to analyze the interrelationship of cost and risk of electricity generation alternatives.

^{Huisman, Mahieu, and Schlicther (2007)} propose using the Markowitz Model as a measure of coverage in day-ahead energy markets, selecting the appropriate peak and off-peak contracts, which provide coverage in price and volume, depending on the aversion or propensity to risk of the buyer.

^{Beltrán (2009)} used the Markowitz Model to evaluate the electric power generation portfolio of Mexico. The study compares the costs and variance of different generation options, including hydropower, combined cycle, coal, nuclear, wind, fuel oil, and geothermal, among others. The author concludes that using lower cost methods to analyze each generation option separately may entail missing opportunities for lower risk, which does not occur if they are analyzed as a portfolio.

On the other hand, ^{Roques, Hiroux, and Saguan (2010)} used the Markowitz Model to analyze the effect of the geographical diversity of wind generation fields located in five European countries (Austria, Denmark, France, Germany, and Spain). It was concluded that the Markowitz Model is a tool that can help in the best planning of a wind generation system. In order to optimize the generation mix in Belgium, ^{Delarue, De Jonghe, Belmans, and D’haeseleer (2009)} suggest a model in which the initial cost is distinguished from the variable cost of operation (including maintenance, fuels, and operation in general), unlike other models in which a levelled cost is used. In this way, a portfolio of alternatives is analyzed with sources such as nuclear, coal, gas, oil, and wind. It was concluded that by adding wind generation to the portfolio its risk decreases.

Roques, Newbery, and Nuttall (2007) use simulation techniques and the Markowitz Model to assess the impact of fuel price, electricity, and CO2 emission risks to demonstrate the usefulness of diversification for an investor.

^{Liu and Wu (2010)} made an analysis from the perspective of a generator, using the Markowitz Model. The approach suggests that there are two ways of managing risk in energy trade, through diversification and through hedging. The study concluded that by using the Markowitz Model it is possible to control risk by making the relevant allocation in the different marketing options, through the wholesale market and through bilateral contracts. The optimal solution would be the one that gives the best rate of return for a given level of risk.

^{Rodoulis (2010)} utilizes the Markowitz Model to analyze the feasibility of using natural gas, wind energy, and coal, together with oil, which until then was the main source of electricity generation in Cyprus. The objective was to determine which generation portfolio mix would give the best cost-risk ratio for the island. It was concluded that the planned diversification of power generation sources helps to reduce the portfolio risk generated by oil price fluctuations.

^{Yan (2011)} uses the Markowitz Model to develop the proposal for an energy portfolio, considering social cost as a fundamental element for consumers and regulators, in the case of the United States.

^{Cucchiella, D’Adamo, and Gastaldi (2012)} applied the Markowitz Model to analyze different renewable energies in the Italian market. They concluded that it is possible to diversify the sources of energy generation with renewable ones, mitigating portfolio return risk.

^{Ziegler, Schmitz, and Weber (2012)} analyzed the German domestic market through the Markowitz Model to find the optimal generation portfolio. For the analysis they used the average net present value, based on the modeling of fuel prices.

^{Lüttgens and Antons (2013)} combine a qualitative management model of technology portfolios with the Markowitz Model to evaluate an energy generation portfolio, highlighting the value from a holistic perspective, not purely qualitative and not purely in risk evaluation.

In an analysis of the case of Albania, where 100% of electricity was generated by hydraulic sources, ^{Tola (2015)} compared wind and photovoltaic sources in the same portfolio. This comparison made it possible to visualize how the diversification of energy generation sources had an effect on risk reduction, the way it happens when considering a portfolio of financial assets.

Similarly, ^{De Llano Paz, Calvo Silvosa, and Portos García (2012)} used the Markowitz Model in a case in Spain.

^{Kumar, Mohanta, and Reddy (2015)} analyzed information from the Indian electricity market with the aim of optimizing a portfolio of electricity generation alternatives that represents the lowest price, given a certain level of risk and a certain level of CO₂ emissions. In line with other studies, it is concluded that considering different electricity generation options reduces the risk associated with variations in fuel prices for certain technologies.

A study carried out in the United Kingdom by ^{Adams and Jamasb (2016)} analyzed the generation portfolio that should be in place to ensure the supply of electricity. Using the Markowitz Model, they analyzed the portfolio that includes generation through gas (combined cycle), nuclear, coal, and wind, considering the investment, fuel, and, CO₂ bonus costs. It was concluded that a diversified mix can mitigate the risk associated with the uncertainty of electricity price, fuel costs, and carbon offset costs. This study proved that considering a portfolio of wind and photovoltaic sources as a complement to hydraulic generation may be adequate to reduce the risk.

Finally, ^{Gökgöz and Atmaca (2016)} compared methodologies, based on the Markowitz Model, in an analysis to maximize return and minimize risk in the electricity market in Turkey. The price information for the day-ahead was analyzed and compared with the generation cost of that same hour to calculate the return; this was done on an hourly basis. Each of the 24 hours of the day is considered an asset in the study. With the proposed methodologies, the hours with the most weight in the optimal portfolio are obtained. The authors recognized that the Markowitz Model was valuable to support investment decision making and to define regulations in an electricity market.

This literature review showed different areas within the electricity sector in which the Markowitz Model has been used. By virtue of the above, the objective of this work was to use these concepts to conform an optimal portfolio of electrical energy supply from the perspective of a user in Mexico. With this approach it is intended that users, instead of only making decisions based on the minimization of costs, consider, at the same time, the relevant risk considerations in order to strengthen the decision-making process on the supply of electricity.

Methodological Framework

The ^{Markowitz Model (1952)} proposes, as an analysis and decision strategy, an essentially statistical and mathematical methodology, also known as media-variance analysis, to seek the best investment strategy, from the perspective of an investor, to balance profits and mitigate risks.

^{Markowitz (1952)} defines that the expected yield from an investment portfolio is given by:

Where w_ is the weighting vector, the elements of which represent the ratio to be invested in each of the n financial instruments that make up the portfolio, while ER_ is the expected yield vector, in which each element represents the expected yield of each instrument in the portfolio.

The standard deviation of portfolio yields, which is normally a measure of portfolio volatility, is given by:

Where VarR_ is the matrix of variances and covariances of the yields associated with the instruments that make up the portfolio.

From this context, it is clear that the main objective of the analysis of an investment portfolio, based on these ideas, is to maximize its return, assuming a certain level of risk or volatility or to minimize the volatility by accepting a certain level of return. These ideas are expressed below as quadratic programming models:

Subject to:

Subject to:

According to the ^{Markowitz Model (1952)}, the diversification of an investment portfolio makes it possible to reduce the variance of the expected yield of the portfolio, given the correlation between the yields of the instruments considered. However, the portfolio that reduces the yield variance is not, as a matter of principle, the portfolio that maximizes the return of the investment. These ideas can best be understood in terms of the efficient frontier, which shows the relationship between the expected return of a portfolio and its level of risk, measured from the standard deviation, as shown in Figure 1.

This graph illustrating the relationship between Yield and Risk was constructed by simulating a portfolio of three assets to which different weights were assigned. The blue portion of Figure 1 represents the different portfolios, constructed from different ratios assigned to the instruments that make it up. The efficient frontier corresponds to the upper perimeter of the graph and represents the optimal set of portfolios in which to invest, since it is guaranteed that the more the assumed risk, the greater the reward in return. In other words, the efficient frontier represents the maximum return in the set of portfolios given a certain level of risk.

From this graph it can be observed that it is possible to maximize the performance of the portfolio conditioned to assume a certain level of risk.

A variant in the application of these ideas is the use of the Markowitz Model in the planning of an electric power generation portfolio, in a country or in a region. The argument is that the usual method of planning a generation portfolio analyzes the cost of each generation source separately. In such an analysis, the selected generation source will always be the one with the lowest cost. The costs of traditional technologies, based on coal and gas, for example, depend on fuel prices and, as these are volatile, the costs of these technologies become high risk.

Instead of analyzing each generation source separately, the entire portfolio of alternatives is analyzed using the Markowitz Model. Thus, there is an opportunity to consider the correlations between the various generation sources. Renewable sources of generation (such as solar or wind) are characterized by having a high initial investment but a low operating cost, resulting in a predictable cost of energy. Conversely, traditional sources of generation have a high operating cost and a relatively low initial cost, resulting in a volatile cost dependent on the price of fuels. The Markowitz Model has made it possible to include renewable generation sources in the optimal portfolios, because their inclusion mitigates the risk of the cost of energy of the entire portfolio.

Description of the context to be studied

Today, practically all users receive electricity from the transmission and distribution networks of the new State Productive Enterprises: CFE-Transmission and CFE-Distribution. The energy they consume is fully or partially generated by one of the CFE-Generation companies. At least in the short-term, the Federal Electricity Commission (CFE for its acronym in Spanish) will continue to provide part of the energy consumption and other services that users require.

As mentioned above, there are generators that have Legacy Interconnection Contracts, which are generation permits granted by the CFE before the issuance of the Electricity Industry Law (2014). These suppliers have self-supply or cogeneration permits that may be offered, fully or partially, to third parties through private contracts.

Finally, a user registered as Qualified User Participating in the Market may also have access to the wholesale market, and purchase electricity and other services in the auctions of the market itself.

Basically, there are two types of contracts: pay as you use and take or pay. In a pay as you use contract, users pay what they consume. The price of electricity is agreed and paid as much as it is consumed.

In a take or pay contract, the rate paid is normally lower, but the users buy the agreed capacity and it is fully liquidated, whether it is consumed or not.

Both types of contract, with their respective adaptations, are the types of contract to which a user will be able to access, either under the regime of the new Electric Energy Law or under the previous law, with generators with legacy permits.

Supply Portfolio from the Markowitz Model

The ^{Markowitz Model (1952)} was used to optimize a portfolio of electricity supply alternatives. The model considers the perspective of a user who currently has electricity supply from the CFE.

The user is presented with energy supply options which, in general terms, can be grouped as follows:

In the case of a portfolio with n electricity supply options, the expected cost per kW-h is given by:

Where w_ is the vector of weights, the elements of which represent the ratio of the total supply that each of the n suppliers will supply, while EC_ is the vector of expected costs, where each element represents the expected cost of the corresponding supplier.

The standard deviation of the portfolio costs is described by:

Where is the VarC_ matrix of variances and covariances of the costs of the different supply alternatives.

The objective is to minimize the cost of the project subject to an appropriate level of risk for the user, which is equivalent to:

Subject to:

Or, minimize the risk of the portfolio subject to an appropriate cost level for the user, which implies:

Subject to:

Case Study

Based on real information, a hypothetical scenario was created with fictitious companies in which a manufacturing company faces the decision-making concerning the supply of electric energy. This was done using the current context in Mexico since the Energy Reform.

The company Manufacturas de Occidente is headquartered in the United States of America. Its operation in Mexico includes several loading points located in several states of the Mexican Republic.

In 2016, the operation at these loading points had an electric power supply with a contracted demand of 3,500 kW in HM rate with the Federal Electricity Commission. It had an annual consumption of 12,400 MW-h, with a simple rate of MXN$1.34 per kW-h.

In the search for new electricity supply offers that could give a better price for 2017 and the following years, Manufacturas de Occidente received different proposals. Now, the decision must be made on which of the following options would be most convenient:

First, the average expected costs for each of the options were estimated. In order to estimate means, variances, and covariances, monthly data were considered from January 2011 to October 2016. For the analysis, an independent calculation was made for each loading point and then the estimated rates per loading point were weighted to obtain a final rate for Western Manufactures. For the construction of the cost matrix, the following methodology was followed for each of the supply options:

From a graph of the costs of the different alternatives throughout time, it can be observed that the Nueva Era dollar proposal behaves differently from the other options that are referenced to the HM rate of the CFE. It can be observed that this rate, in dollars, tends to increase in recent months, as can be observed in Figure 2.

The means and standard deviations calculated from the cost matrix are included in Table 2.

The variance and covariance matrix is shown in Table 3.

Figure 3 shows the Expected Cost graph, E(C_p), according Cost Volatility, σ(C_p), which was built by simulating different weights to the different supply options that make up the portfolio.

In addition to a low price per kW-h of electricity, Manufacturas de Occidente also seeks to reduce the risk. It wants to ensure a reasonably stable price that allows it to make medium- and long-term plans reliably, but it is willing to assume a moderate risk that allows obtaining a better price than the minimal variance portfolio. Therefore, a low risk level is the first quartile. The methodology used would not change if the company wanted to assume more or less risk.

Figure 3 shows that the standard deviation of the costs of possible portfolios varies from 0.05 to 0.21 pesos. The standard deviation of the first quartile is 0.0900 (a variance of 0.0081).

The mathematical approach to the problem is:

Subject to: