One of the most important goals of research presented in this article was establishing whether the portfolio theory, as a conceptual tool, is any good for optimising portfolios of wind plants. Our working hypothesis is that the portfolio theory can in principle be used as a tool for minimization of generation variability.

To investigate this, we carried out a simulation on 2014 measured generation data in 15 minute intervals at nine locations of wind plants in Croatia. It is important to stress that the results of the analyses “have really nothing to do with these power plants themselves”. The measurements from them were only used to calculate, in the first step, the percentage of the plant’s output relative to the amount of energy that could have been produced if a plant operated at full installed capacity during a given 15 minute interval. The time series obtained in that way were used as a starting point for portfolio analysis.

In this investigation only the generation variability was used as parameter to minimise. In reality, there are other costs related to locations of wind plants, so the future research should include these parameters, as well. However, it was outside the intended scope of this paper.

Say now that one wants to allocate a fixed amount of new generation capacity, C, to install in N available locations with known wind statistics during a certain period of time. For the sake of simplicity, let C = 1 (that is, the measurement unit for electrical power does not matter). The basic form of portfolio theory optimisation problem can then be stated as follows:

where r_i is a weight factor, a share in total installed capacity allocated to the location i, E_i is expected hourly production expressed as a percentage of capacity allocated to the location i, σ_i is the standard deviation of production (defined in the above described manner) in percentage points, ρ_ij is the correlation coefficient between time series of production at locations i and j, E_p is expected total hourly production of the portfolio, also expressed in percentages of total installed capacity (thus E_p is the load factor of the portfolio), and σ_p² is portfolio’s variance. The first constraint states that all the weight factors must add up to 1 (that is, actual installation capacities must add up to C). The second constraint means that a wind plant cannot be installed with negative capacity. The third constraint is simply a definition of expected load factor of a portfolio of wind plants. The definition of portfolio’s variance, σ_p², is given in the problem statement line, because it is the function to be minimised.

In the research presented here, analytical solutions without the second constraint were also calculated using standard methods (Merton [21]; Lando and Poulsen [22]), because they form a hyperbolic “efficient frontier” in σ-E plane (the loci of points with the lowest possible σ_p at any given level of E_p) which is impossible to beat. Although with the locations used in this research it was not possible to combine them efficiently without negative weights of at least one of them (which is a trait of nature of wind statistics at them), this absolute efficiency border is a good orientation regarding the portfolios that actually can be achieved.

As regards the problem with non-Gaussian character of wind generation, we carried out portfolio optimisation with four different goal parameters: ordinary standard deviation of the portfolio (σ_p), left semi-deviation (σ_l), right semi-deviation (σ_r), and skewness (S). Left semi-deviation is a square root of average square of all negative deviations of generation, while right semi-deviation is a square root of average square of all positive ones. Skewness is simply (σ_r/σ_l)². The solutions were then compared to assess the appropriateness of portfolio theory approach to the minimisation of wind generation variability.

As the second constraint brings along mathematical difficulties (that is, application of Kuhn-Tucker conditions instead of plain Lagrange multipliers), a numerical approach to solving optimisation problem was used. The portfolios were randomly generated by repeating random assignments of weight factors to the nine plant locations one million times. The population of portfolios obtained in that way was then searched for the best solutions for a given value of E_p (that is, within a narrow strip of 0.99 E_p < E_p < 1.01 E_p). By “the best” we understood a solution with the lowest σ_p, σ_l, σ_r, or S, depending on which parameter was to be minimised. Sometimes, a subset of one hundred thousand portfolios was used for purposes of comparison of results obtained by different number of stochastically generated portfolios. The random generation of weight factors was performed like this: a sequence of N – 1 random numbers were drawn from an interval [0, 1], using a uniform distribution. They were then ordered by ascending value. Then the interval [0, 1] was divided into N sub-intervals delimited with that points. The length between each pair of adjacent points was then assigned as weight factor to a respective location. Such randomisation strategy was proven to generate diversely valued weight factors, which was good from the computational efficiency point of view.

The nine locations used for study will be kept in anonymity, as their identity is not important for the analyses. However, Tables 1-3 list their basic statistical properties.

In the continuation of this article the results will be displayed in the briefest possible form. Then the discussions on contributions, as well as conclusions, will be given.

A very important issue to tackle is stability of numerical estimation of efficient portfolios. Namely, if there were no non-negativity constraints, the relation between any of the r_i’s and the E_p would be linear, as it can be seen from the third constraint of the above stated optimisation problem, or from Figure 3, where analytically calculated values are displayed. Clearly, in non-constrained case the r_i values change ideally smoothly along the efficiency border. However, in our case, where we must forbid negative values, the resulting weight factors of efficient portfolios cannot change linearly, anymore. However, none of the r_i values should change too much, especially not in an abrupt way, when one moves along the efficient frontier from one efficient portfolio to a near another one. Note that in the presented example it is not possible to achieve unconstrained efficiency without at least one negative weight factor. This means that constrained efficiency frontier assumes strictly larger σ_p values for any given E_p than the non-constrained one. Figure 4 gives an example of the course of r_i values in the vicinity of E_p = 25%. Note that all the transitions are smooth, indicating that the “process of portfolio selection is stable” even with relatively coarse resolution along the efficient border (that is, relatively wide strips) for portfolio search. That would improve still more with growing number of random portfolios.

Figure 3.

Weight factors of nine locations resulting from a non-constrained optimisation, note that there are no portfolios without at least one r_i negative

Figure 4.

Weight factors of nine locations resulting from a constrained optimisation in the vicinity of E_p = 25%, note there is no abrupt changes in any of the r_i’s

An interesting feature of optimisation for minimum variability is that a process of finding most efficient portfolios itself selects individual plant locations that are present with high weights in almost all portfolios. See for example r₉ or r₇ on Figure 3. On the other hand, there are locations that contribute only minutely to any of the efficient portfolios, like r₅. In the constrained case the “blocks” of “good” and “bad” locations are even more distinctive, because, naturally, the non-negativity constraint pushes at least the locations which assume negative values in non-constrained optimisation pretty close to zero. One way to quantify this phenomenon is to define “average participation” of each location in efficient portfolios, mean values of weight factors r_i over the entire observed range of E_p values. Table 4 gives the average participation across the set of nine locations, as well as standard deviations of their respective weight factors. Then, the correlation coefficient between the two nine-component row vectors were also calculated. As a rule of thumb, “the higher the i-th location’s average participation, the lower the spread of its r_i values”. Thus, there are some locations that, “in the context of all observed ones”, are considerably “better” than the others (here, especially Loc. 9 and 7), and there are some that are notably “worse” (here, especially Loc. 6 and 5). It should, however, be stressed that this is a consequence of “both” individual “and” group statistical behaviour of wind at the locations in question. In some other group’s context the situation regarding any of the individual plant locations could in principle turn out to be different. To conclude, the portfolio selection algorithm “produces information on location quality”, within the context of all available locations.

The next question to address, as it comes to application of portfolio optimisation theory to wind plants, pertains to the fact that distributions of energy produced by a system of wind plants are notoriously non-Gaussian (Sabolić [18]). Markowitz’s theory, on the other hand, relies on an assumption of normal distribution of data. It is important to stress that this fact itself is not important if we use a numerical (or analytical, if we can) method to minimise some other parameter, rather than “plain” deviation σ_p. However, the question is whether the standard Markowitz’s theory applies in case of skewed distribution type encountered in wind generation. To investigate this, we compared standard deviations of optimal portfolios found by searching for the ones with various parameters minimised. Ideally, there should be no difference, at all, because if the minimisation criteria were not important, minimisation of any of them would lead to exactly the same solutions, with exactly the same standard deviations. Consider Figure 5, where outcomes of optimisations by different criteria are illustrated. Let σ_p^* be standard deviation of a portfolio that has some of the other three parameters minimised, rather than “normal” σ_p. In the ideal case, all the curves would be constant and equal to 1. One can see on the graph that minimisation by skewness leads to almost the same results as the one by standard deviation. Next, portfolios optimal by criterion of minimal right semi-variance are also very near (sometimes not different, at all) to the ones with minimised standard deviation, σ_p. However, minimisation of left semi-variance gives somewhat more notable differences in resulting portfolios.

Figure 5.

Ratio σ_p^*/σ_p along the efficiency frontier for different optimisation criteria used, σ_p is standard deviation of portfolio with minimal σ_p, while σ_p^* is standard deviation of portfolio with minimal σ_l (semisig left), σ_r (semisig right), or S (skew)

To assess by what degree the actual solutions for optimal portfolios differ, one can devise several methods. Here, the following one will be applied: Take the solution r^* = [r₁^* r₂^* … r_N^*], obtained by minimisation of one of the parameters σ_r, σ_l, or S. Then, take the solution obtained by minimisation of “plain” standard deviation: r = [r₁ r₂ … r_N]. Compute the following quantity: MPD = (1/N) Σ_i |r_i^* – r_i|, where summation goes from i = 1 to N. Here, MPD stands for Mean Portfolio Distance. It measures how far actual solutions are from each other (that is, what is the average absolute difference of a weight factor obtained by minimisation of one of the three alternative parameters from the one obtained by minimisation of standard deviation). The ray chart for nine locations observed in this research is given on Figure 6. Note that optimisations by either skewness, or right semi-variance, give solutions very close to those coming out of the σ_p minimisation. The left semi-variance optimisation again leads to somewhat bigger average differences in r factors. Yet, as the maximum possible difference equals 1, they are generally not too big.

Figure 6.

Mean portfolio distances for various minimisation criteria: semisig left (σ_l), semisig right (σ_r), skew (S)

Table 5 shows correlations between standard deviations of portfolios optimised by any of the aforementioned criteria, including the “ordinary” minimisation of variance without non-negativity constraints imposed. Note quite high values, although a bit lower when σ_l is involved. These results suggest that “optimisation by any of the criteria yields similar results”.

Table 6 enables one to compare an outcome of Croatian policy regarding the nine locations observed in this article to what would be optimal allocation of capacity between them for minimised variance. Croatian authorities did not have any particular strategy regarding capacity allocations, so that the policy outcome may be safely deemed random. Random decision may lead to any of the outcomes within the feasibility set visible on Figure 1. In the example studied here, the allocation yielded portfolio’s standard deviation about ten percent worse than optimal. It does not look too much, but still, the cost of regulation is about ten percent higher than it could have been. This is a cost of pure managerial inefficiency as there is no other principle reason for it. The financial resources dissipated in such a way could have been used, for example, to regulate more wind plants, or to do anything else useful. In Table 6 a comparison between mean-variance ratios is given, too. This ratio, E_p/σ_p², is often used as a measure of satisfaction with the portfolio. By that criterion, Croatian policy outcome turned out to be “slightly worse than the optimal solution”, but it is also slightly better than “centre of gravity” of feasibility set (which was estimated by calculating mean values of E_p, and σ_p, of all randomly generated portfolios from Figure 1). The latter is only relevant as a benchmark for expected outcome of random portfolio selection.

The next issue to address is how the optimal solution changes if one uses more coarse time series of data to extract statistical properties (means, variances, and co-variances) of wind plant locations. The original 15 minute time series for 2014 were converted to 60 minute ones simply by summing each four consequent data points. Then, to compare outcomes, the non-constrained efficient frontiers were calculated for both data sets. Figure 7 shows the two, while Figure 8 displays error of σ_p of an efficient portfolio. As one can see on both pictures, using rougher time resolution leads to a bit too optimistic results, meaning that the standard deviation of any efficient portfolio would be slightly underestimated. “In reality, deviation, and associated regulation costs, would be slightly higher than what optimisation results would suggest. However, the error is rather minute”, and given other possible sources of imprecision, it seems not to have too much practical significance. Naturally, it is always better to use finer data if one has them available.

Figure 7.

Comparison of unconstrained efficient frontiers calculated for the same nine wind plant locations, based on their 2014 data, with 15 minute or 60 minute time resolution

Figure 8.

Error of σ_p estimated from 60 minute time series, relative to the one computed from 15 minute ones

Finally, one can ask what if there is already a set of already operating wind plants, and there are plans to build more. Usually, a policy maker cannot deprive owners of existing plants of the rights granted to them in the past. They must be taken into account as given. The set of existing plants as a whole has historically measured data on energy production, probably in 15 minute resolution. So, the total installed power, average production, and deviations from average, are known. If the policy maker had data of wind speeds at new locations in the same roster, he could easily “optimise allocation of intended new capacity between new locations by taking the pre-existing situation as just another constraint” in problem formulation, that is:

where r₀ is the weight factor assigned to the set of pre-existing plants, C₀ is the total installed capacity of the same set, and C is new capacity to be assigned to the new locations. Now, in the problem formulation (1), all summations must run from 0 to N (instead from 1 to N, as before).