Power system portfolio selection under uncertainty

Abstract

We present a general methodology for power system portfolio selection under uncertainty in which fossil fuels and CO$_2$ market prices as assumed as the main sources of risk. The planning problem is developed by considering the power system as a whole in its interactions between dispatchable sources and intermittent renewables, under load demand and power capacity constraints. The portfolio selection is performed taking into account costs and benefits of the power system from a societal perspective. Efficient frontiers and optimal generation portfolios are derived and discussed. Based on USA data, an empirical analysis is developed to illustrate the main features of this approach.

Appendices

Appendix A: Data and sources

Table 10 Technical data and costs

In the empirical analysis we use technical data and costs reported in Table 10. Data are collected from the ‘Annual Energy Outlook 2016’ [12] as reported in ‘Capital Cost Estimates for Utility Scale Electricity Generating Plants’ [13] and in ‘Levelized Cost and Levelized Avoided Cost of New Generation Resources in the Annual Energy Outlook 2016’ [14] provided by the U.S. Energy Information Administration. Costs are denominated in US dollars referred to the base year 2015. We use a real discount rate of $3.0\%$. Such a value is in agreement with the US Department of Energy prescriptions for evaluating costs and benefits of energy systems from a societal perspective [44].

Appendix B: The dynamic model

Figure 6 shows monthly returns of gas and coal market prices since 1980 until 2015. Depicted data have been downloaded from the US Energy Information Administration at site http://www.eia.doe.gov/totalenergy.

Fig. 6

Fossil fuels monthly returns since 1985 until 2015. Left panel: Gas returns. Right panel: Coal returns

We use a dynamic model in which the time evolution of (real) fossil fuel prices, $X^\text {ga}_t$ and $X^\text {co}_t$, are both described by geometric Brownian motions as discussed by Hogue [21]

$$\begin{aligned} \frac{dX^\text {ga}_t}{X^\text {ga}_t}=\pi ^\text {ga}dt+\sigma ^\text {ga}dZ^\text {ga}_t, \end{aligned}$$

(B.1)

and

$$\begin{aligned} \frac{dX^\text {co}_t}{X^\text {co}_t}=\pi ^\text {co}dt+\sigma ^\text {co}dZ^\text {co}_t, \end{aligned}$$

(B.2)

where

• $\pi ^\text {ga}$ and $\pi ^\text {co}$ are, respectively, the natural logarithm of one plus the real escalation rate of gas and coal prices, whose values are reported in Table 10;

• $\sigma ^\text {ga}$ and $\sigma ^\text {co}$ are, respectively, the volatilities of gas and coal prices;

• $Z^\text {ga}_t$, $Z^\text {co}_t$ are independent standard Brownian motions.

The numerical values of the dynamics parameters are reported in Table 11.

Table 11 Dynamical parameters

Fig. 7

EEC distributions in the four carbon volatility scenarios. The arrow indicates the maximum EEC value in each distribution. Left panel: $S^\text {ga}(\omega )$. Right panel: $S^\text {co}(\omega )$

The real escalation rates parameters assumed and displayed in Table 11, are forecast expected rate of growth of fossil fuels prices, as given in AEO 2016 [12]. The volatility parameters are chosen according to the estimates reported in [21], obtained by using a geometric Brownian motion to simulate the fuels prices dynamics on wellhead prices from 1950–2011 for natural gas, and from 1950–2010 for coal.

The dynamics of carbon (real) prices is modeled according to a geometric Brownian motion of the type

$$\begin{aligned} \frac{dX^\text {ca}_t}{X^\text {ca}_t}=\sigma ^\text {ca} dZ^\text {ca}_t, \end{aligned}$$

(B.3)

where $\sigma ^\text {ca}$ is the carbon volatility and $Z^\text {ca}_t$ is a standard Brownian motion which is assumed to be independent on $Z^\text {ga}_t$ and $Z^\text {co}_t$. The dynamics of $X^\text {ca}_t$ affects both gas and coal stochastic EEC, thus introducing positive correlation between the two stochastic variables. The entity of such a correlation, as well as the values of the volatility of EEC for both gas and coal generating technologies, can be obtained by using Monte Carlo techniques. Simulations have been performed in four different scenarios, namely assuming a carbon volatility equal to $0, 20, 30, 40\%$. These assumptions try to depict a zero volatility, a low, a medium, and a high volatility scenario, respectively, in order to capture the relevance of hedging effects [15]. The sample consists of one hundred thousand of randomly generated trajectories obtained using the antithetic variable method. For each run of the Monte Carlo simulation, an evolution path for fossil fuel prices and carbon credits prices is obtained and, along such paths, EEC values have been calculated. Figure 7 shows EEC sample distributions in the four considered scenarios. Table 12 reports the first two moments and the correlation coefficient values of the EEC simulated distribution in each carbon volatility scenario. Computations are performed using the nominal capacity factors reported in Table 10.

Table 12 First two central moments of the $S^x(\omega )$ (EEC) distribution

As the carbon volatility increases, the coupling between gas and coal stochastic EECs strengthens. In particular, the standard deviation of the gas EEC is greater than the EEC coal standard deviation in the $\sigma ^\text {ca}=0, 0.2$ scenarios but this relation reverses in the $\sigma ^\text {ca}=0.3, 0.4$ scenarios. Moreover, as the carbon volatility increases the correlation increases as well, making stronger the coupling between gas and coal stochastic EECs. Finally, we notice that as the carbon volatility increases, EEC distributions become more asymmetric with long fat tails. As it is shown in the text, such features have strong consequences on systemic portfolios frontiers.

Appendix C: CVaRD: a brief review

Consider a random variables Y (e.g. the stochastic EEC of a systemic portfolio) with probability density p(y), a threshold h and a probability value $\alpha$ (see Fig. 8).

Fig. 8

Risk measures associated to a skewed, long tailed generic distribution

As conventional in risk theory notation, losses (as e.g. adverse values of the stochastic EEC) are considered as right tail values. The CVaR of the portfolio at confidence level $\alpha$ is defined as the conditional expectation on losses

$$\begin{aligned} \text {CVaR}_{\alpha }(Y) = \frac{1}{1-\alpha } \int _{y \ge h^*} y \, p(y)\, dy, \end{aligned}$$

(C.1)

when $h^*=\text {VaR}_{\alpha }(Y)$. $\text {CVaR}_{\alpha }$ can thus be seen as the expectation over the residual $1-\alpha$ cases, the most adverse ones (so that $\text {CVaR}_{\alpha } \ge \text {VaR}_{\alpha }$). In this way, CVaR fully takes into account tail risk, but in an asymmetric way, being defined on the most adverse tail only. In turn, CVaRD at confidence level $\alpha$ is defined in terms of CVaR [46] as

$$\begin{aligned} \text {CVaRD}_{\alpha }(Y) \equiv \text {CVaR}_{\alpha } (Y-\mu ) = \text {CVaR}_{\alpha } (Y) - \mu , \end{aligned}$$

(C.2)

where $\mu =E[Y]$. In Eq. (C.2) the first equality shows that CVaRD is the deviation associated to CVaR, like standard deviation $\sigma$, from $\sigma ^2 = E\big [(Y-\mu )^2\big ]$, is associated to the mean. $\text {CVaRD}$ is non-negative (like the standard deviation), whereas this is not necessarily true for CVaR (and the mean). If c is a constant, the following equalities hold

$$\begin{aligned} \text {CVaRD}_{\alpha }(Y+c)=\text {CVaRD}_{\alpha }(Y), \end{aligned}$$

(C.3)

and

$$\begin{aligned} \text {CVaRD}_{\alpha }(cY)=c \;\text {CVaRD}_{\alpha }(Y). \end{aligned}$$

(C.4)

Intuitively, the relationship between VaR and CVaRD (or CVaR) is displayed in Fig. 8. Being a deviation, CVaRD has a different field of application than CVaR. The measures useful to manage the risk we have in mind are indeed deviation measures, among which we selected standard deviation (i.e. variance) and CVaRD, with different tail properties.