Medium-term planning for thermal electricity production

Abstract

In the present paper, we present a mid-term planning model for thermal power generation which is based on multistage stochastic optimization and involves stochastic electricity spot prices, a mixture of fuels with stochastic prices, the effect of CO$_2$ emission prices and various types of further operating costs. Going from data to decisions, the first goal was to estimate simulation models for various commodity prices. We apply Geometric Brownian motions with jumps to model gas, coal, oil and emission allowance spot prices. Electricity spot prices are modeled by a regime switching approach which takes into account seasonal effects and spikes. Given the estimated models, we simulate scenario paths and then use a multiperiod generalization of the Wasserstein distance for constructing the stochastic trees used in the optimization model. Finally, we solve a 1-year planning problem for a fictitious configuration of thermal units, producing against the markets. We use the implemented model to demonstrate the effect of CO$_2$ prices on cumulated emissions and to apply the indifference pricing principle to simple electricity delivery contracts.

Appendices

Appendix A: Construction of the hourly price forward curves

For the derivation of the HPFCs, we follow the approach introduced by Fleten and Lemming (2003). At any given time, the observed term structure at EEX is based only on a limited number of futures/forward products. Hence, a theoretical hourly price curve, representing forwards for individual hours, is very useful but must be constructed using additional information. We model the hourly price curve by combining the information contained in the observed bid and ask prices with information about the shape of the seasonal variation.

Let $f_t$ be the price of the forward contract with delivery at time $t$, where time is measured in hours, and let $F(T_1,T_2)$ be the price of forward contract with delivery in the interval $[T_1, T_2]$. Since only bid/ask prices can be observed, we have:

$$\begin{aligned} F(T_1,T_2)_\mathrm{bid}\le \frac{1}{\sum _{t=T_1}^{T_2} \exp (-rt/a)}\sum _{t=T_1}^{T_2}\exp (-rt/a)f_t\le F(T_1,T_2)_\mathrm{ask}, \end{aligned}$$

(23)

where $r$ is the continuously compounded rate for discounting per annum and $a$ is the number of hours per year. A realistic price forward curve should capture information about the hourly seasonality pattern of electricity prices. For the derivation of the seasonality shape of electricity prices, we follow Blöchlinger (2008) (chapter 6). Basically we fit the HPFC to the seasonality shape by minimizing

$$\begin{aligned} \min \left[ \sum _{t=1}^{T}(f_t-s_t)^2\right] \end{aligned}$$

(24)

subject to constraints of the type of Eq. (23) for all observed instruments. $f_t$ is the value of the HPFC at time $t$ and $s_t$ is the seasonality curve at hour $t$.³ For details see Fleten and Lemming (2003). To keep the optimization problem feasible, overlapping contracts as well as contracts with delivery periods which are completely overlapped by other contracts with shorter delivery periods are removed. For the derivation of the shape $s_t$, we follow the procedure discussed in Blöchlinger (2008), see pp. 133–137.

Appendix B: Estimation procedure for the Merton model

The model parameters $\psi =(\alpha , \sigma , \lambda , \mu , \delta )$ in Eq. (17) are estimated by maximum likelihood. $S_t$ denotes the price of a commodity at time $t$. Observations are available at equally distributed time points $t_i = i\cdot \varDelta$ for $i=0, \ldots , T$, where $\varDelta$ is the sampling frequency. For simplification, let an observation at time $t_i$ be denoted by $S_i$. The density function of the log return $x_{i+1}=\ln S_{i+1}-\ln S_i$ is

$$\begin{aligned} p(x; \psi ) = \sum _{j=0}^\infty \frac{e^{-\lambda \varDelta } (\lambda \varDelta )^j}{j!} \phi \left( x; \left( \alpha - \frac{1}{2} \sigma ^2\right) \varDelta + j \mu , \sigma ^2 \varDelta + j \delta ^2\right) \!, \end{aligned}$$

(25)

where $\phi (x;m,v)$ is the density function of the normal distribution for mean $m$ and variance $v$. The density of the log returns is evaluated by an infinite sum as in the probability function of the Poisson distribution. For practical reasons, it is approximated by the first 100 terms of the sum in the estimation. The log-likelihood function becomes

$$\begin{aligned} l(x_1, \ldots , x_T; \psi ) = \sum _{i=1}^{T} \ln p(x_i;\psi ). \end{aligned}$$

(26)

However, it has been pointed out by Honoré (1998) that the likelihood function (26) may become unbounded for some parametric specifications if it is maximized without further restrictions on the parameter space. As a solution, it is proposed to link the variance of the standard Brownian motion $\sigma ^2$ and the variance of the jump diffusion amplitudes $\delta ^2$, i.e., to set $\delta ^2 = m \sigma ^2$ for a fixed positive $m\in M$, where $M$ is a compact set on $\mathbb{R }^+$. Then, the new log-likelihood function

$$\begin{aligned} l_m(x_1, \ldots , x_T;\psi ^*) = l(x_1, \ldots , x_T;(\alpha , \sigma , \lambda , \mu , \sqrt{m}\sigma )) \end{aligned}$$

(27)

can be maximized with respect to the reduced parameter vector $\psi ^* = (\alpha , \sigma , \lambda , \mu )$ for a fixed value of $m$. As argued in Honoré (1998), $l_m(\cdot ;\psi ^*)$ is bounded in contrast to $l(\cdot ;\psi )$. Finally, a consistent estimator $\psi$ is obtained by choosing the value of $m$ which maximizes $l_m(\cdot ;\psi _m^*)$.

Appendix C: Tree formulation of the optimization model

See Fig. 8.

	Unit	Description
Sets and indices
${\fancyscript{I}}=\{ 1,\ldots ,I\}$		Set of thermal units
$i\in {\fancyscript{I}}$		Thermal unit $i$
${\fancyscript{J}}=\{ 1,\ldots ,J\}$		Set of fuels
$j\in {\fancyscript{J}}$		Fuel $j$
${\fancyscript{T}}=\{ \tau _{0},\ldots ,\tau _{t}\ldots ,\tau _{T}\}$		Considered points in time
$t\in \{0,\ldots ,T\}$
$\varDelta _{t}=\tau _{t+1}-\tau _{t}$	hours (h)	Length of time intervals
${\fancyscript{N}}=\{0,\ldots ,N\}$		Set of nodes in a stochastic tree with root $0$
$n\in {\fancyscript{N}}$		Node $n$
${\fancyscript{N}}_{T}\subseteq {\fancyscript{N}}$		Set of leaf nodes (scenarios)
${\fancyscript{N}}_{\lnot T}\subseteq {\fancyscript{N}}\setminus {\fancyscript{N}}_{T}$		Set of all nodes excluding the leaf nodes
${\fancyscript{N}}_{\lnot 0}$		Set of all nodes excluding the root node
${\fancyscript{N}}^b_{\lnot 0}$		Set of all nodes with more than one successors, excluding the root node
Decision variables
$x_{t,ij}$	MWh	Electricity produced by unit $i$ with fuel $j$ during period $(\tau _{t},\tau _{t+1}]$
$y_{t}$	MWh	Electricity bought from spot market during period $(\tau _{t},\tau _{t+1}]$
$f_{t,j}$	MWh	Fuel of type $j$ bought at time $t$
$c_{t}$	(metric) tons	CO$_2$ certificates bought at time $t$
Calculated variables
$s_{t,j}$	MWh	Stored amount of fuel $j$ at time $t$
$w_{t}$	EUR	Cash position
$w^+_{t}, w^-_{t}$	EUR	Positive and negative parts of the cash position
$v_{t}$	EUR	Asset value
$a_{t}$	(metric) tons	Cumulated amount of CO$_2$ certificates, bought up to time $t$
$e_{t}$	(metric) tons	Cumulated amount of CO$_2$, emitted up to time $t$
$u^+,u^-$	(metric) tons	Positive and negative parts of the difference between emitted CO$_2$ and the emission certificates held at time $T$

	Unit	Description
Random factors
$P_{t,j}^\mathrm{f}$	EUR/MWh	Mean spot price of fuel $j$ over period $(\tau _{t},\tau _{t}+\varDelta _{t}]$
$P_{t}^\mathrm{x}$	EUR/MWh	Mean electricity spot price over period $(\tau _{t},\tau _{t}+\varDelta _{t}]$
$P_{t}^\mathrm{c}$	EUR/tonne	Mean spot price for CO$_2$ certificates over period $(\tau _{t},\tau _{t}+\varDelta _{t}]$
Parameters
$\eta _{ij}$		Efficiency of burning fuel $j$ with generator $i$
$\varepsilon _{ij}$	tons/MWh	Amount of CO$_2$ emitted by unit $i$ per MWh of fuel burnt
$\beta _{j}$	MWh	Maximum power that can be produced by generator $i$
$\gamma _{i}$	EUR/h	Variable operating costs of machine $i$ when fuel $j$ is used
$\kappa _{i}$	EUR/h	Fixed operating costs of machine $i$
$\varsigma _{j}$	EUR/MWh	Storage costs for fuel $j$
$\sigma _{j}$	MWh	Maximum storage for fuel $j$
$\vartheta$	EUR/tonne	Penalty for excessive CO$_2$ emissions
$\lambda$		Mixing factor for the objective function
$\alpha$		Parameter of the average value at risk, $\hbox {AV}@\hbox {R}_{\alpha }$

Fig. 8

Tree formulation of the optimization problem

Appendix D: Estimation results

See Tables 2, 3, 4, 5, 6, 7, 8 and Figs. 9, 10, 11, 12, 13, 14.

Table 2 Descriptive statistics of gas, coal, oil and EUA spot prices (P) and logarithmic returns (LR)

Table 3 Unit root test results for gas, coal, oil and EUA logarithmic spot prices

Table 4 PCA of commodity prices: correlation matrix

Table 5 PCA of commodity prices: total variance explained

Table 6 PCA of commodity prices: component matrix and rotated component matrix

Table 7 Estimates of the regime switching model for the night hours

Table 8 Estimates of the regime switching model for the day hours

Fig. 9

50,000 oil scenarios quantiles with start in 01/12/2011 for 300 days horizon

Fig. 10

50,000 EUA scenarios quantiles with start in 01/12/2011 for 300 days horizon

Fig. 11

50,000 gas scenarios quantiles with start in 01/12/2011 for 300 days horizon

Fig. 12

50,000 coal scenarios quantiles with start in 01/12/2011 for 52 weeks horizon

Fig. 13

Occurrence of negative prices during Sept. 2008–Dec. 2011 on different hours

Fig. 14

50,000 EEX Phelix spot prices in sample scenarios quantiles starting in 01/09/2008 on a horizon of 1 month