On the construction of hourly price forward curves for electricity prices

Abstract

There are several approaches in the literature for the derivation of price forward curves (PFCs) which distinguish among each other by the procedure employed for the derivation of seasonality shapes, smoothing technique and by the design of the optimization procedure. However, a comparative study to highlight the strengths and weaknesses of different methods is missing. For the construction of PFCs we typically incorporate the information about market expectation from the observed futures prices and the deterministic seasonal effects of electricity prices. In most existing approaches, the seasonality shape is fitted to historically observed spot prices, and it is an exogenous input to the optimization procedure. As seasonal effects on electricity prices differ between markets, our model allows a more general and flexible definition of the seasonality shape. In this study, we propose an alternative calibration procedure for the seasonality shape, where the level of futures as well as historical spot prices are simultaneously taken into account in a joint optimization approach. We discuss comparatively the features of existing methods for PFCs, and highlight the advantages of our optimization procedure.

Appendix

1.1 Testing Procedure Table 1

The tests for Table 1 is done as follows: We obtain our estimated prices by taking the real price for each day d multiplied by the hourly profiles to get an estimated price for each hour h in day d. Then we take the mean of the absolute differences between this estimate and the observed price for hour h at day d for all days in 2008–2015. The same is done for the squared differences.

$$\begin{aligned} \text {Absolute}\_\text {Error}\_\text {Hour}_h= & {} \frac{1}{n}\sum _{d=1}^n|\textit{Day\,Price}_{d}\cdot \textit{Hourly\,Profile}_{h}-\textit{Hour\,Price}_{d,h}| \nonumber \\\end{aligned}$$

(11)

$$\begin{aligned} \text {Squared}\_\text {Error}\_\text {Hour}_h= & {} \frac{1}{n}\sum _{d=1}^n(\textit{Day\,Price}_{d}\cdot \textit{Hourly\,Profile}_{h}-\textit{Hour\,Price}_{d,h})^2\nonumber \\ \end{aligned}$$

(12)

1.2 Constraints in the method by Benth et al. (2007)

Throughout the periods, the following Eqs. (13)–(15) need to hold:

$$\begin{aligned}&(a_{j+1}-a_j)t_j^4+ (b_{j+1}-b_j)t_j^3+ (c_{j+1}-c_j) t_j^2 + (d_{j+1}-d_j)t_j+e_{j+1}-e_j= 0 \end{aligned}$$

(13)

$$\begin{aligned}&4 (a_{j+1}-a_j)t_j^3+3 (b_{j+1}-b_j)t_j^2 +2 (c_{j+1}-c_j)t_j +d_{j+1}-d_j=0 \end{aligned}$$

(14)

$$\begin{aligned}&12 (a_{j+1}-a_j)t_j^2+6 (b_{j+1}-b_j)t_j+2 (c_{j+1}-c_j)=0 \end{aligned}$$

(15)

To ensure that the curve is flat in the long end, we set the first derivative in the end point equal to 0, ensured by Eq. (12). To account for settlement of the contracts throughout the period, one can include a function w(r; t) as shown in Eq. (13). For settlement only at the end points, one sets $w (r;t)=1/ (T_i^e-T_i^s)$.

$$\begin{aligned} \varepsilon ' (t_n;x)&=0 \end{aligned}$$

(16)

$$\begin{aligned} F_i^C&=\int _{T_i^s}^{T_i^e}w (r;t) (\varepsilon (t)+s (t))dt \end{aligned}$$

(17)

1.3 Novel modeling approach

The term S(m) is chosen to insure continuity of the curve between the transition times of the months. As an example, when going from January to February, we get $M(1)=31$ and $M(2)=28$, where the transition between January and February takes place when $t=31$. This means that for the curve to be continuous, we need that:

$$\begin{aligned} \frac{31+S(1)}{31}=\frac{31+S(2)}{28} \end{aligned}$$

The correction term $S (\cdot )$ that ensures the continuity is given by the vector defined as:

$$\begin{aligned} S= (0, -\,3, 3, 0, 4, -\,1, 5, 5, -\,3, 6, -\,4, 7) \end{aligned}$$

where S(m) is element number m in the vector S.

The matrices A needed for the optimization in the novel modeling approach are defined as follows:

$$\begin{aligned} A=\left[ A_1,\ldots ,A_6,B_1,\ldots ,B_6,A_4^1,\ldots ,A_4^4,B_{4}^1,\ldots ,B_{4}^4\right] \end{aligned}$$

(18)

where:

$$\begin{aligned} A_j= & {} \left( \sin \left( \frac{2\pi \cdot j \cdot 1}{12\cdot 31}\right) ,\ldots ,\sin \left( \frac{2\pi \cdot j \cdot 31}{12\cdot 31}\right) ,\sin \left( \frac{2\pi \cdot j \cdot 25}{12\cdot 28}\right) ,\ldots ,\sin \left( \frac{2\pi \cdot j \cdot 341}{12\cdot 31}\right) \right) \\ B_j= & {} \left( \cos \left( \frac{2\pi \cdot j \cdot 1}{12\cdot 31}\right) ,\ldots ,\cos \left( \frac{2\pi \cdot j \cdot 31}{12\cdot 31}\right) ,\cos \left( \frac{2\pi \cdot j \cdot 25}{12\cdot 28}\right) ,\ldots ,\cos \left( \frac{2\pi \cdot j \cdot 341}{12\cdot 31}\right) \right) \end{aligned}$$

and $A_j^Q$ for $1\le Q \le 4$ is the vector $A_j$ in quarter Q, and 0 otherwise, giving:

$$\begin{aligned} A_j^1=\large \left( \sin \left( \frac{2\pi \cdot j \cdot 1}{12\cdot 31}\right) ,\ldots ,\sin \left( \frac{2\pi \cdot j \cdot 93}{12\cdot 31}\right) ,0,\ldots \ldots ,0\right) \end{aligned}$$

and the same for $B_j^Q$.

The matrix for the constraints, named C is as said separated into two different matrices, H, G, where H makes sure the PFC fits to the Futures, and G takes care of the continuity of the spline part of the PFC. The constraints coming from the Futures are given as:

$$\begin{aligned} V_i=\frac{1}{T_E-T_S}\int _{T_S}^{T_E}f (t)dt \end{aligned}$$

(19)

where we divide by the length of the period the Futures product is covering since the Futures price is denoted by the average price for that period. Assuming we have a Futures product covering January with price $V_1$, the corresponding constraint for term j of the Fourier series is:

$$\begin{aligned} V_1= & {} \frac{1}{31}a_j\int _0^{31}\sin \left( \frac{2\pi \cdot j \cdot t}{12\cdot 31}\right) +b_j\cos \left( \frac{2\pi \cdot j \cdot t}{12\cdot 31}\right) dt\\= & {} \frac{31\cdot 12}{31\cdot 2\pi \cdot j}\left[ -a_j \left( \cos \left( \frac{2\pi \cdot j}{12}\right) -\cos \left( 0\right) \right) +b_j \left( \sin \left( \frac{2\pi \cdot j}{12}\right) -\sin \left( 0\right) \right) \right] \end{aligned}$$

Which shows two of the advantages with changing the seasonality corresponding to the length of the months: Firstly, one can cancel the terms coming from the dividing by the length of the period directly against the term coming from multiplying with the denominator in the $\sin / \cos$ term. Secondly, one only need to evaluate $\sin / \cos$ in values that are a multiple of $2\pi /12$, where one has nice analytic values for the result. One gets similar results if the Futures products covers more than 1 month. By denoting:

$$\begin{aligned} \left. F \left( i,j\right) _C=\frac{12}{2\pi \cdot j}\cos \left( \frac{2\pi \cdot j\cdot \left( i+1\right) }{12}\right) -\cos \left( \frac{2\pi \cdot j\cdot \left( i\right) }{12}\right) \right) \end{aligned}$$

(20)

and similar for the $\sin$ function by $F (i,j)_S$. Then the matrix C, if all monthly Futures are given, is defined by:

$$\begin{aligned} H= \begin{bmatrix} F (1,1)_C&F (1,2)_C&\ldots&F (1,6)_C&F (1,1)_S&\ldots \\ F (2,1)_C&\ldots&\ldots&\ldots&F (2,1)_S&\ldots \\ \ldots&\ldots \\ F (12,1)_C&\ldots \end{bmatrix} \end{aligned}$$

Where the pattern is similar as for the matrix A with the spline coefficients. For the matrix G that ensures continuity and continuity of the derivatives of the splines, one has these constraints:

Continuity:

$$\begin{aligned}&b_4^1+b_{12}^1\\ =&b_4^2+b_{12}^2\\&b_4^2+b_{12}^2\\ =&b_4^3+b_{12}^3\\&b_4^3+b_{12}^3\\ =&b_4^4+b_{12}^4 \end{aligned}$$

differentiability:

$$\begin{aligned}&\frac{1}{31} \left( a_4^1+3a_{12}^1\right) \\ =&\frac{1}{30} \left( a_4^2+3a_{12}^2\right) \\&\frac{1}{30} \left( a_4^2+3a_{12}^2\right) \\ =&\frac{1}{31} \left( a_4^3+3a_{12}^3\right) \\&\frac{1}{30} \left( a_4^3+3a_{12}^3\right) \\ =&\frac{1}{31} \left( a_4^4+3a_{12}^4\right) \end{aligned}$$

Giving us 6 constraints for the 26 parameters, making G a $6 \times 26$ matrix.