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A Large-Scale Spatial Optimization Model of the European Electricity Market

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Abstract

In this paper, we present a large-scale spatial model of the European electricity market including both generation and the physical transmission network (DC Load Flow approach). The model was developed to analyze various questions on market design, congestion management, and investment decisions, with a focus on Germany and Continental Europe. It is a bottom-up model combining electrical engineering and economics: its objective function is welfare maximization, subject to line flow, energy balance, and generation constraints. The model provides simulations on an hourly basis, taking into account variable demand, wind input, unit commitment, start-up costs, pump storage, and other details. Various forms of spatial price discrimination can be implemented, such as locational marginal pricing (“nodal pricing”), or zonal pricing. With over 2,000 nodes and over 3,000 lines, this is one of the largest models developed to date, and allows a highly differentiated spatial analysis. We report modeling results regarding efficient congestion management for Germany and Europe, optimal network expansion under the aspect of increased wind energy production, and the impact of network constraints on location decisions of generation investments.

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Notes

  1. Note that even these ‘simple’ perfect competition models can become computationally challenging if certain nonconvex characteristics of electricity markets such as start-up (e.g., Gabriel et al. 2004; García-Bertrand et al. 2005) or minimum profit constraints (e.g., García-Bertrand et al. 2006) are included.

  2. The name ‘DC load flow’ is due to historical origins and does not refer to the use of direct current in the electricity network.

  3. Overbye et al. (2004) come to the conclusion that the DCLF is adequate for modeling nodal prices albeit there are some buses with a certain price deviation. The latter occurs particularly on lines with high reactive power and low real power flows.

  4. If the model includes more than one voltage level as it is the case within ELMOD, the standardization works by choosing a reference voltage level and then convert all other line parameters by a conversion factor. For example, the factor to express 220 kV parameters in 380 kV terms would be approximately 0.58. Hence, resistances and reactances of the 220 kV lines would have been divided by this factor. However, as the maximum power capacity has to be converted, too, one could also define a reference power flow level, relate a line’s power capacity to this predefined level, and convert all parameters accordingly. In any case, regarding the conversion factors, one has to be aware that the power capacity is a quadratic function of the voltage magnitude.

  5. The line susceptances BL l are real parameters that can be observed using voltage level, length, number of circuits, and material of a transmission line (compare Section 4).

  6. It can be easily seen that ELMOD could also be run as cost minimization model by fixing the reference demand values at each node.

  7. The definition of positive and negative direction is arbitrary.

  8. In the case of including grid losses, the line losses for l are assigned equally to the nodes connected by this line via the net input: \(ni_{nt}= \sum_i B_{ni} \theta_{it} - 0.5 \sum_l (IM_{ln}p_{lt})^2R_l\).

  9. However, another constraint of the form \(\theta_{i^{'}t} = 0\) needs to be implemented in the model which has been left out above for simplicity reasons. This constraint is used to define an arbitrary node i ′ within the system to be the so-called slack bus or hub. Consequently, all θ it values in the system are relative in respect to this slack bus which ensures that all single net inputs ni nt sum up to zero for the entire system and, thus, a system-wide energy balance is established. An exhaustive discussion of this modeling approach can be found in Schweppe et al. (1988).

  10. This is irrelevant for the time constraint but important for the cost estimation.

  11. According to Müller (2001), modern PSPs have an average efficiency between 70% and 80%.

  12. The maximum timeframe modeled with ELMOD for the time time being is 24 h. Hence, modeling the storage behavior might be simplified as the storage process also takes place at weekend nights. In addition, the hourly increment used in ELMOD may result in a biased representation of PSPs as one of their main tasks is to react in case of rapidly changing conditions. However, these simplifications could be included within the model framework by extending the timeframe beyond 24 h.

  13. Compare DEWI (2007).

  14. Compare EWEA (2007).

  15. This constraint can become critical if the grid is not capable of transporting all wind energy. Then the only way to fulfill the energy balance constraint is the increase of local demand even if prices become negative. For the time being, in reality other measures are taken in order to avoid such situations. Possibilities in order to manage such extreme cases are the shut-down of certain wind parks and other technical measures. Such short-term measures are not included in ELMOD.

  16. For clarity, we do not describe the problem sizes of all possible applications. However, the following problem sizes might help to get a rough idea: ELMOD for one hour without unit commitment and PSP for the German network has about 2,000 variables; ELMOD for one hour without unit commitment and PSP for the European network has about 10,000 variables; ELMOD for 24 h including unit commitment and PSP for the European network has about 400,000 continuous and 40,000 discrete variables.

  17. It must be noticed that the implementation of neighboring countries has an impact on the welfare calculation. As they are part of the overall optimization problem, their demand and generation adds to the total system welfare. Due to energy exports and imports, it is not possible to calculate the welfare for Germany only when including neighboring countries. This must be taken into account while regarding welfare effects. However, as long as only Germany is modeled in detail and the other countries are aggregated to a few nodes, the values should largely reflect changes in Germany.

  18. Heat demand curves are not included; the actual output is approximated via seasonal factors.

  19. The actual generation costs are derived from the plants efficiency η s and the input fuel price fp st : \(\widehat{C}_{st} = \frac{fp_{st}}{\eta_s(g_{nst})}g_{nst}+su_{nst}\).

  20. A simple example reveals the impact: Assume a 1,000 MW fossil plant with generation costs of 10 €/MWh that has to reduce its output because 200 MW wind energy are available and need to be fed into the grid. Running at 80% of optimal output causes the efficiency to drop and thereby the costs to rise to 10.07 €/MWh. The cost reduction therefore is not 2,000 €/h, but only 1944 €/h. The difference could be considered as the indirect marginal cost of wind energy. In reality, a clear cost allocation of wind energy is not possible, because changes in demand modify the operation of the fossil plants. Furthermore, the indirect cost of wind generation is not constant but changes with the load situation of the fossil power plants.

  21. Compare Gampe (2004).

  22. This may lead to biased results in the long run, but should not influence the price and welfare calculation within the modeled reference timeframe.

  23. Compare ISET/IWES (2002).

  24. Compare DWD (2005).

  25. Further data are derived from EMD (2005), EWEA (2005), IGW (2005), and WSH (2005).

  26. The remaining electricity consumption is used by agriculture, transportation, the energy sector and others. Since these sectors amount only for a small part of the overall consumption, they are not taken into account separately.

  27. Compare VDEW (1999).

  28. NUTS (Nomenclature des Unites Territoriales Statistiques) is a geographical code standard developed by the EU for statistical reasons: http://ec.europa.eu/eurostat/ramon/nuts/introduction_regions_de.html.

  29. Green (2007) includes different assumptions about demand point elasticities in his nodal pricing analysis of a simplified network of the UK. Based on his study, the default demand elasticity in ELMOD is −0.25. However, this value can be altered easily for different model applications - normally between 0 and −0.25.

  30. In case no national price is available, a European average price is calculated based on the existing national prices.

  31. For this application, inter-temporal and integer constraints were not included and the grid coverage was restricted to Germany. Thus, the model is solved as a NLP using the CONOPT solver in GAMS.

  32. For this application, the model is solved as a two-stage process. First the unit commitment under fixed demand and ignoring line losses is conducted as MIP using the CPLEX solver in GAMS. After fixing the plant statuses, the welfare maximizing market clearing is conducted as a NLP using the CONOPT solver in GAMS.

  33. For this application, inter-temporal and integer constraints were not included. Thus, the model solved as a NLP using the CONOPT solver in GAMS.

  34. For this application, inter-temporal and integer constraints were not included and the grid coverage was restricted to Germany and in a simplified way to its neighboring countries. Thus, the model is solved as a NLP using the CONOPT solver in GAMS.

  35. For this application, inter-temporal and integer constraints were not included. The model is solved as a NLP using the CONOPT solver in GAMS. The investment decisions are made in an iterative process solving the NLP repeatedly.

  36. For this application, the model is solved under fixed demand and ignoring losses as MIP using the CPLEX solver in GAMS.

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Acknowledgements

An earlier version of this paper was presented at the 9th IAEE European Energy Conference (June 10–13, 2007, Florence, Italy). The authors thank Kristin Dietrich, Ina Rumiantseva, Franziska Holz, Christian Todem, Till Jeske, Jürgen Apfelbeck, and Bert Willems for their help and advice on various stages of the model development process.

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  1. Faculty of Business and Economics, Chair of Energy Economics and Public Sector Management, Dresden University of Technology, 01069, Dresden, Germany

    Florian U. Leuthold, Hannes Weigt & Christian von Hirschhausen

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  1. Florian U. Leuthold
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Correspondence to Christian von Hirschhausen.

Appendix: The linear inverse demand function

Appendix: The linear inverse demand function

Assume a linear inverse demand function p nt (q nt ) of the general known form as given in Eq. 14a with the slope M ≤ 0 and axis intercept A ≥ 0. Rearranging yields the demand function Eq. 14b while Eq. 14c displays the standard equation for calculating the price elasticity of the demand.

$$ p_{nt} = A + Mq_{nt} \label{eq:df1} $$
(14a)
$$ q_{nt} = -\frac{A}{M}+\frac{1}{M}p_{nt} \label{eq:df2} $$
(14b)
$$ \varepsilon = \frac{\partial q_{nt}}{\partial p_{nt}} \frac{p_{nt}}{q_{nt}} = \frac{p_{nt}}{Mq_{nt}} $$
(14c)

In order to derive the prohibitive price A and slope M, we assume a known demand elasticity ε at the observed reference point \((P_{nt}^{\text{ref}},Q_{nt}^{\text{ref}})\). Prohibitive price A and slope M for each node n and time period t can, thus, be calculated on the basis of the given reference values for price and demand according to Eqs. 15a and 15b.

$$ M = \frac{P_{nt}^{\text{ref}}}{Q_{nt}^{\text{ref}}} \frac{1}{\varepsilon} $$
(15a)
$$ A = P_{nt}^{\text{ref}} - MQ_{nt}^{\text{ref}} $$
(15b)

Inserting Eqs. 15a and 15b into Eq. 14a yields Eq. 16 which is the equation that is used in ELMOD to determine the linear inverse demand functions per node n and time period t.

$$ p_{nt} = P_{nt}^{\text{ref}} - \frac{P_{nt}^{\text{ref}}}{MQ_{nt}^{\text{ref}}} \frac{1}{\varepsilon} Q_{nt}^{\text{ref}} + \frac{P_{nt}^{\text{ref}}}{MQ_{nt}^{\text{ref}}} \frac{1}{\varepsilon} q_{nt} = P_{nt}^{\text{ref}} + \frac{1}{\varepsilon} P_{nt}^{\text{ref}} \left( \frac{q_{nt}}{Q_{nt}^{\text{ref}}} - 1 \right) $$
(16)

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Leuthold, F.U., Weigt, H. & von Hirschhausen, C. A Large-Scale Spatial Optimization Model of the European Electricity Market. Netw Spat Econ 12, 75–107 (2012). https://doi.org/10.1007/s11067-010-9148-1

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