Congestion management in power systems

Abstract

In liberalized power systems, generation and transmission services are unbundled, but remain tightly interlinked. Congestion management in the transmission network is of crucial importance for the efficiency of these inter-linkages. Different regulatory designs have been suggested, analyzed and followed, such as uniform zonal pricing with redispatch or nodal pricing. However, the literature has either focused on the short-term efficiency of congestion management or specific issues of timing investments. In contrast, this paper presents a generalized and flexible economic modeling framework based on a decomposed inter-temporal equilibrium model including generation, transmission, as well as their inter-linkages. The model covers short-run operation and long-run investments and hence, allows to analyze short and long-term efficiency of different congestion management designs that vary with respect to the definition of market areas, the regulation and organization of TSOs, the way of managing congestion besides grid expansion, and the type of cross-border capacity allocation. We are able to identify and isolate implicit frictions and sources of inefficiencies in the different regulatory designs, and to provide a comparative analysis including a benchmark against a first-best welfare-optimal result. To demonstrate the applicability of our framework, we calibrate and numerically solve our model for a detailed representation of the Central Western European (CWE) region, consisting of 70 nodes and 174 power lines. Analyzing six different congestion management designs until 2030, we show that compared to the first-best benchmark, i.e., nodal pricing, inefficiencies of up to 4.6% arise. Inefficiencies are mainly driven by the approach of determining cross-border capacities as well as the coordination of transmission system operators’ activities.

Appendices

Appendix 1: Notation list

See Table 2.

Table 2 Model sets, parameters and variables

Appendix 2: Derivation of the load flow equations by means of PTDFs

Power Transfer Distribution Factors (PTDFs) are a well-established method to account for load flows in meshed electricity networks by means of linearization. They can be derived from the network equations in an AC power network that write as follows:²⁹

$$\begin{aligned}&\displaystyle P_i = \ U_i \sum _{j\in {\mathcal {I}}} U_j (g_{i,j} \cos (\varphi _i - \varphi _j) + b_{i,j} \sin (\varphi _i - \varphi _j)) \end{aligned}$$

(5)

$$\begin{aligned}&\displaystyle Q_i = \ U_i \sum _{j\in {\mathcal {I}}} U_j (g_{i,j} \sin (\varphi _i - \varphi _j) - b_{i,j} \cos (\varphi _i - \varphi _j)) \end{aligned}$$

(6)

$$\begin{aligned}&\displaystyle P_{i,j} = \ U_i^2 g_{i,j} - U_i U_j g_{i,j} \cos (\varphi _i - \varphi _j) - U_i U_j b_{i,j} \sin (\varphi _i - \varphi _j) \end{aligned}$$

(7)

$$\begin{aligned}&\displaystyle Q_{i,j} = \ - U_i^2 (b_{i,j} + b_{i,j}^{sh}) + U_i U_j b_{i,j} \cos (\varphi _i - \varphi _j) - U_i U_j g_{i,j} \sin (\varphi _i - \varphi _j).\qquad \end{aligned}$$

(8)

$P_i$ and $Q_i$ represent the net active and reactive power infeed (i.e., nodal power balances), and $P_{i,j}$ and $Q_{i,j}$ the active and reactive power flows between node i and j. Voltage levels U and phase angles $\varphi$ of the nodes as well as series conductances g and series susceptances b of the transmission lines determine active and reactive power flows in a highly nonlinear way.

In order to linearize the above equations, a number of assumptions are made:

• All voltages are set to 1 p.u.

• Voltage angles are all similar (and hence, $\sin (\varphi _i - \varphi _j)\approx \varphi _i - \varphi _j$).

• Reactive power is neglected (i.e., $Q_{i}=Q_{i,j}=0$).

• Losses are neglected and line reactances are much larger than their resistance, such that $x \gg r \approx 0$.

Under these assumptions and using Kirchoff’s power law, the network equations can be simplified to

$$\begin{aligned} P_{i,j}&\approx \frac{1}{x_{i,j}}(\varphi _i - \varphi _j) \end{aligned}$$

(9)

$$\begin{aligned} P_{i}&\approx \sum _{j\in {\varOmega }_i} \frac{1}{x_{i,j}} (\varphi _i - \varphi _j) , \end{aligned}$$

(10)

with ${\varOmega }_i$ representing the nodes adjacent to i. If there are multiple nodes and branches, this can be written in a more convenient matrix notation as $\varvec{{\tilde{P}}_{i}} = \varvec{{\tilde{B}}} \cdot \varvec{{\tilde{{\varTheta }}}}$, with $\varvec{{\tilde{P}}_{i}}$ being the vector of net active nodal power balances $P_i$, $\varvec{{\tilde{{\varTheta }}}}$ the vector of phase angles, and $\varvec{{\tilde{B}}}$ the nodal admittance matrix with the following entries:

$$\begin{aligned} {\tilde{B}}_{i,j}&= -\frac{1}{x_{i,j}} \end{aligned}$$

(11)

$$\begin{aligned} {\tilde{B}}_{i,i}&= \sum _{j\in {\varOmega }_i} \frac{1}{x_{i,j}}. \end{aligned}$$

(12)

By deleting the row and column belonging to the reference node (thus assuming a zero reference angle at this node), the previously singular matrix $\varvec{{\tilde{B}}}$ becomes ${\varvec{B}}$, the vector of phase angles $\varvec{{\varTheta }}$, and the vector of net active nodal power balances $\varvec{P_{i}}$. We can now solve for $\varvec{{\varTheta }}$ by matrix inversion:

$$\begin{aligned} \varvec{{\varTheta }} = {\varvec{B}} ^{-1} \cdot \varvec{P_{i}}. \end{aligned}$$

(13)

Defining $H_{ki}=1/x_{i,j}$, $H_{kj}=-1/x_{i,j}$ and $H_{km}=0$ for $m \ne i,j$ (with k running over the branches i, j), Eq. 9 can be rewritten in matrix form as $\varvec{P_{i,j}} = {\varvec{H}} \cdot \varvec{{\varTheta }}$. Inserting $\varvec{{\varTheta }}$ from Eq. 13 finally yields

$$\begin{aligned} \varvec{P_{i,j}} = {\varvec{H}} \cdot \varvec{{\varTheta }} = {\varvec{H}} \cdot {\varvec{B}} ^{-1} \cdot \varvec{P_{i}} = PTDF \cdot \varvec{P_{i}} \end{aligned}$$

(14)

The elements of PTDF are the power transfer distribution factors that constitute a linear relationship between nodal power balances and load flows. Note that the size of the PTDF matrix is determined by the size of the system, with the number of matrix lines corresponding to the number of transmission lines, and the number of matrix columns representing the number of nodes. The matrix entry $PTDF_{k,i,j}$ represents the impact of the power balance in node k on power flows on line between node i and j. Also note that PTDF essentially depends (only) on the line impedances $x_{i,j}$ in the system that in turn depend primarily on the respective line capacities ${\overline{P}}_{i,j}$. Hence, as done, e.g., in Hogan et al. (2010), we apply the law of parallel circuits to adjust line reactances when altering transmission capacities, i.e.,

$$\begin{aligned} x_{i,j} = \frac{{\overline{P}}^0_{i,j}}{{\overline{P}}_{i,j}} x^0_{i,j}, \end{aligned}$$

(15)

where $\{{\overline{P}}^0_{i,j}, x^0_{i,j}\}$ is a point of reference taken from the original configuration of the transmission network. Overall, this yields a functional dependency of power flows on nodal balances (determined by generation $G_k$ and load $d_k$ in all nodes) as well as line capacities ${\overline{P}}_{k,l}$ of all lines in the system, i.e., $P_{i,j} = P_{i,j}({\overline{P}}_{k,l},G_k,d_k)$.

Appendix 3: Equivalence of Problem P1 and P1 $^\prime$

To show the equivalence of the optimal solution of P1 and P1 $^\prime$, we compare the problems by means of their Karush–Kuhn–Tucker (KKT) conditions. If they are equal, the optimal solution has to be equal, too (e.g., Bazaraa et al. 2006). For the derivations, note that trade is a function of line capacity, generation and demand, i.e., $T_{i,j,t} = T_{i,j,t}({\overline{P}}_{k,l},G_{k,t},d_{k,t})$, and that $T_{i,j,t} = -T_{j,i,t}$. The following is the Lagrangian function belonging to Problem P1:

$$\begin{aligned} \begin{aligned} L({\overline{G}}_{i},G_{i,t},T_{i,j,t},{\overline{P}}_{i,j},\lambda _{i,t}, \tau _{i,t},\kappa _{i,j,t})&=\sum _{i} \delta _{i} {\overline{G}}_{i} + \sum _{i,t} \gamma _{i,t} G_{i,t} + \sum _{i,j} \mu _{i,j} {\overline{P}}_{i,j}\\&\quad +\,\sum _{i,t}\left( \lambda _{i,t}\left( G_{i,t} - \sum _{j} T_{i,j,t} - d_{i,t}\right) \right. \\&\quad +\left. \,\tau _{i,t}\left( G_{i,t} - {\overline{G}}_{i}\right) \right) \\&\quad +\,\sum _{i,j,t}\left( \kappa _{i,j,t}\left( \vert {T_{i,j,t}}\vert - {\overline{P}}_{i,j} \right) \right) \nonumber \end{aligned}\\ \end{aligned}$$

(16)

The corresponding KKT conditions are:

The Langragian functions for P1 $^\prime$ are:

$$\begin{aligned} \begin{aligned} L'^{a}({\overline{G}}_{i},G_{i,t},T_{i,j,t},\lambda _{i,t},\tau _{i,t})&=\sum _{i} \delta _{i} {\overline{G}}_{i} + \sum _{i,t} \gamma _{i,t} G_{i,t} + \sum _{i,j,t} \kappa _{i,j,t} T_{i,j,t}\\&\quad +\,\sum _{i,t}\left( \lambda _{i,t}\left( G_{i,t} - \sum _{j,t} T_{i,j,t} - d_{i,t}\right) \right. \\&\quad +\left. \,\kappa _{i,j,t}\left( G_{i,t} - {\overline{G}}_{i}\right) \right) \\ \end{aligned} \end{aligned}$$

(18)

$$\begin{aligned} L'^{b}({\overline{P}}_{i,j},\kappa _{i,j,t})= & {} \sum _{i,j} \mu _{i,j} {\overline{P}}_{i,j} + \sum _{i,t}\left( \lambda _{i,t}\left( G_{i,t} - \sum _{j,t} T_{i,j,t} - d_{i,t}\right) \right) \nonumber \\&+ \sum _{i,j,t} \left( \kappa _{i,j,t}\left( \vert {T_{i,j,t}}\vert -{\overline{P}}_{i,j}\right) \right) \end{aligned}$$

(19)

The KKT conditions of P1 $^\prime$ a are:

The KKT conditions of P1 $^\prime$ b are:

Comparing the KKT conditions of problem P1 to the ones of P1a and P1b, we can conclude that the problems are indeed equivalent.

Appendix 4: Model of Setting III: coupled zonal markets with zonal TSOs and zonal redispatch

Mathematically, the model of Setting III, representing coupled zonal markets with zonal TSOs and zonal redispatch, is formulated as follows:

P3a Generation

P3b Transmission

In problem P3, there are now separate optimization problems for each zonal TSO (indicated by $Y_{m}$), with the objective to minimize costs from zonal grid and cross-border capacity extensions as well as from zonal redispatch measures (Eq. 22e). For the redispatch, TSOs have to consider the same restrictions as in the previous setting (Eqs. 22h and 22i). TSOs are assumed to negotiate about the extension of cross-border capacities according to some regulatory rule that ensures the acceptance of a unique price for each cross-border line by both of the neighboring TSOs. For instance, the regulatory rule may be specified such that both TSOs are obliged to accept the higher price offer, or, equivalently, the lower of the two capacities offered for the specific cross-border line. Corresponding costs from inter-zonal grid extensions are assumed to be shared among the TSOs according to the cost allocation key $\sigma _{i,j}$. According to Eq. 22j, prices for transmission between zones that are provided to the generation stage ($\kappa _{m,n,t}$) are determined just as in the previous Setting II with only one TSO, depending on the type of market coupling, i.e., the specification of function g. The only difference is that line-specific prices $\kappa _{i,j,t}$ may now deviate from Setting II as they result from the separated activities of each zonal TSO (specifically, from Eq. 22g, i.e., the restriction of flows on intra-zonal and cross-border lines).

Appendix 5: Model of Setting IV: coupled zonal markets with zonal TSOs and g-component

Mathematically, the model of Setting IV, representing coupled zonal markets with zonal TSOs and g-component, is formulated as follows:

P4a Generation

P4b Transmission

Problem P4a is almost identical to P2a (and P3a), with the exception of one term in the objective function (23a). With a g-component, generators pay nodal instead of zonal prices for transmission ($\kappa _{i,j,t}$ instead of $\kappa _{m,n,t}$), depending on the impact of their nodal generation level on the grid infrastructure (by means of ${T}_{i,j,t} = {T}_{i,j,t}(G_{k,t},d_{k,t})$). These prices are determined by the zonal TSOs via their flow-restriction (23g).

Appendix 6: Numerical algorithm for NTC-coupled zonal markets, zonal TSOs, and zonal redispatch

In Section 3, we have shown the numerical implementation of the nodal pricing regime. For the sake of clarifying the major changes needed to represent the alternative Settings II–IV, we here present the model for m zonal (instead of nodal) markets that are coupled via NTC-based capacity restrictions, along with multiple zonal TSOs (instead of only one), all having the possibility to deploy zonal redispatch as an alternative to grid expansion. Hence, the model corresponds to Setting III with NTC-based market coupling. Compared to nodal pricing, no more nodal or time-specific information about grid costs is provided. Instead, an aggregated price $\kappa _{m,n,t}^{(v)}$ for each border is calculated via a function $g_{NTC}$ and passed on to the generation level. The model with flow-based market coupling works in the same way, only that the price $\kappa _{m,n,t}^{(v)}$ is calculated via a different function $g_{FB}$.

Appendix 7: Numerical assumptions for the large-scale application

To depict the CWE region in a high spatial resolution, we split the gross electricity demand per country among the nodes belonging to this country according to the percentage of population living in that region (Tables 3–6).

Table 3 Assumptions for the gross electricity demand [TWh]

Table 4 Assumptions for the generation technology investment costs [€/kW]

Table 5 Assumptions for the gross fuel prices [€/MWh$_{th}$]

Table 6 Assumptions for the grid extension and FOM costs

Appendix 8: Convergence analysis

To illustrate the convergent behavior of our problem, Fig. 6, left hand side, shows the development of the optimality error (relative difference between the upper and lower bound of the optimization), along with the (absolute) rate of change of the lower bound obtained during the iterative solution of the nodal pricing setting. The lower bound is observed to change only slightly, reaching change rates smaller than 0.01% after some 40 iterations. Moreover, as can be derived from the interpolation curves presented in Fig. 6, left hand side, the relative error decreases at much faster rates with a ratio of approximately 200 for an estimated exponential trend and an iteration count of 60. Based on the fact that in a Benders decomposition the lower bound is non-decreasing (i.e., change rates are always positive as demonstrated in Fig. 6, left hand side), and the empirically observed behavior of the lower bound, it can be concluded that the error further decreases mainly due to changes in the upper bound. Hence, we argue that the lower bound can be taken as a good approximation of the optimal objective value as soon as our convergence criterion is met. To support this argument and to deepen our insights, we closely analyzed optimized levels of the variables, observing that they reach fairly stable levels in the last iterations before reaching the convergence criterion.³⁰ As an example, the right hand side of Fig. 6 shows aggregated AC line capacities obtained in the final runs of the nodal pricing setting.

Fig. 6

Development of lower bound, optimality error and aggregated AC line capacities during the iteration in Setting I

Based on the interpolation curves estimated from the observed changes in the optimality error, a 1% threshold is expected to be reached after around 150 iterations. The estimated increase of the lower bound and hence, the improvement of the optimal solution, will then be around 0.21% higher compared to our obtained value. At around 300 iterations, the optimal solution will deviate by about 0.24% from our obtained value, and further improvements of the optimal solution would be negligible. Considering the extensive computational burden as well as the expected limited improvements, we do not consider a smaller convergence threshold and rather accept some level of uncertainty regarding the different levels of optimality achieved in the different settings.