Why zonal pricing does not ignore intra-zonal constraints

There is a common misconception with zonal electricity pricing which is that it ignores intra-zonal constraints. The aim of this blog post is to explain why it is not the case by sticking to the fundamental principles of zonal pricing.

First, let me insist on the fact that when I refer to zonal pricing, I mean the fundamental principles of zonal pricing, which is that the price of electricity is the same for nodes of the same zone. I do not refer to specific implementation of the zonal pricing paradigm in exisiting market (e.g. the day-ahead market in Europe).

With this clarification made, I can go even further in my claim: zonal pricing does not discriminate between intra and inter-zonal lines. It treats them exactly in the same way.

In order to demonstrate this, and to discuss why there are often misconceptions on this matter, I will present below what I believe is the purest model of economic dispatch with zonal pricing. For that, let me start with the most simple model of transmission-aware economic dispatch with nodal pricing. This problem can be written as follows:

\begin{aligned} \min_{q, r, f} &\sum_{g \in G} MC_g \cdot q_{g} \\ (\mu_{g}): \, &P_{g} - q_{g} \geq 0, g \in G \\ (\rho_{n}): \, &-r_{n} + \sum_{g \in G_n(n)} q_{g} - D_{n} = 0, n \in N \\ (\psi_{k}): \, &f_{k} - \sum_{n \in N} PTDF_{kn} \cdot r_{n} = 0, k \in K \\ (\phi): \, &\sum_{n \in N} r_{n} = 0\\ (\lambda^-_k, \lambda^+_k): \, &- TC_k \leq f_{k} \leq TC_k, k \in K \\ &q \geq 0 \end{aligned}

The notation is as follows: $q_g$ is the electricity production by generator $g \in G$ , $P_g$ is the capacity of generator $g$ and $MC_g$ corresponds to its marginal cost. $r_n$ is the injection at node $n \in N$ while $D_n$ is the inelastic demand at that node. $f_k$ is the power flow on transmission line $k \in K$ which is limited by a thermal limit $TC_k$ . PTDF is the PTDF matrix corresponding to the transmission network and the variables shown between brackets on the left of teach constraints corresponds to their dual variables. The goal of this optimization problem is to minimize the cost of production of the electricity for serving a fixed demand $D$ while respecting the DC approximation of the power flow equations associated to the transmission grid.

The question is now : what is the corresponding model under the zonal pricing paradigm ? This question, unlike in nodal pricing, is ambiguous: there has been several zonal pricing models proposed in the literature and implemented in real life (Weibelzahl (2017)). Let me propose the following: let us go back to the fundamental principle of zonal pricing and impose that principle, and only that principle, on the nodal model (1). As I said earlier, the fundamental principle of zonal pricing is that the price of electricity is the same for nodes of the same zone. We thus have to add constraints on the prices. In order to that, one should go the dual space of our market clearing problem. The dual of (1) can be written as follows:

\begin{aligned} \max_{\rho, \mu, \psi, \phi, \lambda} & \sum_n D_n \rho_n - \sum_{g} P_{g} \mu_{g} - \sum_{k} TC_k (\lambda_k^+ + \lambda_k^-) \\ (r_n): \, &\rho_n + \sum_k PTDF_{kn} \psi_k - \phi = 0, n \in N\\ (q_{g}): \, &MC_g - \rho_{N_g(g)} + \mu_{g} \geq 0, g \in G\\ (f_k): \, &-\psi_k - \lambda_k^- + \lambda_k^+ = 0, k \in K\\ &\mu, \lambda^+, \lambda^- \geq 0 \end{aligned}

Applying directly the fundamental property of zonal pricing, which is that the prices within the same zone should be equal, we obtain a natural zonal extension of nodal pricing by adding to model (2) the following constraints:

\rho_{n_1} = \rho_{n_2}, \, \, \, \forall (n_1, n_2) \in z, \forall z \in Z

This is equivalent to introducing a new variable $\rho_z$ for each zone $z$ , corresponding to the price of the zone, and imposing the following constraints:

\rho_{n} = \rho_{z}, \, \, \, \forall n \in N(z), \forall z \in Z

This in turn is equivalent to replacing every occurrence of $\rho_n$ by $\rho_{Z(n)}$ in (2), yielding:

\begin{aligned} \max_{\rho, \mu, \psi, \phi, \lambda} & \sum_n D_n \rho_{Z(n)} - \sum_{g} P_{g} \mu_{g} - \sum_{k} TC_k (\lambda_k^+ + \lambda_k^-) \\ (r_n): \, &\rho_{Z(n)} + \sum_k PTDF_{kn} \psi_k - \phi = 0, n \in N\\ (q_{g}): \, &MC_g - \rho_{N_g(g)} + \mu_{g} \geq 0, g \in G\\ (f_k): \, &-\psi_k - \lambda_k^- + \lambda_k^+ = 0, k \in K\\ &\mu, \lambda^+, \lambda^- \geq 0 \end{aligned}

Model (5) will produce zonal market clearing prices and is thus a model for clearing the market under the zonal pricing paradigm. In order to obtain the equivalent of model (2) under the zonal paradigm, we can simply go back to the primal space:

\begin{aligned} \min_{q, r, f} &\sum_{g \in G} MC_g \cdot q_{g} \\ (\mu_{g}): \, &P_{g} - q_{g} \geq 0, g \in G\\ (\rho_{z}): \, &-\sum_{n \in N(z)} r_{n} + \sum_{g \in G_z(z)} q_{g} - \sum_{n \in N(z)} D_{n} = 0, z \in Z \\ (\psi_{k}): \, &f_{k} - \sum_{n \in N} PTDF_{kn} \cdot r_{n} = 0, k \in K \\ (\phi): \, &\sum_{n \in N} r_{n} = 0\\ (\lambda^-_k, \lambda^+_k): \, &- TC_k \leq f_{k} \leq TC_k, k \in K \\ &q \geq 0 \end{aligned}

The two models, i.e. models (1) and (5), are impressively similar. The only equation that differs is the energy balance equation, imposed on every node in the nodal model but only on every zone in the zonal model.

In order to improve our understanding of the model, one can define the following variable without modifying the model:

p_z = \sum_{n \in N(z)} r_{n}

The model becomes:

\begin{aligned} \min_{q, r, f} &\sum_{g \in G} MC_g \cdot q_{g} \\ (\mu_{g}): \, &P_{g} - q_{g} \geq 0, g \in G\\ (\rho_{z}): \, &-p_z + \sum_{g \in G_z(z)} q_{g} - \sum_{n \in N(z)} D_{n} = 0, z \in Z \\ (\tilde{\rho_z}): \, &p_z - \sum_{n \in N(z)} r_{n} = 0, z \in Z \\ (\psi_{k}): \, &f_{k} - \sum_{n \in N} PTDF_{kn} \cdot r_{n} = 0, k \in K \\ (\phi): \, &\sum_{n \in N} r_{n} = 0\\ (\lambda^-_k, \lambda^+_k): \, &- TC_k \leq f_{k} \leq TC_k, k \in K \\ &q \geq 0 \end{aligned}

This transformation shows that the fundamental variables on which the transmission equations are defined are the zonal net positions $p_z$ . The feasible set of zonal net positions is then defined as the projection of the feasible set of nodal net injections on the space of zonal net positions. It is now clear that the only thing that matters for the zonal model is the set of zonal net positions. Zonal pricing will only consider inter-zonal exchanges. However, when it comes to transmission lines, it treats them exactly in the same way, whether they are intra or inter-zonal. If an intra-zonal line is largely affected by cross-zonal trade, it could be binding.

Short references

Weibelzahl, Martin, Nodal, zonal, or uniform electricity pricing: how to deal with network congestion, 2017.