Generalizing Connection Charges to Any Radial Network
The same pricing procedure as use in this example on connection charges extends naturally from a one-link or two-link example to any radial electricity network. The core idea is unchanged: each link in the network must be sized to support the aggregate load of all nodes downstream of that link, and node prices are then obtained by assigning each node its contribution to required link capacity along the unique path from the root.
A Radial Network as a Graph
Let the network be represented by a directed radial graph \(G=(V,E)\), where \(V\) is the set of nodes and \(E\) is the set of links. Because the graph is radial, each node has a unique path from the root node to that node.
For each node \(n \in V\), let \(\mathcal P(n) \subseteq E\) denote the unique set of links on the path from the root to node \(n\).
For each link \(e \in E\), define the downstream set
\[ D(e) := \{n \in V : e \in \mathcal P(n)\}. \]
This is the set of nodes whose withdrawal rights use link \(e\).
Withdrawal Rights, Loads, and Peaks
Let \(H(n)\) denote withdrawal rights sold at node \(n\). For any subset of nodes \(S \subseteq V\), define total withdrawal rights by
\[ H(S) := \sum_{n \in S} H(n). \]
Let \(\mathcal I(S)\) denote the set of consumers located at nodes in \(S\). For each consumer \(i\), let \(q_{it}\) denote consumption at time \(t\), and define the individual peak
\[ \hat q_i := \max_t q_{it}. \]
For any set of nodes \(S\), define aggregate load and peak load as
\[ Q(S,t) := \sum_{i\in\mathcal I(S)} q_{it}, \qquad \hat Q(S) := \max_t Q(S,t). \]
Let the time of peak load be
\[ \hat t(S) := \arg\max_t Q(S,t). \]
Utilization and Coincidence
For any set of nodes \(S\), define the utilization rate by
\[ u(S) := \frac{\sum_{i\in\mathcal I(S)} \hat q_i}{H(S)}. \]
This measures the extent to which purchased withdrawal rights exceed realized individual peak demand.
Define the coincidence factor by
\[ \delta(S) := \frac{\hat Q(S)}{\sum_{i\in\mathcal I(S)} \hat q_i}. \]
This measures how much simultaneity there is within the set \(S\).
Combining the two definitions gives the capacity coefficient
\[ \psi(S) := \delta(S)u(S), \]
so that
\[ \hat Q(S) = \psi(S)H(S). \]
General Capacity Constraints in a Radial Network
For each link \(e\), required capacity must support the aggregate peak load of the downstream set \(D(e)\). This gives the general constraint
\[ K(e) \ge \hat Q(D(e)). \]
Using the coefficient \(\psi(D(e))\), this can be written as
\[ K(e) \ge \psi(D(e))H(D(e)) = \psi(D(e))\sum_{n\in D(e)} H(n). \]
This is the most direct generalization of the single-node and two-node setup. It implies that all downstream nodes are treated symmetrically at a given link.
The Capacity Matrix
Collecting the link constraints across all links gives the linear system
\[ AH \le K. \]
Under the symmetric aggregate construction, the matrix entries are
\[ A_{en} = \begin{cases} \psi(D(e)) & \text{if } n \in D(e),\\ 0 & \text{otherwise.} \end{cases} \]
Thus each row corresponds to a link, and each column corresponds to a node. A coefficient is nonzero exactly when the node lies downstream of the link.
Cost Minimization and Cost Recovery
Let \(r(e)\) denote the unit capacity cost on link \(e\). Total capacity cost is then
\[ r^\top K = \sum_{e\in E} r(e)K(e). \]
Given withdrawal rights \(H\), the network solves
\[ C(H) := \min_K \{r^\top K \mid AH \le K\}. \]
Because capacity is costly, the constraints bind at optimum, implying
\[ K = AH. \]
The resulting cost function is therefore
\[ C(H) = r^\top AH. \]
If prices exactly recover cost, then node prices satisfy
\[ p = A^\top r. \]
Equivalently, the price at node \(n\) is
\[ p(n) = \sum_{e\in\mathcal P(n)} A_{en}r(e). \]
This shows that the node price is the sum of weighted link-capacity costs along the unique path from the root to that node.
A More Flexible Decomposition
The symmetric aggregate formulation is convenient, but it is not the only possible construction. A more flexible approach allows each node to contribute differently to required capacity on a given link.
For each link \(e\), define node-specific alignment coefficients relative to the peak time of the downstream set:
\[ \gamma(\{n\},D(e)) := \frac{Q(\{n\},\hat t(D(e)))}{\hat Q(\{n\})}, \qquad n \in D(e). \]
Then the peak load on link \(e\) can be decomposed as
\[ \hat Q(D(e)) = \sum_{n\in D(e)} \gamma(\{n\},D(e))\hat Q(\{n\}). \]
Substituting \(\hat Q(\{n\})=\psi(\{n\})H(n)\) gives the share-based capacity constraint
\[ K(e) \ge \sum_{n\in D(e)} \gamma(\{n\},D(e))\psi(\{n\})H(n). \]
This yields the more general matrix
\[ A_{en} = \begin{cases} \gamma(\{n\},D(e))\psi(\{n\}) & \text{if } n \in D(e),\\ 0 & \text{otherwise.} \end{cases} \]
Under this formulation, nodes downstream of the same link need not be treated symmetrically. Their coefficients can differ according to how strongly each node contributes to the realized peak of that link.
Interpretation
The general radial-network procedure can therefore be summarized in four steps.
- First, identify for each link the set of downstream nodes \(D(e)\).
- Second, estimate how withdrawal rights at those nodes translate into aggregate peak load.
- Third, use these coefficients to construct the capacity matrix \(A\).
- Fourth, compute node prices from \(p=A^\top r\).
The economic structure is therefore the same in any radial network. What changes from one specification to another is the empirical construction of the matrix \(A\), that is, the rule that maps sold withdrawal rights into required link capacity.
Conclusion
Yes, the procedure generalizes to any radial network graph. The radial structure guarantees a unique path from the root to each node, which makes it possible to define downstream link usage unambiguously. Once the matrix \(A\) is specified, cost minimization, exact cost recovery, and node pricing follow directly from the same formulas as in the simpler examples.