Mathematical formulation

This document summarises the mathematical foundations of ElectricityNetworkReduction.jl. The model builds a reduced electrical network that preserves transfer-capacity behaviour between representative nodes while greatly reducing network size.

1. Network admittance matrix

The network is represented by its nodal admittance matrix:

\[Y = G + jB\]

Each AC branch between buses $i$ and $j$ has series admittance $y_{ij} = 1/(R_{ij} + jX_{ij})$. Line charging and bus shunts are included in the diagonal terms. Transformers are treated as AC elements and are merged into the line set before $Y_{bus}$ assembly.

DC interconnectors are excluded from $Y_{bus}$. Their capacity contributes additively to TTC after the AC reduction is complete.

2. PTDFs

Under the DC approximation, active power flow is:

\[P = B\theta\]

For a transaction from bus $a$ to bus $b$, the PTDF on line $l=(i,j)$ is:

\[PTDF_l^{a \to b} = b_{ij}(\theta_i^{a \to b} - \theta_j^{a \to b})\]

For multi-island AC networks, one reference bus is chosen per AC component. Transactions between different AC islands have $TTC_{AC} = \infty$.

3. Original-network TTC

For transaction $t$, the AC transfer capacity is:

\[TTC_{AC,t} = \min_l \frac{C_l}{|PTDF_{t,l}|}\]

Lines with $|PTDF| < \varepsilon$ are ignored, where $\varepsilon$ is CONFIG.ptdf_epsilon.

DC interconnectors are additive:

\[TTC_t = TTC_{AC,t} + C_{DC,t}\]

For cross-island transactions:

\[TTC_t = \begin{cases} C_{DC,t}, & \text{if a DC bridge exists} \\ \infty, & \text{otherwise} \end{cases} $$ --- ## 4. Kron reduction Representative nodes are retained and all other nodes are eliminated: $$Y = \begin{bmatrix} Y_{RR} & Y_{RE} \\ Y_{ER} & Y_{EE} \end{bmatrix} $$ The reduced admittance matrix is: $$Y_{red} = Y_{RR} - Y_{RE}Y_{EE}^{-1}Y_{ER}\]

In multi-island cases, a small diagonal regularisation is used when needed:

\[Y_{EE}^{-1} \approx (Y_{EE} + \epsilon I)^{-1}\]

The implementation uses a direct linear solve instead of forming an explicit inverse.

5. Equivalent capacity fitting

The model identifies synthetic AC capacities $C_l^{eq}$ for the reduced network. DC capacities remain outside the AC equivalent and are added additively to total TTC.

High-resolution zone mode fixes synthetic lines between two retained high-res buses to their original physical capacity. Other synthetic capacities are optimised or constructed according to CONFIG.optimisation_type.

5.1 Optimisation modes

CONFIG.optimisation_type controls how the synthetic AC capacities are obtained:

Type	Main purpose
`QP`	Fast continuous capacity fitting. Use it when you want a practical balance between TTC accuracy and computation time.
`LP`	Deterministic capacity-envelope construction. Use it when you want a solver-light capacity set based directly on PTDF envelopes.
`MIQP`	Binding-line formulation. Use it when identifying the limiting synthetic line and obtaining physically consistent capacities is more important than speed.

In all modes, DC capacity remains additive:

\[TTC_t^{eq} = TTC_{AC,t}^{eq} + C_{DC,t}\]

For cross-island transactions, the transfer limit is the DC bridge capacity when a bridge exists, or Inf otherwise.

5.2 QP

QP minimises squared error against the original AC TTC component:

\[ \min \sum_t (TTC_{AC,t}^{eq} - TTC_{AC,t}^{orig})^2 + \lambda \sum_l (C_l^{eq})^2 $$ subject to: $$TTC_{AC,t}^{eq}|PTDF_{t,l}| \le C_l^{eq} \quad \forall t,l\]

and non-negativity of capacities and equivalent AC TTCs. DC capacity is not used as an AC floor; it is added back only in reporting.

5.3 LP

LP mode uses a deterministic capacity envelope rather than an iterative Ipopt solve. For each synthetic line:

\[C_l^{eq} = \max_{t \in \mathcal{T}_l} \left(TTC_{AC,t}^{orig}|PTDF_{t,l}|\right)\]

The resulting TTC follows the same thermal-limit relation, $\min_l C_l^{eq}/|PTDF_{t,l}|$. This is fast and deterministic. It guarantees support for the original AC target when no fixed physical capacity is lower than the envelope, but it can overestimate some transaction TTCs.

5.4 MIQP / MILP binding formulation

MIQP mode is implemented as a HiGHS MILP that selects a binding synthetic line for each transaction from a pruned candidate set.

Decision variables:

\[C_l^{eq} \ge 0\]
: synthetic capacities
\[TTC_{AC,t}^{eq} \ge 0\]
: equivalent AC TTC
\[z_{t,l} \in \{0,1\}\]
: binding-line indicator for candidate lines
\[V_t^{abs} \ge 0\]
: absolute TTC error

Objective:

\[ \min \sum_t V_t^{abs} + \lambda\sum_l C_l^{eq} $$ with: $$V_t^{abs} \ge TTC_{AC,t}^{eq} - TTC_{AC,t}^{orig}\]

\[V_t^{abs} \ge TTC_{AC,t}^{orig} - TTC_{AC,t}^{eq}\]

Physical upper-bound constraints:

\[C_l^{eq} \ge TTC_{AC,t}^{eq}|PTDF_{t,l}| \quad \forall t,l\]

Binding candidate selection:

\[\sum_{l \in \mathcal{K}_t} z_{t,l} = 1\]

where $\mathcal{K}_t$ is the pruned set of likely limiting lines for transaction $t$.

Big-M binding logic:

\[C_l^{eq} \le TTC_{AC,t}^{eq}|PTDF_{t,l}| + M_l(1-z_{t,l}) \quad \forall l \in \mathcal{K}_t\]

Controls:

CONFIG.miqp_binding_candidates_per_txn
CONFIG.miqp_time_limit_sec
CONFIG.miqp_mip_rel_gap

6. Implementation notes

ptdf_epsilon removes near-zero PTDF entries to avoid unstable divisions.
Optimisation is performed in per-unit and exported in MW using CONFIG.base.
Parallel AC circuits are aggregated consistently: admittances are summed in $Y_{bus}$ and capacities are summed for TTC limits.
lambda discourages unnecessarily large equivalent capacities.
Equivalent reactance is back-computed from the Kron-reduced matrix as $X^{eq} = 1 / \operatorname{Im}(Y_{kron,ij})$.
DC lines are appended to Equivalent_Capacities with is_dc = true and X_pu = 0.