Mathematical formulation

This document presents the mathematical foundations of the ElectricityNetworkReduction.jl model. The objective of the model is to construct a reduced electrical network that preserves the power transfer characteristics of the original system while significantly reducing its size.

1. Network admittance matrix ($Y_{bus}$)

The electrical network is represented by its nodal admittance matrix:

\[Y = G + jB\]

In the DC power flow approximation, we focus on the susceptance matrix ($B$), assuming conductance ($G$) is negligible. Each transmission line between buses $i$ and $j$ is defined by its series admittance $y_{ij} = \frac{1}{R_{ij} + jX_{ij}}$. Including shunt susceptance $B_{ij}$, the $Y_{bus}$ is assembled as:

Diagonal elements ($Y_{ii}$): Represent the sum of all admittances connected to bus $i$, including shunt susceptances: $Y_{ii} = \sum_{j \in \mathcal{N}(i)} \left( y_{ij} + j\frac{B_{ij}}{2} \right) + (G_{i}^{shunt} + jB_{i}^{shunt})$
Off-diagonal elements ($Y_{ij}$): Represent the negative of the admittance between buses $i$ and $j$: $Y_{ij} = -y_{ij}$

2. DC Power Flow and PTDFs

Under the DC approximation (negligible resistance, flat voltage magnitudes, small angles), active power flow is:

\[P = B \theta\]

Power Transfer Distribution Factors (PTDF)

The PTDF for a line $l$ (from bus $i$ to $j$) during a transaction between buses $a$ and $b$ is:

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

In the code, this is solved by calculating the sensitivity of bus angles to injections at buses $a$ and $b$ relative to a slack bus.

3. Total Transfer Capacity (TTC)

The TTC for a transaction $t$ (from $a$ to $b$) is the maximum power flow possible before any line hits its thermal limit $C_l$:

\[TTC_{t} = \min_{l \in \text{Lines}} \left( \frac{C_l}{|PTDF_{t,l}|} \right)\]

4. Network Reduction (Kron)

To reduce the system size, the network nodes are partitioned into Representative Nodes (R), which are kept, and Eliminated Nodes (E), which are removed:

\[Y = \begin{bmatrix} Y_{RR} & Y_{RE} \\ Y_{ER} & Y_{EE} \end{bmatrix}\]

Using the Kron Reduction method, we derive a reduced admittance matrix $Y_{reduced}$ that maintains the exact electrical behavior seen from the representative nodes:

\[Y_{reduced} = Y_{RR} - Y_{RE} Y_{EE}^{-1} Y_{ER}\]

5. Optimization of Equivalent Capacities

The package identifies synthetic capacities $C_l^{eq}$ for the reduced network to match the original $TTC^{orig}$. Based on optimization.jl, three methods are available:

5.1 Quadratic Programming (QP)

Objective: Minimize the squared error between original and equivalent TTC, with a regularization term to keep capacities realistic.

\[\min \sum_{t \in \mathcal{T}} (TTC_t^{eq} - TTC_t^{orig})^2 + \lambda \sum_{l \in \mathcal{L}_{red}} (C_l^{eq})^2\]

Constraints:

Flow Limit: For every transaction $t$ and every synthetic line $l$: $TTC_t^{eq} \cdot |PTDF_{t,l}| \le C_l^{eq} \quad \forall t, \forall l$
Non-negativity: $C_l^{eq} \ge 0, \quad TTC_t^{eq} \ge 0$

5.2 Linear Programming (LP)

Objective: Maximize the total transfer capacity of the reduced network while ensuring it never exceeds the original limits.

\[\max \sum_{t \in \mathcal{T}} TTC_t^{eq}\]

Constraints:

Upper Bound: $TTC_t^{eq} \le TTC_t^{orig} \quad \forall t$
Physical Limit: $TTC_t^{eq} \cdot |PTDF_{t,l}| \le C_l^{eq} \quad \forall t, \forall l$

5.3 MIQP (Original Mixed-Integer Quadratic Programming)

This formulation is used when we want to identify which synthetic line is binding (limiting) for each transaction.

Decision Variables

Synthetic line capacities: $C_l^{eq} \ge 0$
Equivalent TTC values: $TTC_t^{eq} \ge 0$:
Binding indicator: ($z_{t,l} = 1$ means line $l$ limits transaction $t$) $z_{t,l} \in \{0,1\}$: b

Objective

\[\min \sum_{t\in\mathcal{T}} (TTC_t^{eq}-TTC_t^{orig})^2 + \lambda\sum_{l\in\mathcal{L}} (C_l^{eq})^2\]

Constraints

General physical TTC upper bound: $TTC_t^{eq} \le \frac{C_l^{eq}}{|PTDF_{t,l}|}\qquad \forall t,l$
Exactly one binding line per transaction: $\sum_{l\in\mathcal{L}} z_{t,l} = 1 \qquad \forall t$
Big-M binding constraint (enforcing equality on chosen line): $\frac{C_l^{eq}}{|PTDF_{t,l}|} - TTC_t^{eq} \le M(1-z_{t,l}) \qquad \forall t,l$

5.4 Linearized MIQP → MILP

To make the MIQP practical for large networks, the implementation uses a linearized MILP solved by HiGHS.

Decision Variables (MILP)

Synthetic line capacity: $C_l^{eq} \ge 0$
Equivalent TTC: $TTC_t^{eq} \ge 0$
Binding selection: $b_{t,l}\in\{0,1\}$
Allocated capacity for transaction $t$ on line $l$: $Z_{t,l}\ge 0$
absolute TTC error (L1 linearization): $V_t^{abs}\ge 0$

if $b_{t,l}=1$, then $Z_{t,l}=C_l^{eq}$
if $b_{t,l}=0$, then $Z_{t,l}=0$

\[Z_{t,l} \le M_Z\, b_{t,l}\qquad \forall t,l\]

\[Z_{t,l} \ge C_l^{eq} - M_Z(1-b_{t,l})\qquad \forall t,l\]

E) Numerical stability bounds

\[TTC_t^{eq} \le M_{TTC}\qquad \forall t\]

Important Implementation Notes

PTDF Epsilon: The optimization ignores lines where $|PTDF| < \epsilon$ (defined in config.jl) to avoid numerical instability (division by zero).
MW Conversion: While the optimization typically occurs in per-unit (p.u.), the main-analysis.jl script converts final $C_l^{eq}$ results to MW using the Sbase defined in CONFIG.
Regularization ($\lambda$): The $\lambda$ parameter is crucial. Without it, the solver may assign infinitely high capacities to lines that don't participate in any binding constraints. A small $\lambda$ (e.g., $10^{-6}$) ensures the smallest sufficient capacity is chosen.