🔌 Power Flow Margin Optimization

This repository implements a Mixed-Integer Linear Programming (MILP) model for max–min thermal margin optimization in a transmission network using linear sensitivity factors.

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📁 Repository Structure

power-flow-margin-optimization/
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|-- solve_pulp.py              # Main optimization model (MILP, PuLP + CBC)
|-- plot_results.py            # Plot generation (base vs optimized comparisons)
|-- data.py                    # Fully synthetic dataset (lines, PSTs, generators, sensitivities)
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|-- requirements.txt
|-- README.md
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|-- results/                   # Generated plots (after running plot_results.py)
    |-- margins_base_vs_opt.png
    |-- absflow_vs_limit.png
    |-- controls_taps.png
    |-- controls_gens.png

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🧾 Problem Description

A transmission operator monitors a set of critical network elements (lines) that must remain within their thermal limits.

The operator can influence line flows using:

• Phase-Shifting Transformers (PSTs) with integer tap positions
• Generator redispatch with minimum/maximum output bounds

The goal is to improve system security by maximizing the worst (minimum) thermal margin across all monitored lines.

Thermal margin of a line ℓ (for all ℓ ∈ ℒ):
margin_ℓ = F_ℓ^max − |F_ℓ|

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📊 Synthetic Input Data

All data in this repo is synthetic and fully editable in data.py.

Line data

• F_ℓ^max : thermal limit of line ℓ (MW), ∀ ℓ ∈ ℒ
• F_ℓ^ref : reference flow before control (MW), ∀ ℓ ∈ ℒ

PST data

• t_p^init : initial tap position of PST p (integer), ∀ p ∈ 𝓟
• R_p : available tap range of PST p, taps allowed in [−R_p, +R_p], ∀ p ∈ 𝓟
• PSDF_{ℓ,p} : MW change on line ℓ per +1 tap on PST p, ∀ ℓ ∈ ℒ, p ∈ 𝓟

Generator data

• G_g^init : initial output of generator g (MW), ∀ g ∈ 𝓖
• G_g^min , G_g^max : output bounds of generator g (MW), ∀ g ∈ 𝓖
• PTDF_{ℓ,g} : MW change on line ℓ per +1 MW redispatch of generator g, ∀ ℓ ∈ ℒ, g ∈ 𝓖

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🧮 Optimization Model

Sets

• ℒ : set of monitored lines, indexed by ℓ
• 𝓟 : set of PSTs, indexed by p
• 𝓖 : set of generators, indexed by g

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Decision Variables

• t_p ∈ ℤ
  Tap position of PST p, ∀ p ∈ 𝓟

• G_g ∈ ℝ
  Output power of generator g (MW), ∀ g ∈ 𝓖

• m ≥ 0
  Minimum thermal margin across all lines (MW)

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🔁 Flow Model (Linear Sensitivity Approximation)

Final power flow on line ℓ is computed as:

F_ℓ = F_ℓ^ref
    + Σ_{p∈𝓟} PSDF_{ℓ,p} · (t_p − t_p^init)
    + Σ_{g∈𝓖} PTDF_{ℓ,g} · (G_g − G_g^init)      ∀ ℓ ∈ ℒ

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🎯 Objective Function

Maximize the worst-case (minimum) margin across all lines:

maximize   m

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📐 Constraints

1️⃣ PST Tap Bounds

−R_p ≤ t_p ≤ R_p            ∀ p ∈ 𝓟

2️⃣ Generator Output Bounds

G_g^min ≤ G_g ≤ G_g^max     ∀ g ∈ 𝓖

3️⃣ Power Balance (Lossless Approximation)

Σ_{g∈𝓖} (G_g − G_g^init) = 0

4️⃣ Thermal Safety (Absolute-Value Linearization)

For every monitored line:

|F_ℓ| ≤ F_ℓ^max − m         ∀ ℓ ∈ ℒ

This is enforced using two linear inequalities:

 F_ℓ ≤ F_ℓ^max − m          ∀ ℓ ∈ ℒ
−F_ℓ ≤ F_ℓ^max − m          ∀ ℓ ∈ ℒ

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▶️ How to Run

pip install -r requirements.txt
python solve_pulp.py
python plot_results.py

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📦 Requirements

pulp>=2.7
matplotlib>=3.7

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👤 Author

Huseyin Ugur Yildiz