Fairness in Robust Unit Commitment Problem Considering Suppression of Renewable Energy

This paper proposes the RE-RPfair model, an extension of the Renewable Energy Robust Optimization Problem, which incorporates the Gini Index to ensure fair allocation of renewable energy suppression among PV sources while addressing uncertainties in unit commitment.

The Gist

Here is an explanation of the paper, translated into simple language with everyday analogies.

The Big Picture: The "Tug-of-War" of Power

Imagine a power company is like a restaurant manager trying to plan the kitchen for the next day. They have to decide:

1. Which ovens (thermal generators) to turn on.
2. How much food to cook.
3. How to handle the fact that they don't know exactly how many customers will show up (electricity demand) or how much "free food" (solar power) they might get from a neighbor.

The problem is that solar power is unpredictable. Sometimes the sun shines brightly (lots of free energy), and sometimes it's cloudy (little energy). If there is too much solar power and not enough demand, the grid gets overloaded. To keep the lights on safely, the manager has to tell some solar farms to turn down their output (a process called "curtailment" or "suppression").

The Old Problem: The "Unfair Lunch"

In the past, the manager used a computer program (called Unit Commitment) to make these decisions. The goal was simply to save money on fuel.

However, this created a fairness problem. Imagine three solar farms (PV1, PV2, and PV3) all sending power to the grid. To save money, the old computer program might decide:

• "We'll shut down PV1 completely today."
• "We'll shut down PV2 completely today."
• "We'll let PV3 run at full power."

Even though PV1 and PV2 lost all their income for the day, and PV3 made a fortune, the computer didn't care. It just wanted the cheapest fuel bill. In a deregulated market where solar farms are owned by different people (not the power company), this feels like unfairness. If owners feel treated unfairly, they might quit the market entirely.

The New Solution: The "Fairness Penalty"

The authors of this paper (from Toshiba and RIKEN) proposed a new, smarter computer program called RE-RPfair.

Think of this new program as a strict referee who adds a new rule: "You can save money, but you must share the burden of turning off the lights equally."

Here is how they did it:

1. The Goal: Minimize the total cost of running the power plant.
2. The Twist: They added a "Fairness Penalty" to the math.
   • Imagine the solar farms are students in a class. The goal is to make sure every student gets roughly the same amount of "homework" (being turned off).
   • If one student gets 100% of the homework and the others get 0%, the "Fairness Penalty" score goes up.
   • Since the computer wants to keep the total score (cost + penalty) as low as possible, it is forced to spread the "turning off" duties evenly among all solar farms.

The Secret Sauce: The "Gini Index"

How do you measure fairness mathematically? The authors used a famous tool from economics called the Gini Index.

• 0 means everyone is exactly equal (perfect fairness).
• 1 means one person has everything and everyone else has nothing (total unfairness).

Usually, the Gini Index is hard to use in computer math because it requires sorting numbers, which breaks the math formulas. The authors found a clever workaround: they used a simpler math trick (summing up the differences from the average) that acts just like the Gini Index but is easy for the computer to solve.

The Result: A Fairer Grid

The team tested their new model on a simulated island power grid (like Miyako Island in Japan). They compared the old "Money-Only" model against their new "Fairness" model.

• The Old Model: One solar farm got shut down 90% of the time, while another got shut down only 10%. The Gini Index (unfairness score) was high.
• The New Model: The shut-downs were spread out. Every solar farm got turned off roughly the same amount of time. The Gini Index dropped significantly.

The Best Part: The new model didn't cost much more money to run. The "penalty" for being fair was so small that it barely affected the total bill, but it made a huge difference in how fair the system felt to the solar owners.

Summary Analogy

Imagine a group of friends going on a road trip where they have to take turns driving the car.

• The Old Way: The friend with the cheapest gas card drives the whole way, and the others never drive. It's efficient, but the others get bored and angry.
• The New Way (RE-RPfair): The computer calculates the route to save gas, but it also ensures that everyone drives for roughly the same amount of time. It might add a tiny bit of extra distance to make sure everyone gets a turn, but everyone leaves the trip happy and feeling treated fairly.

This paper proves that we can build power grids that are not only efficient and safe but also fair to the renewable energy owners who keep the system running.

Technical Summary

Here is a detailed technical summary of the paper "Fairness in Robust Unit Commitment Problem Considering Suppression of Renewable Energy."

1. Problem Statement

The paper addresses the Unit Commitment (UC) problem, where power system operators schedule generation units (thermal and renewable) one day in advance to minimize costs while meeting demand. The study focuses on two critical challenges:

• Uncertainty: The UC problem is subject to uncertainties in electricity load and renewable energy generation (specifically Photovoltaic/PV systems), which are not perfectly predictable.
• Curtailment and Fairness: Due to the rapid spread of PVs, system operators often must suppress (curtail) PV output to maintain system security. In deregulated markets where PV owners are separate from the utility, curtailment causes revenue loss. Existing robust optimization models (specifically the authors' previous RE-RP model) handle uncertainty and suppression but fail to ensure fairness in how suppression is distributed among different PV owners. This lack of fairness can lead to market exit by generators who perceive the allocation as inequitable.

The core objective is to develop a model that minimizes operational costs under uncertainty while ensuring an equitable distribution of PV curtailment among all PV units.

2. Methodology

2.1 Mathematical Formulation: RE-RPfair

The authors propose a new model, RE-RPfair, which extends their previous Renewable Energy Robust Optimization Problem (RE-RP).

• Deterministic Base: The model starts with a mixed-integer linear program (MILP) representing the deterministic UC problem, including thermal unit commitment (on/off, start-up/shut-down) and PV suppression decisions.
• Fairness Metric (Gini Index): To quantify fairness, the authors utilize the Gini Index, a standard metric for income inequality. However, the Gini Index requires a sorting operation, making it non-linear and incompatible with standard mathematical optimization.
• Linearization Strategy: To make the problem solvable, the authors replace the Gini Index in the optimization objective with the Sum of Absolute Deviations ($L_1$ norm) of PV outputs from their mean.
  • They define $s_l$ as the total power of PV $l$ over the planning horizon.
  • The penalty term added to the objective function is $\chi \sum |s_l - \bar{s}|$, where $\chi$ is a weighting parameter.
  • This non-linear absolute value term is linearized using auxiliary variables ($a^+, a^-$) and constraints, allowing the problem to remain a Mixed-Integer Linear Program (MILP).
• Two-Stage Adaptive Robust Optimization: The model is formulated as a two-stage robust problem:
  • Stage 1 (Here-and-Now): Decisions on unit commitment ($x$) and PV suppression signals ($r$).
  • Stage 2 (Wait-and-See): Economic dispatch ($y$) and recourse actions based on realized uncertainty (load $\zeta$ and PV output $\eta$).
  • The objective is to minimize the worst-case cost (maximizing over the uncertainty set $\mathcal{D} \times \mathcal{Z}$) while including the fairness penalty.

2.2 Solution Algorithm: Benders Decomposition

The authors prove that the Benders Decomposition algorithm, previously used for RE-RP, is applicable to RE-RPfair.

• Convexification: They demonstrate that the recourse function $R(x)$ (the worst-case cost for a fixed first-stage decision) can be convexified.
• Binary Uncertainty: By analyzing the structure of the dual problem, they show that the optimal uncertainty variables ($\zeta, \eta$) occur at the extreme points of the uncertainty set, effectively behaving as binary vectors.
• Linearization of Bilinear Terms: The interaction between dual variables and binary uncertainty variables is linearized using standard "big-M" constraints.
• Algorithm: The problem is solved iteratively. The master problem minimizes cost subject to Benders cuts (optimality cuts) generated from the sub-problem (recourse problem), which identifies the worst-case scenario and the fairness penalty for the current commitment plan.

3. Key Contributions

1. Novel Fairness Model: Introduction of RE-RPfair, the first robust UC model to explicitly incorporate fairness constraints regarding PV curtailment allocation using a mathematically tractable proxy for the Gini Index.
2. Theoretical Proof: A rigorous proof that the Benders Decomposition algorithm remains valid and convergent for this expanded model, despite the added complexity of the fairness penalty and bilinear terms.
3. Linearization Technique: A specific method to linearize the sum of absolute deviations within a robust optimization framework, enabling the use of standard MILP solvers.

4. Experimental Results

The authors conducted numerical experiments using an artificial model of an isolated island power system (based on Miyako Island data) with 3 thermal generators, 3 PV units, and a 24-hour horizon.

• Setup: They compared RE-RP (baseline, no fairness) against RE-RPfair (with fairness penalty) under three PV output patterns: Low (LP), Medium (MP), and High (HP).
• Parameter Tuning: The penalty weight $\chi$ was tuned. Results showed that increasing $\chi$ reduced the Gini Index (improving fairness) with a negligible impact on total operational cost. A value of $\chi = 100.0$ was selected for main experiments.
• Performance Metrics:
  • Gini Index Reduction: RE-RPfair consistently achieved lower Gini Indices than RE-RP across all scenarios.
    • LP Case: 0.27 (RE-RP) $\to$ 0.25 (RE-RPfair)
    • MP Case: 0.30 $\to$ 0.20
    • HP Case: 0.31 $\to$ 0.26
  • Statistical Significance: Using Shapiro-Wilk (normality), F-test (variance equality), and Student's t-test, the authors confirmed that the reduction in the Gini Index for RE-RPfair is statistically significant ($p < 0.001$) compared to the baseline.
• Cost Impact: The increase in total system cost due to the fairness constraint was found to be minimal, suggesting that fairness can be achieved without significant economic penalty.

5. Significance

• Market Stability: By ensuring equitable curtailment, the model helps prevent PV owners from exiting the market due to perceived unfairness, which is crucial for the long-term stability of deregulated energy markets.
• Operational Feasibility: The paper demonstrates that incorporating complex social objectives (fairness) into technical grid operations (UC) is computationally feasible using existing robust optimization frameworks.
• Scalability: The successful application of Benders Decomposition suggests the approach can be scaled to larger, more realistic power systems, though the authors note future work is needed to test with real-world data due to privacy constraints.

In conclusion, RE-RPfair provides a robust, mathematically sound, and computationally efficient framework for power system operators to manage renewable energy uncertainties while maintaining social equity among distributed energy resource owners.