# Pricing in non-convex markets with quadratic deliverability costs

## TL;DR

This paper extends market-clearing price methods to include quadratic costs, such as ramping costs in renewable energy, providing a more accurate pricing framework for non-convex electricity markets.

## Contribution

It introduces a novel approach to incorporate quadratic costs into market-clearing prices in non-convex markets, especially electricity markets with renewable ramping costs.

## Key findings

- Derived new pricing formulas accounting for quadratic costs.
- Analyzed the impact of ramping costs on market prices.
- Validated results with a classical market example.

## Abstract

The problem of obtaining market-clearing prices for markets with non-convexities has been widely studied in the literature. This is particularly the case in electricity markets, where worldwide deregulation leads to markets in which non-convexities arise from the decisions of market operators regarding which generators are committed to provide electricity power. Here, we extend seminal results in this area to address the problem of obtaining market-clearing prices for markets in which beyond non-convexities, it is relevant to account for convex quadratic market costs. In a general market, such costs arise from quadratic commodity costs or transactions costs. In an electricity market, such quadratic costs arise when ramping costs need to be considered due to the presence of renewable energy sources, which continue to increase their participation in electricity markets. To illustrate our results, we compute and analyze the clearing prices of a classical market problem with the addition of ramping costs.