Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Applied and Computational Mathematics

Team Research Publications Teaching

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Multilevel Monte-Carlo for Uncertainty Quantification in Pricing Green PPAs

Masterthesis Econo-Mathematics

Supervision

• Prof. Dr. Matthias Ehrhardt

Cooperation

• Dr. Daniel Oeltz, Fraunhofer-Institut für Algorithmen und Wissenschaftliches Rechnen SCAI
• Angelina Steffens, Fraunhofer-Institut für Algorithmen und Wissenschaftliches Rechnen SCAI

Description

Renewable Power Purchase Agreements (PPAs) are contracts between a renewable energy developer and a buyer, typically a corporation or a utility, that stipulate the terms of sale for renewable energy generated by the developer. The goal of these agreements is to facilitate the development of new renewable energy projects, as they provide a stable and predictable revenue stream for the developer and help the buyer meet their sustainability goals.
Under a Renewable PPA, the developer agrees to sell the buyer a specified quantity of renewable energy at an agreed-upon price over a set period of time. The contract will typically include details such as the location of the project, the type of renewable energy being produced (e.g., wind, solar, geothermal), the capacity of the project, the duration of the contract, and the price per unit of energy sold.
PPAs can take different forms, but generally fall into two categories: off-site and on-site. Off-site PPAs involve the buyer purchasing renewable energy from a project located elsewhere and typically require the use of the transmission grid to deliver the energy to the buyer. On-site PPAs, on the other hand, involve the buyer purchasing renewable energy directly from a project located on their property, such as a solar installation on the roof of a building.
Renewable PPAs are beneficial for both the developer and the buyer. evelopers are able to secure long-term revenue for their projects, which provides certainty and stability in financing and enables them to scale up their operations. Buyers are able to meet their sustainability goals, often at a lower cost than traditional fossil fuel-based energy, while also demonstrating their commitment to corporate social responsibility.
Overall, Renewable PPAs are an important tool in the transition to a more sustainable energy system and have played a significant role in the growth of renewable energy in recent years.

Assessing the value and risk of Renewable PPAs is an important step for both developers and buyers before entering into a contract. Here are some factors that should be considered:
Value:

• Price: The price of the energy sold under the PPA is a key factor in determining the value of the contract. Buyers will want to ensure that the price is competitive with other energy sources, while developers will want to ensure that the price is sufficient to cover their costs and generate a profit.
• Contract duration: The length of the contract can impact the value of the PPA. Longer contracts provide more stability and predictability for both parties, but shorter contracts may provide more flexibility and the opportunity to renegotiate terms.
• Volume: The volume of energy sold under the PPA is also important, as it can impact the revenue generated for the developer and the energy savings for the buyer.

• Creditworthiness: The creditworthiness of the buyer is an important factor in assessing risk, as it impacts the likelihood of the buyer defaulting on the contract. Developers should consider conducting a credit analysis of the buyer to assess their financial stability and ability to meet their contractual obligations.
• Regulatory risk: Regulatory changes, such as changes in tax incentives or renewable energy mandates, can impact the value of the PPA. Developers should consider the potential for regulatory changes when assessing the risk of the contract.
• Project risk: The risk associated with the development of the renewable energy project, such as construction delays or equipment failure, can impact the value of the PPA. Buyers should consider conducting due diligence on the project to assess the risk associated with the development process.

Uncertainty quantification and multi-level Monte Carlo (MLMC) are computational techniques that can be used to assess the value and risk of Renewable PPAs by accounting for various sources of uncertainty.
Uncertainty quantification involves identifying sources of uncertainty in the PPA and quantifying their impact on the value and risk of the contract. For example, uncertainties in renewable energy resource availability, electricity demand, and energy prices can all impact the revenue generated by the project and the value of the PPA. By quantifying the impact of these uncertainties, developers and buyers can better understand the risk associated with the contract and adjust the terms of the PPA to mitigate that risk.
MLMC is a computational method that can be used to estimate complex mathematical functions, such as those used to model energy prices or renewable energy production, with high accuracy and efficiency. MLMC works by using a hierarchy of models with varying levels of resolution and complexity, and then estimating the function at each level using Monte Carlo simulation. This approach can significantly reduce the computational resources required to accurately estimate the value and risk of the PPA, making it more feasible to perform large-scale risk assessments.
Together, uncertainty quantification and MLMC can be used to provide more accurate and comprehensive assessments of the value and risk of Renewable PPAs. By accounting for uncertainty and using efficient computational methods, developers and buyers can make more informed decisions about whether to enter into a PPA, how to structure the contract, and how to manage risk over the life of the contract.

Determining the uncertainty in the local volatility using MLMC involves estimating the expected value of the local volatility function at different levels of resolution, and then using this information to quantify the uncertainty in the function. Here are the general steps involved in this process:

• Define the local volatility function: The local volatility function is a key input to options pricing models and is often subject to uncertainty. It is a function of both time and the underlying asset price, and can be estimated using historical data or other methods.
• Define the hierarchy of models: MLMC involves using a hierarchy of models with varying levels of resolution and complexity. In the case of local volatility, this might involve using different discretizations of the time and asset price domains, with finer discretizations at higher levels of resolution.
• Simulate the local volatility function: At each level of resolution, use Monte Carlo simulation to estimate the expected value of the local volatility function. This involves simulating the underlying asset price and using the local volatility function to estimate the volatility at each point in time and price.
• Calculate the variance: Use the estimates of the expected value of the local volatility function to calculate the variance at each level of resolution. This can be used to estimate the uncertainty in the function, with higher levels of resolution providing more accurate estimates of the variance.
• Determine the optimal sampling allocation: MLMC involves allocating computational resources optimally between different levels of resolution, to minimize the overall computational cost while achieving a desired level of accuracy. This involves determining the relative variances at each level of resolution, and then allocating computational resources accordingly.

Problem

Keywords

Green PPA, Multi-Level Monte-Carlo Method, Uncertainty Quantification

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