# Decision Support Models for Operations and Maintenance for Offshore Wind Farms: A Review
## 4 Modeling Techniques
### Types of Models
#### 4.2.3 Differential Equations  
Ordinary differential equations (ODEs) are equations, where the relationship \(F(t,x,x^{\prime},\ldots)\) between a variable t, an unknown function \(f(\cdot)\) of this variable \(x=f(t)\) and the derivatives of this unknown function \((x^{\prime},x^{\prime\prime},\ldots)\) is known. Often, this independent variable \(t\) is time. This means that all variables in the function \(F(\cdot)\) are dependent on time. When more than one unknown functions of the same variable exist, a system of ODEs can be used. Depending on the nature of the relationship \(F(\cdot)\), different mathematical approaches to finding a solution exist, providing exact solutions for integrable functions or numerical results using e.g., the Euler or Runge-Kutta method. Software that can be used for solving ODEs includes MATLAB and Maple.  
In contrast to the ODEs, partial differential equations (PDEs) can have two or more independent variables. In addition to the time, this can be e.g., distance from a certain point. Examples for time-independent variables in wind farm maintenance modeling are the quality of repair or quality of a weather forecast. PDEs could be used for maintenance modeling; however, solving PDEs is not as straight forward as solving ODEs and the benefits most likely do not outweigh the complications in solving.  
Stochastic differential equations (SDEs) are differential equations, where one or more terms are stochastic processes. The SDEs can be written in different forms. One of the most commonly used forms in physics are Langevin equations. These consist of ODEs using a deterministic function and an additional random term. The random term can take different forms, most commonly (Gaussian) white noise. Other forms of presenting SDEs include the Smoluchovski and Fokker-Planck equations, based on PDEs and the Ito equation, similar to the Langevin equation in differential notation. SDEs can be numerically solved by using e.g., the Euler-Maruyama method, the generalized Runge-Kutta method or Monte Carlo simulation. In the given problem setup of maintenance scheduling, SDEs can be a good way to incorporate the uncertainties in the variables. Existing models often use the mean for different input, such as the mean annual failure rate, mean times to repair or mean wind speed during downtime. With SDEs, these means can be used for the deterministic function and uncertainties in the observations can additionally be modeled by the random term. One of the drawbacks is the solving of the equations. Numerical methods to solve the equations are limited and often suffer from poor numerical convergence. Therefore, the main challenge when using SDEs for modeling the scheduling of maintenance will be to investigate the existence of solutions to the equations and the (numerical) calculation of these. Since the SDEs will depend largely on the input parameters, they cannot easily be used for different wind farm layouts and site, without changing the equation structure. This is another drawback, since a maintenance model should be able to capture the properties of different wind farms without the hassle of fitting a new set of equations to each case.