The Modeling and Control of (Renewable) Energy Systems by Partial Differential Equations—An Overview

Abstract

Mathematical models of energy systems have been mostly represented by either linear or nonlinear ordinary differential equations. This is consistent with lumped-parameter dynamic system modeling, where dynamics of system state variables can be fully described only in the time domain. However, when dynamic processes of energy systems display both temporal and spatial evolutions (as is the case of distributed-parameter systems), the use of partial differential equations is necessary. Distributed-parameter systems, being described by partial differential equations, are mathematically (and computationally) much more difficult for modeling, analysis, simulation, and control. Despite these difficulties in recent years, quite a significant number of papers that use partial differential equations to model and control energy processes and systems have appeared in journal and conference publications and in some books. As a matter of fact, distributed-parameter systems are a modern trend in the areas of control systems engineering and some energy systems. In this overview, we will limit our attention mostly to renewable energy systems, particularly to partial differential equation modeling, simulation, analysis, and control papers published on fuel cells, wind turbines, solar energy, batteries, and wave energy. In addition, we will indicate the state of some papers published on tidal energy systems that can be modelled, analyzed, simulated, and controlled using either lumped or distributed-parameter models. This paper will first of all provide a review of several important research topics and results obtained for several classes of renewable energy systems using partial differential equations. Due to a substantial number of papers published on these topics in the past decade, the time has come for an overview paper that will help researchers in these areas to develop a systematic approach to modeling, analysis, simulation, and control of energy processes and systems whose time–space evolutions are described by partial differential equations. The presented overview was written after the authors surveyed more than five hundred publications available in well-known databases such as IEEE, ASME, Wiley, Google, Scopus, and Web of Science. To the authors’ best knowledge, no such overview on PDEs for energy systems is available in the scientific and engineering literature. Throughout the paper, the authors emphasize novelties, originalities, and new ideas, and identify open problems for future research. To achieve this goal, the authors reviewed more than five hundred journal articles and conference papers.

1. Introduction

This overview paper presents several categories of renewable energy systems, covering topics on modeling, analysis, simulation, and control based on the usage of distributed-parameter mathematical models obtained using partial differential equations (PDEs). PDEs model the dynamics of distributed-parameter systems whose state variables change in both time and space domains (temporal and spatial evolutions), and in general, provide more complete descriptions of system dynamics than modeling based on ordinary differential equations (ODEs).

The overview paper is mostly limited to renewable energy systems described by PDEs, which are the recent trends in engineering and sciences. In independent sections, the paper will cover fuel cells, wind turbines, solar cells, batteries, wave energy processes and systems, and indicate some other classes of renewable energy systems like tidal energy that can be efficiently modelled, simulated, analyzed, and controlled using PDEs. Renewable geothermal and biomass and biogas energy sources will not be covered in this overview since hardly any papers exist that use PDE for their modeling or simulation or analysis or control.

This overview paper does not include the coverage of processes and systems that can be described by fractional differential equations. Fractional calculus modeling appears to be a recent trend in mathematics and engineering that potentially has many applications. The overview also does not go into the mathematical theory of partial differential equations or present the most efficient approaches and methods for their solutions, since PDEs are a huge and well-established field with many books and overview papers already published, even for particular PDEs.

Due to a large number of papers published on these topics and the space limitation of one journal paper, this work is by no means an exhaustive overview. It represents the authors’ snapshot of these areas based on their long-term involvement in theoretical research on the modeling and control of distributed-parameter systems and theoretical and practical research experience in energy systems, particularly fuel cells, solar cells, wind turbines.

Throughout the paper, we will present our views, emphasize new ideas, indicate open research problems, and generally discuss the novelties of particular papers on the modeling, analysis, simulation, and control of energy systems and processes using PDE-based mathematical models.

The presentation is self-contained so that researchers on these topics can find full descriptions of corresponding mathematical models in this paper with pin-pointed references for further studies. The authors surveyed more than five hundred publications and selected about two hundred of them to be included in this overview paper. The selection was based on the authors’ experience in performing applied research with practical applications on renewable energy systems such as fuel cells, solar cells, and wind energy, and theoretical research on the modeling and control of different classes of distributed-parameter systems.

The authors expect that this paper will motivate researchers of energy systems to rely more on, and utilize the full power of, PDE modeling and eventually write their own overviews on the use of PDEs for each particular class of renewable energy systems considered in this paper.

If something can be indicated as a general weakness of the presented results from the surveyed papers, it certainly can be the lack of newly developed mathematical techniques for analytical and numerical solutions of the derived PDE mathematical models for energy systems. Only in some papers did the authors present the corresponding mathematical techniques used, which was noted in this survey paper in places where the corresponding papers were reviewed. At the same time, the presented overview paper motivates researchers in engineering and applied mathematics to work on many open mathematical problems that new partial differential equations derived in the process of modeling energy processes and systems have posed.

2. The Modeling and Control of Fuel Cells Using PDEs

Fuel cells are triodes composed of an anode, membrane, and cathode. Their inputs are the gasses hydrogen (provided either by a hydrogen gas reformer from hydrogen-rich fuels or supplied by a tank) and oxygen (supplied by air that is mostly oxygen mixed with nitrogen) and the outputs are electricity (or heat) and water. There are numerous lumped-parameter mathematical models of various dynamic processes of several types of fuel cells, especially for the two most important classes of fuel cells: proton exchange membranes, also known as polymer electrolyte membrane fuel cells (PEM fuel cells, the most developed and the best understood type of fuel cells; used predominantly for vehicular and residential applications) and solid oxide fuel cells (SOFCs, which produce large amounts of electricity and huge amounts of heat; used for residential and industrial applications). See, for example, papers [1,2,3,4,5,6,7,8,9,10,11], very comprehensive books [9,12,13], and the references therein. In the follow-up of this section, we will review the distributed-parameter models of PEMFCs and SOFCs, which provide a more accurate description of dynamic processes in these fuel cells than lumped-parameter models.

2.1. The Modeling and Control of PEMFCs Using PDEs

In a very comprehensive paper [14], the distributed-parameter modeling equations of PEMFCs (PEM fuel cells) developed before 2008 and based on the physical laws of heat and mass transfer were summarized. Papers [14,15] also showed originality in describing mathematically general dynamic processes in PEMFCs using PDEs. The fundamental PDEs were presented and derived in [14,15] for the main physical and chemical processes in PEMFCs, primarily the conservation of mass, conservation of momentum, conservation of energy, and species transport. The obtained mathematical model was used for the simulation of a PEM and produced results that agreed very well with the physical reality, even though the simulation time was considerably longer than the simulation time of the corresponding lumped-parameter model. The simulations were successfully demonstrated using MATLAB.

The conservation of mass equation was defined in [14,15] and is given by the following PDE

$$\frac{\partial }{\partial t}\left(\epsilon \rho \right)+\nabla \left(\epsilon \rho \overrightarrow{v}\right)=S_m,$$

(1)

where $\epsilon$ is the membrane porosity, $\rho$ represents density, $\overrightarrow{v}$ is the velocity, and $S_m$ is the mass source term.

The conservation of momentum PDE is defined as

$$\frac{\partial }{\partial t}\left(\epsilon \rho \overrightarrow{v}\right)+\nabla \left(\epsilon \rho \overrightarrow{v}\overrightarrow{v}\right)=-\nabla p+\nabla \cdot \left(\epsilon \mu \nabla \overrightarrow{v}\right)+S_u,$$

(2)

where $p$ is the pressure, $\mu$ is the dynamic velocity, and $S_m$ is the momentum source.

The energy PDE for the considered PEM fuel cell is given by

$$\frac{\partial }{\partial t}\left(\epsilon \rho c_{p}T\right)+\nabla \left(\epsilon \rho c_p\overrightarrow{v}T\right)=\nabla \cdot \left(k^{eff}\nabla T\right)+S_Q,$$

(3)

with the newly introduced variables representing the following quantities: $c_p$—specific heat at constant pressure, $T$—temperature, and $S_Q$—energy source term.

The species transport PDE is defined as

$$\frac{\partial }{\partial t}\left(\epsilon c_k\right)+\nabla \left(\epsilon \overrightarrow{v}{c}_k\right)=\nabla \cdot \left(D_k^{eff}\nabla c_k\right)+S_k,$$

(4)

where $c_k$ represents the species concentration, $D_k^{eff}$ stands for the species effective thermal conductivity, and $S_k$ is the species source term. Simulation results were presented in [15] for the PEM fuel cell-generated voltage as a function of the current density step changes.

Paper [16] developed a PEMFC membrane PDE model that considered proton conduction, water flow, heat generation and transport, and hydration-dependent conductivity. The finite element method was employed in a simulation of the obtained PDE. The model was used to study imperfections and inhomogeneities in the fuel cell membrane in three dimensions. The simulation tools for a PEMFC model, obtained using distributed-parameter modeling, were presented in [17] for the purpose of studying time and space evolutions in gas channels.

In [18], the proper generalized decomposition method was used for the ionic transport modeling of a PEMFC by PDEs. The paper theoretically developed the complete methodology for the one-dimensional model, including its simulation. A computational fluid dynamic (CFD) model of a PEMFC was considered in [19]; see also [20] for the corresponding simulation results, where results for 2D (two-dimensional) and 3D (three-dimensional) models were compared. A discussion regarding the development of a multi-physical two-dimensional PEMFC model, which is used for real-time fuel cell control, can be found in [21]. Paper [22] performed fluidic domain modeling for gas supply channels, and derived the following PDE for the fluid mean velocity $v(x,t)$ and the fluid pressure gradient $\partial p(x,t)/\partial x$

$$\begin{array}{l}\frac{\partial v(x,t)}{\partial t}+\frac{1}{\rho }\frac{\partial p(x,t)}{\partial x}+\frac{f}{2d}v(x,t)\left|v(x,t)\right|,\\ v(x,t)=\frac{q(x,t)}{\rho A},\hspace{1em}\hspace{1em}f=\frac{64\mu }{\rho dv(x,t)},\end{array}$$

(5)

where $v(x,t)$ is the fluid mean velocity in [m/s], $q(x,t)$ is gas mass flow in [kg/s], $\rho$ is the density, $A$ is the cross-sectional area in [${\mathrm{m}}^2$], $d$ is the hydraulic diameter of the channel in [m], and $\mu$ is the fluid viscosity in [$\mathrm{kg}/\mathrm{s}\cdot \mathrm{m}$].

In addition to the analytical (mathematical) model development, the modeling process can be performed experimentally using the model identification technique known as the input–output diffusive approach, as was performed in [23] for a PEMFC. The measurements of the fuel cell current (input) and measurements of the fuel cell voltage (output) were utilized in [23] to develop a diffusion (PDE) model that was well approximated by a finite-dimensional ordinary differential equation model. The results obtained agree very well with the experimental data. A finite element discretization methodology was utilized in [24] to design a nonlinear observer for a PEMFC model described by a PDE. The state estimates generated by the observer were used in [24] for an efficient continuous-time sliding mode controller design whose purpose was to reduce the observer estimation error to zero.

Despite recent advances, several open research problems remain, first of all concerning the use of control engineering techniques to improve and optimize processes in PEMFCs described by PDEs. Another set of open research problems concerns extending modeling via PDEs to the modeling of PEMFCs done using ODEs and to most PEMFC ODE models that have been successfully used in practice, for example, for vehicular applications, like the ODE models of [1,4,7,9].

2.2. The Modeling of SOFCs Using PDEs

Fundamental PDE models for the corresponding dynamic processes in SOFCs [2], representing hydrogen conservation, water conservation, oxygen conservation, nitrogen conservation, and momentum conservation equations for the anode and cathode channels, were derived in [25,26]. A summary of the obtained partial differential equations can also be found in [2]. Paper [2] discussed both the lumped- and distributed-parameter modeling of SOFCs with the goal of developing a computationally efficient lumped-parameter model used for the control of SOFCs.

In more recent papers [27,28], PDEs were used for the modeling of processes in syngas-supplied tubular SOFCs and reversible tubular SOFCs. It appears that the models obtained are convenient for simulation as well as for control purposes. The models were verified experimentally in both papers [27,28]. The simulation of a PDE control-oriented SOFC model, under realistic operating conditions, was presented in [29]. In general, the modeling and control problems of SOFCs appear to be more complex than the corresponding modeling and control problems of PEMFCs.

In [30,31], PDE mathematical models for fundamental processes in SOFCs were presented. The energy equation in [30] was modelled by

$$\rho \frac{\partial E}{\partial t}+\rho u\nabla E=-\nabla Q+S_q,$$

(6)

where $E$ is the energy, $u$ represents the gas velocity, $Q$ is the conduction heat flux vector, $S_q$ is the volumetric heat source, t represents time, and $\rho$ is the ratio of the amount of substance (n) to the substance (gas) volume (V) obtained from the ideal gas equation

$$p=\rho RT,\hspace{1em}\rho =\frac{n}{V},$$

(7)

where $p$ is the gas pressure, $T$ is the gas temperature, and $R$ is the universal gas constant. The gas velocity satisfies the momentum conservation equation given by

$$\rho \frac{\partial u}{\partial t}+\rho u\nabla u=-\nabla p+\mu {\nabla }^{2}u+\rho f.$$

(8)

In this PDE, $\mu$ is the dynamic viscosity, and $f$ represents the generic body forces. The temperature distribution within the gas can be obtained from the energy equation and Fourier’s law ($Q=-\lambda \nabla T$; $\lambda$ is the thermal conductivity coefficient), which leads to the following PDE:

$$\frac{\partial (\rho cT)}{\partial t}+\rho u\nabla (\rho cT)=\nabla (\lambda \nabla T)+S_q,$$

(9)

where $c$ represents the heat capacity.

Since a fuel cell is composed of bipolar plates, for a plate of thickness $h_p$ and thermal conductivity ${\lambda }_p$ (assuming constant isotropic conditions), paper [31] derived the following PDE for the temperature distribution

$$\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}=\frac{1}{h_p{\lambda }_p}(q_{\alpha }-q_{\beta }),$$

(10)

where $q_{\alpha }$ and $q_{\beta }$ are the plate incoming and leaving fluxes. The bipolar plate voltage $V$ was modelled in [31] by the three-dimensional Laplace partial differential equation defined as

$$\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}=0.$$

(11)

Modeling, analysis, simulation, and control of SOFCs are modern research trends within the fuel cell research community, first of all due to their large electric power generation and huge amount of produced heat, and the fact that many open research problems remain as far as SOFCs are concerned.

It is important to point out that in all surveyed papers on the modeling of fuel cells via partial differential equations, no reference to any mathematics or applied mathematics publication (conference papers, journal papers, or books) can be found, which brings us to the conclusion that either only rather basic methods were used for the analytical and numerical considerations, or only rather standard computer software packages were run while conducting the simulations of the developed PDE models. This fact opens a door for finding the most suitable analytical approaches and corresponding numerical methods to study the considered mathematical models of solid oxide fuel cells.

3. The Modeling and Control of Wind Energy Systems Using PDEs

Connecting a wind turbine or wind farms to an electric power grid is a challenging engineering task due to the fact that the produced wind electric energy is rather volatile, which is caused by the stochastic nature of wind speed and wind intensity [32,33], since they change in time like stochastic processes.

For a wind farm composed of n generators lined up along an x-axis, paper [33] derived a damped hyperbolic wave PDE for the aggregated generator angle $\delta (x,t)$

$$\frac{{\partial }^2\delta (x,t)}{\partial t^2}+\xi \frac{\partial \delta (x,t)}{\partial t}-v^2\frac{{\partial }^2\delta (x,t)}{\partial x^2}=W(x,t),$$

(12)

where $W(x,t)$ represents the net wind power injection, $\xi$ is the average damping density, and $v$ is the wave speed. The system power flow formula is given by

$$P(x,t)=-\frac{1}{\gamma }\frac{\partial \delta (x,t)}{\partial x},$$

(13)

where $\gamma$ is the average reactance density. For the PDE model presented in (12) and (13), paper [34] developed an adaptive control technique to suppress inter-area oscillations in a grid-connected wind farm-integrated power system.

In [35], the wind speed was modelled as a stochastic process and represented by the well-known Fokker–Planck–Kolmogorov (FPK) PDE. The Crank–Nicholson difference scheme was used in [35] to calculate the probability density of the wind speed increments from the derived stochastic partial differential equation that describes variations in wind speed in time series. Namely, the stochastic differential equation for the wind speed $X_t$, assuming the probability density function $f(x)$ with a finite variance and the expected value $\mu =E\left(x\right)$, is given by

$$\frac{d{X}_t}{dt}=-\theta \left(X_t-\mu \right)dt+\sqrt{v\left(X_t\right)}d{W}_t,$$

(14)

where $\theta \ge 0$, $W_t$ is Brownian motion, and

$$\nu (x)=\frac{2\theta }{f(x)}{\displaystyle \underset{}{\overset{x}{\int }}(\mu -\tau )f(\tau )d\tau }.$$

(15)

The FPK PDE for the probability of increment $h$ at $t$, denoted by $I(t,h)$, was obtained in [35] as

$$\frac{\partial I(t,h)}{\partial t}=\frac{1}{2}\frac{{\partial }^2\left(v(h)I(t,h\right)}{\partial h^2}+\frac{\partial \left(\theta (h-\mu )I(t,h)\right)}{\partial h}.$$

(16)

Paper [36] derived a PDE model for the power generation management of a wind farm and used it to design a hierarchical feedback controller with the turbine pitch and yaw angles playing the roles of feedback variables. The concentration of the wind turbines, denoted by $Q(\beta ,\gamma ,t)$ that have pitch angles in the range $\left[\beta -\partial \beta ,\beta +\partial \beta \right]$ and yaw angles in the range $\left[\gamma -\partial \gamma ,\gamma +\partial \gamma \right]$, was modelled in [36] by the following PDE

$$\frac{\partial Q(\beta ,\gamma ,t)}{\partial t}=-\frac{\partial Q(\beta ,\gamma ,t)}{\partial \beta }u(t)-\frac{\partial Q(\beta ,\gamma ,t)}{\partial \gamma }v(t),$$

(17)

where $u(t)$ and $v(t)$ are control input signals that satisfy

$$\frac{d\beta }{dt}=u(t),\hspace{1em}\hspace{1em}\hspace{1em}\frac{d\gamma }{dt}=v(t).$$

(18)

Since the energy generated by a single wind turbine is given by

$$P=0.5\rho A{C}_p(\lambda ,\beta )v_{eff}^3,$$

(19)

where $\rho$ is the density of the air, $A$ is the area of the turbine blades, $v_{eff}$ is the effective wind speed at the turbine, and $C_p$ is the power coefficient, the total power generated by the wind farm is given by the following double integral

$$P_{total}={\displaystyle \underset{{\gamma }_{\min }}{\overset{{\gamma }_{\max }}{\int }}d}\gamma {\displaystyle \underset{{\beta }_{\min }}{\overset{{\beta }_{\max }}{\int }}0.5\rho A{C}_p(\lambda ,\beta )v_{eff}^{3}Q\left(\beta ,\gamma ,t\right)}d\beta$$

(20)

The controller design based on the PDE model in [36] included dynamic inversion, predictive control, and linear matrix inequality controller design techniques. The paper used the so-called DTA (design-then-approximate) approach, in which a controller that guarantees the tracking of a desired power trajectory is first designed, and then the overall system is approximated via discretization. The simulation results, obtained using the standard Lax–Friedrichs finite volume method, demonstrated the effectiveness of the designed controller.

Paper [37] estimated wind velocity, wind direction, and absolute pressure, all obtained using the Navier–Stokes equation that models viscous incompressible flows. The obtained infinite dimensional model was approximated in [37] by a finite dimensional model consisting of a finite number of differential and algebraic equations. The Reynolds-averaged Navier–Stokes equations were also used in [38] for the purpose of studying the aerodynamic performance of five rotors of the Archimedes spiral wind turbine.

In a recent paper [39], the forecasting of wind-produced power as a function of wind speed and chaotic atmospheric processes (modeled in general by stochastic PDEs) was considered via the polynomial decomposition of the general (stochastic) differential equation, using the methodology developed in [40]. To that end, a differential polynomial neural network was designed.

It is important to emphasize that there are many conference and journal publications on the control of processes in wind turbines, including wind farms, using concentrated (lumped) parameter systems; see for example [41,42] and the references therein. However, a very small number of papers use PDE-based control techniques to optimize dynamic processes in wind turbines and wind farms, and hence obtain more accurate and more efficient results. Even more, developing the corresponding PDE mathematical models of the well-established ODE mathematical models, like those in [41,42], will be an interesting area for future research.

Wind turbines in cold-climate regions are more efficient than those in warm-climate regions due to higher wind speeds and more dense air. However, cold-climate regions have more ice, so ice accumulation on wind turbine blades may impair turbine efficiency. Paper [43] proposes designing plasma actuators to combat the icing problem, which improve wind turbine aerodynamic performance and perform deicing. To that end, the paper presents a design based on the solution of Poisson’s PDE with the Neumann boundary conditions imposed on the inlet and outlet sides, and the Dirichlet boundary conditions imposed on the other sides. Experimental verification was also presented in [43].

4. The Modeling and Control of Solar Cells Using PDEs

Semiconductors are a critical component of modern technologies, including solar cells. Semiconductors are solid materials with higher conductivity than insulators but far less conductivity than conductors at room temperature [44]. This implies that electrons can jump a small energy gap from the valence and conduction bands given small amounts of energy input. The holes left in the valence band create a positive charge. Silicon, which has an energy gap of 1.1 eV, is an abundant semiconductor material on Earth and is commonly used as a semiconductor, including in Silicon Photovoltaic (SiPV) cells. Silicon can be doped with small impurities to either increase the number of electrons (negatively charged, n-type) or holes (positively charged, p-type). When an n-type semiconductor and a p-type semiconductor are in contact, the extra electrons from the n-type will move into the holes of the p-type in a region surrounding the junction, and the impurities with extra atoms will have a negative charge and those with a missing atom will have a positive charge. In this region surrounding the junction, called the depletion region, an electric field is created where the n-side is positive and the p-side is negative, which prevents more holes or electrons from crossing the depletion region. If photons with energy higher than the band gap penetrate the depletion region, valence electrons move into the conduction band, resulting in electron–hole pairs, which are driven out of the depletion region by the electric field. This creates an excess of electrons in the n-side and holes in the p-side, and if a load is connected across these regions, excess electrons in the n-side will flow through the load to the p-side and recombine with the holes, generating a continuous direct current [44,45].

The Schrodinger equation models the state of quantum particles and can be used to model the transport of electrons in a semiconductor [46]. It is given by

$$\begin{array}{r}\hfill i\hslash {\partial }_t\psi =-\frac{{\hslash }^2}{2m}\mathsf{\Delta }\psi -qV\left(x,t\right)\psi ,\hspace{0.17em}t>0,x\in {ℝ}^3,\end{array}$$

(21)

where $\psi \left(x,t\right)$ is a wave function, $V\left(x,t\right)$ is electric potential, $\hslash$ is the Planck constant, $i=\sqrt{-1}$, and $m$ is the particle mass. Various types of controllability for variations of the Schrodinger equation have been addressed in several papers, for example in [47,48,49,50,51], and controllers have been developed for them in different contexts in [52,53,54,55]. Some of these papers, e.g., [50,51,54], also studied the observability issue of the Schrodinger equation. Paper [52] considered the estimation and the output regulation problems of the linearized Schrodinger equation. In general, the Schrodinger equation has been studied extensively from the control system point of view, especially the controllability property of systems whose dynamics are described by this equation.

Due to the large number of particles present in semiconductors, the probability density functions are used to model the behavior of particles rather than analyzing the position of individual particles. There is a mathematical hierarchy of partial differential equations that are used to model particle flow, divided into microscopic and macroscopic equations. Microscopic equations are the Liouville equation, derived using Newton’s laws; Vlasov equations, which consider Coulomb forces but not collision interactions between particles; and Boltzmann equations. Collisions between particles are considered in the Boltzmann–Poisson system.

The Liouville equation is given below [44]

$$\begin{array}{ll}\hfill & {\partial }_{t}f+\frac{1}{\hslash }{\nabla }_{k}E\cdot {\nabla }_{x}f+\frac{q}{\hslash }{\nabla }_{x}V\cdot {\nabla }_{k}f=0,\hfill \\ \hfill & x\in {ℝ}^{3M},k\in B^M,t>0,\hfill \end{array}$$

(22)

where $f\left(x,k,t\right)$ is the probability density function of an ensemble of particles, $x\left(t\right)$ is the position, $k\left(t\right)$ is the pseudo-wave vector, and $t$ is time. $-qE\left(x,t\right)$ is the electric field, $B$ is the Brillouin zone, and $M$ is the number of particles in the ensemble.

The Vlasov equation with consideration of Coulomb forces is as follows [44]

$$\begin{array}{ll}\hfill & {\partial }_t{f}^{*}+v\left(k\right)\cdot {\nabla }_x{f}^{*}+\frac{q}{\hslash }{\nabla }_{x}V\cdot {\nabla }_k{f}^{*}=0,\hspace{0.17em}x\in {ℝ}^3,k\in B,t>0,\hfill \\ \hfill & {ϵ}_s\Delta V=q\left(n-C\right),\hfill \end{array}$$

(23)

where $f^{*}\left(x,k,t\right)$ is the probability distribution of a single particle, $v\left(k\right)$ is the particle velocity, $V\left(x,t\right)$ is the potential, $n\left(x,t\right)$ is the electron density, $C\left(x\right)$ is the doping concentration, $q$ is the charge, and ${ϵ}_s$ is the permittivity of the semiconductor.

The Boltzmann–Poisson system was defined by [44]

$$\begin{array}{ll}\hfill & {\partial }_{t}f+v\left(k\right)\cdot {\nabla }_{x}f-\frac{q}{\hslash }{E}_{\mathrm{eff}}\cdot {\nabla }_{k}f=Q\left(f\right),\hspace{0.17em}x\in {ℝ}^3,k\in B,t>0,\hfill \\ \hfill & {ϵ}_s\Delta V=q\left(n-C\right),\hfill \end{array}$$

(24)

where $Q\left(f\right)$ is a collision operator.

If information only about particle or energy density is needed, the microscopic equations can be simplified, leading to the macroscopic equations, the simplest being the drift–diffusion equation for particle density combined with the Poisson equation for energy density, given below [46]

$$\begin{array}{ll}\hfill & J_n=q{\mu }_{n}nF+q{D}_n{\nabla }_{x}n,\hspace{0.17em}x\in {ℝ}^3,t>0,\hfill \\ \hfill & J_p=q{\mu }_{p}pF-q{D}_p{\nabla }_{x}p,\hfill \\ \hfill & {ϵ}_s\Delta V=q\left(n-C\right),\hfill \end{array}$$

(25)

where $J_n$ is the current density of electrons and $J_p$ is the current density of holes, $n$ and $p$ are concentrations of electrons and holes, ${\mu }_n$ and ${\mu }_p$ represent the electron and hole mobility, respectively, $D_n$ and $D_p$ are the respective diffusion constants, and $F$ is the electric field. Discussion and derivation of more complex macroscopic equations, such as the energy transport and hydrodynamic semiconductor equations, can be found in [44,56] and the citations within.

The combined drift–diffusion and Poisson equations are parabolic and hyperbolic, respectively, and are often used for semiconductor modeling in the context of solar cells [57,58,59]. They have also been used to model the potential-induced degradation of the shunting type caused by high voltages between solar cells that are connected in series, as demonstrated in article [60].

Organic Photovoltaic (OPV) cells are a recent focus in photovoltaic research. Although they were first reported in 1959 [61], their efficiency was extremely low until the development of bulk heterojunction (BHJ) solar cells [62]. Even though their efficiency is still very low compared to Silicon Photovoltaics (SiPV), their performance has advanced considerably in recent years (see [63,64,65] and the citations within). These advances in efficiency are promising, as OPV has major advantages over SiPV with regards to the ease of fabrication, installation costs, weight, and flexibility [66,67,68]. The drift–diffusion model has been variously studied to account for factors such as charge photogeneration, injection, recombination, and other effects [69], and these models have been valuable in modeling transport in organic solar cells [67,70,71,72,73]. The authors in [74] used machine learning techniques to optimize parameters in a drift–diffusion model developed in [75]. The authors in [76] performed a numerical finite element simulation using the drift–diffusion model of a real 3D heterojunction morphology. Other techniques for numerical simulations were performed in some other papers; see for example [77,78].

In [79], one-dimensional drift–diffusion equations were used to model the steady-state properties of bilayer and bulk heterojunction organic solar devices, with the electron, hole, and charge densities denoted by $n,p,X$, respectively, and with the electron and hole mobilities ${\mu }_n$ and ${\mu }_p$ representing their transport through the acceptor and donor respectively in the z direction. The following PDE was derived in [70] for the electron–hole dynamics in organic solar cells

$$\begin{array}{l}\frac{\partial n}{\partial t}=-\frac{\partial }{\partial z}\left({\mu }_n{K}_{B}T\frac{\partial n}{\partial z}+{\mu }_{n}neE\right)+k_{diss}(E)X-\gamma np,\\ \frac{\partial p}{\partial t}=-\frac{\partial }{\partial z}\left({\mu }_p{K}_{B}T\frac{\partial p}{\partial z}-{\mu }_{p}neE\right)+k_{diss}(E)X-\gamma np,\\ \frac{\partial E}{\partial z}=\frac{1}{{\epsilon }_0{\epsilon }_r}e(p-n),\\ \frac{\partial X}{\partial t}=G-k_{rec}X-k_{diss}(E)X-\gamma np,\end{array}$$

(26)

where $K_B$ is the Boltzmann constant, $T$ is the absolute temperature, $e$ is the elementary charge, $k_{diss}$ is the charge pair dissociation rate constant, $k_{rec}$ is the charge pair recombination rate constant, $E$ the electric field strength, ${\epsilon }_0$ is the permittivity of a vacuum, ${\epsilon }_r$ is the relative permittivity of the material (typically 4 for this type of OSC), $G$ is the generation rate constant, and $\gamma$ is the bimolecular recombination of electrons and holes to form charge pairs. The first two equations represent the coupled drift–diffusion equations for electrons and holes. The third equation represents the change in field inside the device, and the fourth one describes the dynamics of the charge pair.

It was shown in [80] how PDE Equation (26) can be approximated by a lumped-parameter nonlinear system, which can be linearized around its nominal operating points, so linear-quadratic optimal control theory techniques can be used for this type of solar cell to provide the desired constant numbers of electrons and holes. The reader is referred to [81] for the other lumped-parameter techniques used for the optimal control of dynamic processes in solar cells.

This section focused on SiPV and OPV cells due to their particular strengths and relevance in existing infrastructure and emerging research. Two other exciting types of photovoltaic cells with great potential are dye-sensitive solar cells (DSSCs) and Perovskite cells. In DSSCs, the photosensitization occurs in a dye; the working principles for DSSCs can be found in the comprehensive review paper [82], and further reviews discussing advancements and limitations can be found in [83,84,85,86]. Perovskite-based cells use compounds with defined structures, known as Perovskites, as photoanodes, and some recent reviews are available in [87,88].

These types of solar cells (OPV, DSSCs, and Perovskites) are still developing [89,90,91], and their modeling, analysis, simulation, and control are open challenges for future research.

5. The Modeling and Control of Batteries Using PDEs

Electric energy produced by batteries has numerous applications, including laptops, mobile phones, and electric vehicles, to name a few. The research and development of new battery technologies has been a modern trend for some time. Many journal and conference papers and several books have been published on these topics. Several mathematical models obtained using both lumped- and distributed-parameter models have been derived for various types of batteries. Here, we present only the mathematical models of batteries based on PDEs.

An exhaustive review paper from 2015 provided then-state-of-the-art research for electrical energy storage systems, including the latest developments and challenges in the field of rechargeable batteries and flow batteries [92]. The rechargeable battery market is dominated by lead–acid batteries, which have been experiencing continuous growth [93]. Lead–acid batteries are also the oldest type of rechargeable battery and have been continuously used since their invention in 1860 [94]. They are well known for their usage in starting, lighting, and ignition for gas-powered automobiles, but they have numerous other applications including electric vehicles, utility energy storage, grid support, uninterrupted power supplies, and renewable energy storage [92,94,95,96]. Lead–acid batteries use a lead oxide cathode and a lead anode with a sulfuric acid electrolyte, and rely on the following chemical reaction at the anode and cathode [92]

$$\begin{array}{c}\mathrm{Pb}+{\mathrm{SO}}_4^{2-} \Leftarrow \Rightarrow {\mathrm{PbSO}}_4+2{\mathrm{e}}^{-},\\ {\mathrm{PbO}}_2+{\mathrm{SO}}_4^{2-}+4{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}\Leftarrow \Rightarrow {\mathrm{PbSO}}_4+2{\mathrm{H}}_2\mathrm{O}.\end{array}$$

(27)

Ordinary and partial differential equation models for the charge and mass in lead–acid batteries have been developed and discussed in several papers, for example [94,97,98,99,100,101,102]. One such model is as follows,

$$\begin{array}{c}\frac{\partial }{\partial x}\left({\sigma }^{\mathrm{eff}}\frac{\partial {\varphi }_s}{\partial x}\right)=Aj,\\ \frac{\partial }{\partial x}\left(k^{\mathrm{eff}}\frac{\partial {\varphi }_l}{\partial x}\right)+\frac{\partial }{\partial x}\left(k_D^{\mathrm{eff}}\frac{\partial (\ln c)}{\partial x}\right)=-Aj,\\ \frac{\partial \left({ϵ}_{l}c\right)}{\partial t}-\frac{\partial }{\partial x}\left(D^{\mathrm{eff}}\frac{\partial c}{\partial x}\right)=w_2\frac{Aj}{2F}.\end{array}$$

(28)

where ${\varphi }_s\left(x\right)$ and ${\varphi }_l\left(x\right)$ are the potentials in the solid and liquid, respectively, and $c\left(x,t\right)$ is the acid concentration. ${\sigma }^{\mathrm{eff}}$, $k^{\mathrm{eff}}$, and $k_D^{\mathrm{eff}}$ are the material properties of the electrodes related to porosity and tortuosity. ${ϵ}_s$ and ${ϵ}_l$ are the solid and liquid volume fractions, ${ϵ}_s+{ϵ}_l=1$. A is the specific electroactive area and is related to the state of charge of the battery, $F$ is the Faraday constant, $w_2$ is a constant, and $j$ is the transfer current density related to ${\varphi }_l$ and ${\varphi }_s$ [94]. These partial differential equations were reduced to ordinary differential equations in [94].

The state of charge and state of health in lead–acid batteries can be estimated using many different methods, including model-based methods; however, at this time, conventional techniques do not rely on partial differential equation models [103,104,105]. Hopefully, such models will emerge in the near future so that the state of health of a battery can be precisely determined.

The fastest growing rechargeable battery technology is lithium-ion batteries [93], which are increasingly used to power mobile electronic devices, in electric and hybrid vehicles, as a means of energy storage for renewable energy generators, among other important applications [106,107,108]. Lithium-ion batteries use a lithium metal oxide cathode (typically lithium bonded with cobalt), a carbon anode, and a non-aqueous organic liquid electrolyte containing dissolved lithium salts. The chemical reactions at the anode and cathode are as follows [108]

$$\begin{array}{l}\mathrm{Li}\left(\mathrm{C}\right)\Leftarrow \Rightarrow {\mathrm{Li}}^{+}+{\mathrm{e}}^{-},\\ {\mathrm{Li}}^{+}+{\mathrm{e}}^{-}+{\mathrm{CoO}}_2\Leftarrow \Rightarrow {\mathrm{LiCoO}}_2.\end{array}$$

(29)

Unlike a lead–acid battery, the electrolyte is not consumed in the reaction. When the battery is charging, ${\mathrm{Li}}^{+}$ ions are deintercalated from the cathode and flow through the electrolyte, and an equal number of lithium ions from the electrolyte are intercalated in the carbon anode. This process is reversed on discharge, and in both charge and discharge, this results in electron flow through an external circuit [92,109,110]. Due to the cost of lithium cobalt oxide, battery energy storage systems typically use alternatives that dilute the cobalt content with nickel and aluminum or nickel and manganese. A drawback of lithium-ion batteries is that the organic electrolyte is flammable, and therefore, battery-monitoring systems are needed [95,111,112]. An important future research topic will be to use PDEs to precisely model battery dynamics, determine the state of health of a battery, and estimate potential risks when a battery can explode.

A derivation of governing equations for the electrochemical performance of a lithium-ion cell is available in [113]. There are six equations that have been developed specifically for lithium-ion batteries, though they are related to models for other types of batteries including the lead–acid battery model cited above. Of the six equations, three are ordinary differential equations, two are partial differential equations, and one is an algebraic equation. These equations are presented here

$$\begin{array}{c}\nabla {\mathsf{\Phi }}_2=-\frac{i_2}{\kappa }+\frac{2RT}{F}\left(1-t_{+}^0\right)\left(1+\frac{d\ln f_{\pm }}{d\ln c}\right)\nabla \ln c,\\ I-i_2=-\sigma \nabla {\mathsf{\Phi }}_1,\\ ϵ\frac{\partial c}{\partial t}=\nabla \cdot ϵD\left(1-\frac{d\ln c_0}{d\ln c}\right)\nabla c+\frac{t_{-}^0\nabla \cdot i_2+i_2\cdot \nabla t_{-}^0}{z_{+}{v}_{+}F}-\nabla \cdot c{v}_0+a{j}_{-},\\ \frac{\partial c_s}{\partial t}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(D_s{r}^2\frac{\partial c_s}{\partial r}\right),\\ i_n=i_0\left[ \exp \left(\frac{{\alpha }_{a}F\left({\mathsf{\Phi }}_1-{\mathsf{\Phi }}_2-U\right)}{RT}\right)-\exp \left(\frac{{\alpha }_{c}F\left({\mathsf{\Phi }}_1-{\mathsf{\Phi }}_2-U\right)}{RT}\right) \right],\\ \nabla \cdot i_2=a{i}_n.\end{array}$$

(30)

The dependent variables ${\mathsf{\Phi }}_1\left(x\right)$ and ${\mathsf{\Phi }}_2\left(x\right)$ are the potential in the solid and solution, respectively, $x\in {ℝ}^3$. $c_s\left(r,t\right)$ and $c\left(x,t\right)$ are concentrations in the solid and solution, respectively; $i_2\left(x\right)$ is the current. Details about parameters, assumptions that can be made, and boundary and initial conditions can be found in [113,114,115,116,117]. Model (30) was improved to consider capacity fade in [118] using an adaptive observer.

A single-particle model was developed in [119] and adapted for a lithium-ion battery, as described in paper [120]. An adaptive PDE-based observer was designed in [121] to estimate the state of health (the state of charge) of a lithium-ion battery. The model used by these authors contains two uncoupled PDEs for the concentration of lithium in each electrode, $c_s^{-}\left(r,t\right), c_s^{-}\left(r,t\right)$. The PDE model with the corresponding boundary conditions is given by

$$\begin{array}{l}\frac{\partial c_s^{-}}{\partial t}\left(r,t\right)=D_s^{-}\left[\frac{2}{r}\frac{\partial c_s^{-}}{\partial r}\left(r,t\right)+\frac{{\partial }^2{c}_s^{-}}{\partial r^2}\left(r,t\right)\right],\\ \frac{\partial c_s^{-}}{\partial r}\left(0,t\right)=0,\\ \frac{\partial c_s^{-}}{\partial r}\left(R_s^{-},t\right)=\frac{I\left(t\right)}{D_s^{-}F a^{-}A{L}^{-}},\\ \frac{\partial c_s^{+}}{\partial t}\left(r,t\right)=D_s^{+}\left[\frac{2}{r}\frac{\partial c_s^{+}}{\partial r}\left(r,t\right)+\frac{{\partial }^2{c}_s^{+}}{\partial r^2}\left(r,t\right)\right],\\ \frac{\partial c_s^{+}}{\partial r}\left(0,t\right)=0,\\ \frac{\partial c_s^{+}}{\partial r}\left(R_s^{+},t\right)=\frac{I\left(t\right)}{D_s^{+}F a^{+}A{L}^{+}},\end{array}$$

(31)

which has the Neumann type boundary conditions. The single-particle model is so named because each electrode is modeled as a single spherical particle, with radii $R_s^{-}$ and $R_s^{+}$ for the negative and positive electrodes, respectively. The independent variable $r$ is a spherical coordinate $0<r<R_s$, and the independent variable $t$ is time, $t>0$. The authors in [121] use a backstepping technique to estimate the system states $c_s^{-}\left(r,t\right)$, $c_s^{-}\left(r,t\right)$ by voltage measurement.

The authors in [122,123] developed a partial differential equation model to describe lithium-ion diffusion with intercalation stresses, given here with the corresponding boundary conditions

$$\begin{array}{c}\frac{\partial c_s^j}{\partial t}=D_s^j\left[\left(1+{\theta }^j{c}_s^j\right)\left(\frac{{\partial }^2{c}_s^j}{\partial r^2}+\frac{2}{r}\frac{\partial c_s^j}{\partial r}\right)+{\theta }^j{\left(\frac{\partial c_s^j}{\partial r}\right)}^2\right],\\ -D_s^j\left(1+{\theta }^j{c}_s^j\left(R_s^j,t\right)\right)\frac{\partial c_s^j}{\partial r}\left(R_s^j,t\right)=\frac{\pm I\left(t\right)}{F{a}^{j}A{L}^j},\\ \frac{\partial c_s^j}{\partial r}\left(0,t\right)=0,\end{array}$$

(32)

where the authors developed an observer for this model in [122].

The authors in [124] presented a model for the dynamics of a battery that includes mass diffusion, electrical states, thermal dynamics, and battery aging dynamics, which resulted in a combined set of 17 coupled nonlinear PDEs. In that paper, the authors presented a framework for model simplification, and the same authors proposed in [125] a model predictive control technique using their reduced model for optimal charging.

Although this section has focused primarily on lead–acid batteries due to their present ubiquity and utility, and lithium-ion batteries due to the recent growth in research into them, other types of rechargeable batteries such as nickel–metal hydride and flow batteries show promise, and they are the subjects of present and future research [92]. As has been discussed, the fundamental PDEs discussed above can be used to model different types of batteries.

6. The Modeling and Control of Wave Energy and Tidal Energy Using PDEs

Wave energy and tidal energy are ocean-generated electric energy sources. Even though they have the ocean in common, their mathematical models and the principles of electric energy generation differ considerably. It is interesting to observe that for wave energy, there are several mathematical models based on PDEs and many lumped-parameter models. However, for tidal energy, hardly any PDE-based models exist, and there are only a few lumped-parameter models for it available in the journal and conference literature.

6.1. The Modeling and Control of Wave Energy Using PDEs

Waves on the ocean surface are generated primarily by wind, beginning as ripples and increasing in size with sustained winds. If winds are sustained over a sufficient distance with sufficient speed, the waves become fully developed and can travel great distances with very little power loss [126]. Waves are power-dense, carrying potential and kinetic energy that can be harvested by a wide variety of approaches. Among renewable energy sources, ocean wave energy has the highest energy density [127,128,129], but technology to harvest wave energy is far less developed than technology used for other renewables like wind and solar. Consequently, wave energy practically accounts for far less than one percent of world energy generation [130]. Early estimates of the wave energy potential measured as the global resource were about 2.7 TW, though more recent studies have estimated it to be as high as 3.7 TW [126,131].

While many wave energy-harvesting approaches have been proven to be viable in theory and practice, no single candidate has emerged as the obvious leader [130,132]. A 2010 paper claimed that the number of approaches exceeded one hundred and was trending higher, with the number of new approaches considerably outnumbering those estimates [133]. It is apparent that wind energy is an open area for new research, including the development of new approaches and technologies, as well as new mathematical PDE-based models.

A recent comprehensive review paper on wave energy converters categorizes all wave energy devices into three broad categories: (i) oscillating water columns, (ii) overtopping devices, and (iii) oscillating bodies [134].

Oscillating water columns (OWCs) were one of the first wave energy conversion technologies to be developed, wherein waves passing the device act as a piston oscillating in a chamber, which creates a pressure differential in that chamber, causing a wind turbine to spin inside the device. The wind turbine is connected to a generator which converts the mechanical energy into electrical energy, which can be used or stored. An advantage of this design is its relative simplicity, as the only moving part is the turbine, which is not in contact with ocean water [127,135]. This is an important consideration due to sea water corrosion and biofouling issues [136]. While a large number of OWCs have been deployed, many have been decommissioned, and research has slowed on this approach [130].

Oscillating water columns often rely on resonance to maximize energy conversion. While the water inside the column is sometimes modeled as a rigid massless piston, this does not account for other resonance modes that can be activated to improve the efficiency of these devices [137]. A better model relating ocean wave interaction with OWCs was derived from partial differential equations, using the continuity equation and the Navier–Stokes equation

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \overrightarrow{v}\right)=0,$$

(33)

$$\frac{D\overrightarrow{v}}{Dt}\equiv \frac{\partial \overrightarrow{v}}{\partial t}+\left(\overrightarrow{v}\cdot \nabla \right)\overrightarrow{v}=-\frac{1}{\rho }\nabla p_{\mathrm{tot}}+\nu {\nabla }^2\overrightarrow{v}+\frac{1}{\rho }\overrightarrow{f},$$

(34)

where $\rho$ is fluid mass density, $\overrightarrow{v}\left(x,y,z,t\right)$ and $p_{\mathrm{tot}}$ are the velocity and pressure of the fluid, respectively, $\nu$ is the kinematic viscosity coefficient, and $\overrightarrow{f}$ is the external force per unit volume. If the fluid is assumed to be ideal, incompressible, and irrotational, and the only external force is considered to be gravitational, several simplifications can be made. Defining $\varphi$ as the velocity potential,

$$\overrightarrow{v}=\nabla \varphi ,$$

(35)

it can be shown that the following is valid in the fluid domain [138]

$${\nabla }^2\varphi =0.$$

(36)

Boundary conditions can be developed for this partial differential equation by considering the physics of the water column. First, define $p\left(x, y, z, t\right)$ as the hydrodynamic pressure, $p_k\left(x, y, z, t\right)$ as the dynamic portion of this pressure, $\eta \left(x, y, t\right)$ as the elevation of the water–air interface in the open air, and $\eta \left(x, y, t\right)$ as the elevation of the water–air interface inside the column. Assuming the sea bottom is horizontal or the water is sufficiently deep and assuming the waves are harmonic at frequency ω, the velocity potential can be written as

$$\varphi =\varphi \left(x, y, z, t\right)=Re\left\{ \widehat{\varphi }\left(x, y, z\right)e^{i\omega t}\right\},$$

(37)

where $\widehat{\varphi }$ is the complex amplitude of $\varphi$. The variables $p, p_k, \eta$, and ${\eta }_k$ also have time-independent complex amplitudes denoted with the caret notation. Following through with the physical analysis of the oscillating water column, and linearizing assumptions, the following PDEs with the boundary conditions were developed in [138,139,140,141]

$${\nabla }^2\widehat{\varphi }=0 ,$$

(38)

$${\left[\frac{\partial \widehat{\varphi }}{\partial n}\right]}_{S_b}=0 ,$$

(39)

$${\left[-{\omega }^2\widehat{\varphi }+g\frac{\partial \widehat{\varphi }}{\partial z}\right]}_{S_0}=0 ,$$

(40)

$${\left[-{\omega }^2\widehat{\varphi }+g\frac{\partial \widehat{\varphi }}{\partial z}\right]}_{S_k}=-\frac{i\omega }{\rho }{\widehat{p}}_k ,$$

(41)

where $S_b$ is the fixed solid surface, $S_0$ is the free water surface, $S_k$ is the water surface inside the column, $g$ is gravitational acceleration, and $n$ is the normal direction to the solid surface. Other boundary conditions have been developed, for example to account for the scattering of waves along vertical barriers [142].

This theory has been used in the analysis of oscillating water columns, for example in [143]. In [144], the authors analyzed a 40-chamber OWC using numerical analysis results based on this theory. The authors also conducted experiments and comparisons with a single-degree-of-freedom model.

It is frequently assumed in the literature that ocean waves pass over OWC devices normal to the device, which is not always true in practice, and this can affect the performance of these devices [145]. In [146], the authors considered this factor to extend the work of [140], and through semi-analytical and numerical PDE analysis, made conclusions regarding the angle of wave incidence that could lead to more efficient wave energy extraction. The authors in [147] also considered the problem of incidence angle as well as sloshing in the case of dual-chamber OWCs affixed to a coastline (with a secondary function as protectors from erosion and flooding). Other studies considering an array of OWCs on a coastline using partial differential equations can be found in [148,149]. In [149], the authors were able to make several important conclusions regarding the positive and negative effects on energy generation resulting from the coastline array configuration, including that by having channels between the OWCs, the wave amplitude is amplified, which can potentially improve energy extraction.

One alternative to oscillating water columns is overtopping converters, which capture wave water that is above the average surface level of the ocean. The potential energy of this water is then converted to electrical energy by passing it through a turbine. The hydrodynamics of overtopping converters are highly nonlinear, making analysis more difficult [133].

A numerical approach is often taken to study the interaction of air and water in overtopping converters, using the volume of fluids model developed by [150], in which a variable $0 \le {\alpha }_{air} \le 1$ was used to represent the presence of air in a particular volumetric fraction and $0 \le {\alpha }_{\mathrm{water}}\le 1$ represented the presence of water, with

$${\alpha }_{\mathrm{air}}+{\alpha }_{\mathrm{water}}=1.$$

(42)

When considering a volumetric fraction of air and water mixture, the conservation equations are as follows

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \overrightarrow{v}\right)=0 ,$$

(43)

$$\frac{\partial }{\partial t}\left(\rho \overrightarrow{v}\right)+\mathsf{\Delta }\cdot \left(\rho \overrightarrow{v}\overrightarrow{v}\right)=-\nabla p+\nabla \cdot \left(\mu \overline{\overline{\tau }}\right)+\rho \overrightarrow{g}+\overrightarrow{F} ,$$

(44)

$$\frac{\partial \alpha }{\partial t}\nabla \cdot \left(\alpha \overrightarrow{v}\right)=0,$$

(45)

where $\rho$ is the mixture density, $\mu$ is the dynamic viscosity, $\overline{\overline{\tau }}$ is the stress deformation tensor, $\overrightarrow{g}$ is the gravitational acceleration vector, and $\overrightarrow{F}$ is external forces. $p$ and $\overrightarrow{v}$ are pressure and velocity, as above [151]. These equations have been used in numerical analyses of overtopping converters in recent papers, such as [152], where irregular waves were considered, and [153], where the authors were able to recommend a new design configuration for a real overtopping converter in Brazil, for example.

The authors in [152] also used a PDE developed in [154,155,156] describing the directional spectrum of waves to develop realistic wave data

$$\frac{\partial N}{\partial t}+\frac{\partial \left(\dot{x}N\right)}{\partial x}+\frac{\partial \left(\dot{z}N\right)}{\partial z}+\frac{\partial \left({\dot{k}}_{x}N\right)}{\partial k_x}+\frac{\partial \left({\dot{k}}_{z}N\right)}{\partial k_z}=Q\left(k_x,k_z,x,z,t\right) ,$$

(46)

where $N\left(x, y, k_x, k_y, t\right)$ is the directional spectrum of wave action density, $x$ and $y$ are positions, $k_x=k sin \theta$ and $k_y=k cos \theta$ are wave numbers ($\theta$ is the wave direction), and $t$ is time [157]. This PDE could be useful for the analysis of other types of wave energy converters, including oscillating body type converters.

In general, studying dynamics of waves using PDEs isa widelyy accepted approach in many areas of engineering and sciences; see for example [158], where the Zakharov–Kuznetsov PDE was used in the stability analysis of ion acoustic waves in plasma. Another paper presented a PDE that models solitary waves in nonlinear elastic circular rods, [159]. However, not many papers model ocean waves as energy systems, which leaves significant room for research on this topic. Note that we not only lack several additional PDE-based models for wave energy systems, but we also lack techniques for controlling such systems. It should be emphasized that within the control engineering community, many control techniques have been developed for many PDEs, and many articles and conference papers, as well as some books, have been published on these topics.

6.2. The Modeling and Control of Tidal Energy Using PDEs

Tidal energy is caused by the rise and fall of sea-level water as a consequence of the gravitational forces acting between the Earth, Moon, and Sun. Tidal energy represents only a small fraction of the world’s total energy. Tidal energy is underutilized in many countries in the world. As a renewable energy source, it will definitely be the subject of future research and development. As a step in that direction, new PDE models of tidal energy system dynamics will be developed that will facilitate the analysis, simulation, and control of the corresponding processes and systems.

There are many journal articles and conference papers on the modeling of dynamic processes for tidal energy and tidal energy conversion using lumped (concentrated)-parameter models (represented by ordinary differential equations); see for example [160,161,162,163,164,165]. However, there are only a few papers [166,167,168] on tidal energy modeling via partial differential equations. This fact indicates that tidal energy is an open mathematical modeling research field for researchers that develop mathematical models using partial differential equations, and consequently, an open research field for tidal energy simulation, analysis, and control.

It is important to observe that it has been known for some time that tidal dynamics can be described by the so-called Laplace tidal equation [169] (note that this equation is different from the general Laplace partial differential equation presented in Formula (11))

$$\begin{array}{l}\frac{\partial \zeta }{\partial t}+\frac{1}{a\cos (\phi )}\left[\frac{\partial }{\partial \lambda }(uD)+\frac{\partial }{\partial \phi }(vD\cos (\phi )\right]=0,\\ \frac{\partial u}{\partial t}-2v\Omega \sin (\phi )+\frac{1}{a\cos (\phi )}\frac{\partial }{\partial \lambda }(g\zeta +U)=0,\\ \frac{\partial v}{\partial t}+2u\Omega \sin (\phi )+\frac{1}{a}\frac{\partial }{\partial \phi }(g\zeta +U)=0.\end{array}$$

(47)

where $\zeta$ is the vertical tidal elevation, $D$ stands for the average thickness of a sheet of fluid, $u$ and $v$ are horizonal velocity components, $\phi$ is the latitude direction and $\lambda$ is the longitude direction, $g$ is the gravitational constant, $\Omega$ is the angular frequency of rotation, $a$ is the radius, and $U$ is the external gravitational tidal force potential. Hopefully, tidal energy researchers will be able to use this PDE in their future tidal energy analysis, simulation, and control studies.

7. Modeling Other Classes of Energy Systems Using PDEs

Electric power systems and, in general, electric energy transmission lines (though not renewable energy sources) are traditionally considered as distributed-parameter systems, whose dynamic processes are most precisely described by partial differential equations having both a temporal and spatial nature. Quite a few papers have been published on these topics, including modeling, analysis, simulation, and control. Detailed coverage of these topics is outside the scope of this overview paper, first of all due to its space limitation. As a matter of fact, electric energy power systems and their optimization will be covered by one of the coauthors of this article in an upcoming Special Issue of the Energies journal [170], where an overview paper of the considered topic on the optimization of electric energy power systems will be presented.

Dynamic processes in the other classes of renewable energy sources such as geothermal (heat) energy can be also precisely studied, analyzed, simulated, and controlled using distributed-parameter models represented by different classes of partial differential equations.

8. Discussion

The authors presented an overview of research results and applications of partial differential equations (PDEs) used in the modeling, analysis, simulation, and control of renewable energy systems. This is by no means an exhaustive overview. It is based on the authors’ experience in carrying out applied research with practical applications on renewable energy systems and theoretical research on the control of different classes of distributed-parameter systems [55,171,172,173,174], as well as the authors’ survey of more than five hundred papers that can be found in the main scientific databases, such as Elsevier, Scopus, IEEE, Wiley, Google Scholar, and Web of Science. Since the system mathematical models described by PDEs provide a much more complete system description than mathematical models described by ODEs (ordinary differential equations), the authors’ motivation for writing this paper was to attract more researchers, practitioners, and readers to the challenging field of PDEs for energy processes and systems, which are in need of more intellectual efforts but provide more complete and more reliable results, from both the theoretical and practical points of view.

Throughout the paper, we have presented our views, emphasized new ideas, indicated open research problems, and in general, discussed the novelties of particular papers on the modeling, analysis, simulation, and control of energy systems and processes using PDE mathematical models. Despite recent advances, several open research problems remain, first of all regarding the use of control engineering techniques to improve and optimize processes in energy systems described by PDEs.

Another set of open research problems lies in extending modeling via PDEs to the modeling of energy systems using ODEs and to most ODE models that have been successfully used in practice, for example for vehicular applications. An important future research topic will be to use PDEs to precisely model battery dynamics, determine the state of health of a battery, and estimate potential risks when a battery can explode. Similar examples can be found for the other classes of energy systems whose mathematical models described by partial differential equations are reviewed in this paper.

9. Conclusions

Partial differential equations that describe the dynamics of distributed-parameter processes and systems have been extensively used for the modeling of energy processes and systems. The use of the developed models is mostly in simulations and analytical studies of energy systems, and they are rarely used for control purposes in energy processes and systems. Distributed-parameter systems are a recent trend in control systems engineering research [175,176,177,178,179,180,181,182,183,184,185]. Since these and other recent results show the full advantage of using models described by partial differential equations, it is expected that in the near future, many more controllers will be designed and optimized for energy systems (especially renewable energy systems) on the grounds of using the partial differential equation methodology.

Most of the presented models fall into the categories of known classical partial differential equations; see, for example, classic books on PDEs [186,187,188,189]. Discussing approaches for solving the classical PDEs is outside the scope of this already long paper that concentrated its attention towards the PDE models developed and used for energy processes and systems. Moreover, such studies exist in the mathematical theory of PDEs, where many overview papers and books can be found, including overview papers and books on a particular PDE. As a matter of fact, only in some of the papers presented in this overview did the authors consider the corresponding mathematical analytical and numerical techniques used, which was noted in this overview paper at places where the corresponding papers were reviewed.

It should be emphasized that this overview paper does not include coverage of energy processes and systems mathematical models that can be described by fractional differential equations (fractional calculus) [159,190]—the field that has been rapidly growing in recent years, with quite a few applications in engineering and sciences and some potential applications in energy processes and systems.