Linear, Nonlinear, and Distributed-Parameter Observers Used for (Renewable) Energy Processes and Systems—An Overview

Abstract

Full- and reduced-order observers have been used in many engineering applications, particularly for energy systems. Applications of observers to energy systems are twofold: (1) the use of observed variables of dynamic systems for the purpose of feedback control and (2) the use of observers in their own right to observe (estimate) state variables of particular energy processes and systems. In addition to the classical Luenberger-type observers, we will review some papers on functional, fractional, and disturbance observers, as well as sliding-mode observers used for energy systems. Observers have been applied to energy systems in both continuous and discrete time domains and in both deterministic and stochastic problem formulations to observe (estimate) state variables over either finite or infinite time (steady-state) intervals. This overview paper will provide a detailed overview of observers used for linear and linearized mathematical models of energy systems and review the most important and most recent papers on the use of observers for nonlinear lumped (concentrated)-parameter systems. The emphasis will be on applications of observers to renewable energy systems, such as fuel cells, batteries, solar cells, and wind turbines. In addition, we will present recent research results on the use of observers for distributed-parameter systems and comment on their actual and potential applications in energy processes and systems. Due to the large number of papers that have been published on this topic, we will concentrate our attention mostly on papers published in high-quality journals in recent years, mostly in the past decade.

1. Introduction to Observers

This overview paper is concerned with the modeling, design, simulation, and control of energy systems via the use of full- and reduced-order observers. Linear, nonlinear, and distributed-parameter observers (systems described by partial differential equations) in continuous and discrete time domains, in both deterministic and stochastic formulations, will also be considered. Moreover, some papers dealing with nonclassical, non-Luenberger-type observers, such as functional, fractional, and disturbance observers, as well as sliding-mode observers, will also be reviewed.

Due to the large number of papers that have been 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 energy systems using observers and on their theoretical and practical research experience of energy systems, particularly fuel cells, solar cells, and wind turbines. Throughout the paper, we will present our views (after reviewing several hundreds of papers on these topics), emphasize new ideas, indicate open research problems, and generally discuss novelties of particular papers on the modeling, analysis, simulation, and control of energy systems and processes using corresponding mathematical models. We have reviewed papers mostly published in the past decade, the decade when renewable energy sources attracted considerable attention of researchers and practitioners and the whole world in general in the search for clean (green) energy sources. It should at least be mentioned that a significant number of papers have been published on observers since their invention sixty years ago (Luenberger, 1964, [1]), but these papers have been used for electric energy power systems, among many other applications, and not for renewable energy systems, the energy systems that are the central focus of this review.

In this introductory section, we provide the basic observer design steps for linear and nonlinear concentrated (lumped)-parameter systems and for distributed-parameter observers. In the remaining sections of the paper, we point to applications of full- and reduced-order observers to different classes of energy and to specific applications for particular energy systems.

1.1. Design of Full-Order Linear Observers

The design of full- and reduced-order linear observers [1,2,3] is very well documented in the control systems engineering literature [4,5,6,7,8,9,10]. Consider a linear dynamic system defined by its state-space form as follows:

$$\frac{dx(t)}{dt}=Ax(t)+Bu(t),x(t_0)=\mathrm{unkown}$$

(1)

where $x(t)\in {\mathcal{R}}^n$ are the system state-space variables, $u(t)\in {\mathcal{R}}^m$ are the system control inputs, and $A\in {\mathcal{R}}^{n\times n}$ and $B\in {\mathcal{R}}^{n\times m}$ are constant system state-space matrices. Assuming that all state variables are available for feedback, a feedback control signal input is defined by the following:

$$u(x(t))=-Fx(t)$$

(2)

where $F\in {\mathcal{R}}^{m\times n}$ is a feedback matrix gain. The condition that all state-space variables must be available for feedback is a prevalent implementation difficulty and a fundamental challenge for the design of full-state feedback controllers. In practice, not all state variables are available for feedback, or it is very expensive to measure and feed back all of them, so that only an output signal, $y(t)$, can be used for that purpose:

$$y(t)=Cx(t)$$

(3)

where $C\in {\mathcal{R}}^{l\times n}$ is a constant system output (measurement) matrix. In practice, the dimension of the output signal is considerably smaller than the dimension of the state-space variables, $\dim \left\{y\left(t\right)\right\}=l<n=\dim \left\{x\left(t\right)\right\}$. To avoid redundant measurements, it is assumed that the measurement matrix has full rank, that is, $l=c=\mathrm{rank}\left\{C\right\}$, with $y(t)\in {\mathcal{R}}^l$ or $y(t)\in {\mathcal{R}}^c$. For the stated reason, engineers use either full-order or reduced-order observers to estimate all or some of the state-space variables and utilize an estimated signal, $\widehat{x}(t)$, which is in some sense close to the actual state variables, $\widehat{x}(t)\approx x(t)$, to form feedback control:

$$u(\widehat{x}(t))=-F\widehat{x}(t)$$

(4)

A full-order linear observer has the same dimension and structure as the considered linear dynamic system (1). It can be constructed using the classical work of Luenberger as follows [1,2,3]:

$$\begin{array}{l}\frac{d\widehat{x}(t)}{dt}=A\widehat{x}(t)+Bu(t)+K(y(t)-\widehat{y}(t))\\ \widehat{y}(t)=C\widehat{x}(t)\end{array}$$

(5)

where the observer gain, $K$, has to be chosen such that the observation (estimation) error defined as follows:

$$e(t)=x(t)-\widehat{x}(t)$$

(6)

decays to zero with time rather quickly, so that $\widehat{x}(t)\to x(t)$. It can be seen from (5) that the linear observer has the same structure as the system plus a driving feedback term that carries information about the observation error, since $y(t)-\widehat{y}(t)=Cx(t)-C\widehat{x}(t)=Ce(t)$. This simple design principle will also be used in the design of nonlinear and distributed-parameter observers.

In practice, the linear observer defined in (6) can be implemented as a system driven by the system input and the system output:

$$\frac{d\widehat{x}(t)}{dt}=(A-KC)\widehat{x}(t)+Bu(t)+Ky(t)$$

(7)

It can be derived from (1) and (5)–(6) that the observation error satisfies the following differential equation:

$$\frac{de(t)}{dt}=(A-KC)e(t)\Rightarrow e(t)=\{\exp (A-KC)t\}e(0)$$

(8)

Irrespective of the initial condition of the observation error, $e(0)$, if the feedback matrix $A-KC$ is asymptotically stable, the observation error will tend to zero as time goes by. To place all linear observer eigenvalues at the desired asymptotically stable locations, the following assumption is needed [1,2,3,4,5,6,7,8,9,10].

Assumption 1.

The pair $(A,C)$ is observable.

1.2. Design of Reduced-Order Linear Observers

Under the assumption that matrix $C$ has full rank equal to l, Equation (3) provides l equations for n unknown state-space variables. The remaining $n-l$ state-space variables have to be estimated using a reduced-order linear observer of dimensions $n-l$. There are several ways to design reduced-order observers (see, for example, [4,7]). According to [6], the most efficient design for a reduced-order observer is based on the Sylvester algebraic equation.

To perform the reduced-order observer design, we start with an asymptotically stable, reduced-order matrix, $A_{des}^{(n-l)\times (n-l)}$, which contains the desired closed-loop reduced-order observer eigenvalues. The reduced-order observer design algorithm is presented as follows.

Step 1: Select matrix $A_{des}^{(n-l)\times (n-l)}$, which contains the desired set reduced-order observer closed-loop eigenvalues, all of which are different from any other eigenvalue of the system state-space matrix $A$.

Step 2: Guess matrix $K_r^{(n-l)\times l}$ such that the pair $(A_{des}^{(n-l)\times (n-l)},K_r^{(n-l)\times l})$ is controllable.

Step 3: Solve the Sylvester algebraic equation of reduced dimensions:

$$T_r^{(n-l)\times n}{A}^{n\times n}-A_{des}^{(n-l)\times (n-l)}{T}_r^{(n-l)\times n}=K_r^{(n-l)\times l}{C}^{l\times n}$$

(9)

Step 4: Use the solution matrix, $T_r$, and form the matrix:

$${\left[\begin{array}{c}C\\ T_r\end{array}\right]}^{n\times n}$$

(10)

If this matrix is nonsingular, then the state estimate can be obtained by appropriately partitioning the matrix defined in (10) as follows:

$$\widehat{x}(t)={\left[\begin{array}{c}C\\ T_r\end{array}\right]}^{-1}\left[\begin{array}{c}y(t)\\ z_r(t)\end{array}\right]=Ly(t)+L_1{z}_r(t)$$

(11)

If the matrix in (10) is singular, guess a new $K_r$ and repeat Steps 3 and 4. The reduced-order observer differential equation is given by the following:

$$\frac{d{z}_r(t)}{dt}=A_{des}^{(n-l)\times (n-l)}{z}_r(t)+T_{r}Bu(t)+K_{r}y(t)$$

(12)

The outputs of either full- or reduced-order observers provide the state estimates that can be used in (4) for control purposes. Often, the feedback gain defined in (4), F, can be found via optimal control [11,12], in which case the quadratic performance criterion, denoted by $J$, is optimized.

$$J=\frac{1}{2}{\displaystyle \underset{t_0=0}{\overset{t_f=\infty }{\int }}\left[x^T\left(t\right)Qx\left(t\right)+u^T\left(t\right)Ru\left(t\right)\right]dt},Q=Q^T\ge 0,R=R^T>0$$

(13)

where $Q$ and $R$ are penalty (weighted) matrices. Since optimization is performed over an infinite time interval (horizon), the controller to be obtained is the steady-state optimal linear-quadratic (LQ) controller.

Full- and reduced-order observers can be implemented using either passive or active electrical circuits, physical components of the same nature as the system under consideration, or in software when a computer is used in the control loop. Implementation of the full- and reduced-order observers in MATLAB/SIMULINK was considered in [13]. Optimal LQ controllers driven by full- and reduced-order observers were considered in [14,15].

In general, the following are important desirable features that can be achieved with the use of linear observers: (a) globally stabilizing gain exists, and it can be easily found; (b) arbitrarily fast observers can be designed (in practice, we use only sufficiently fast observers, since very fast observers whose eigenvalues are placed too far to the left in the complex plane generate noise); (c) the separation principle is valid (observation and control are two independent tasks, so their designs do not interfere with each other); (d) rational choice of observer initial conditions via a least-squares method is available [13]; (e) reduced-order observers with all of the above four features are feasible.

1.3. Design of Full-Order Nonlinear Observers

There have been a number of nonlinear observer formulations. Here, we present the one that can also be used for the design of observers for distributed-parameter systems that can be called the extended Luenberger formulation. Nonlinear observers can be designed using the same principle used in the design of linear observers: an observer has the same structure as the system under consideration with a driving feedback term that carries information about the observation error. This basic observer design principle is demonstrated in the follow-up formulas.

Consider a nonlinear system

$$\begin{array}{l}\frac{dx(t)}{dt}=f(x(t),u(t)),x(t_0)=\mathrm{unknown}\\ y(t)=g(x(t),u(t))\end{array}$$

(14)

where $x(t)\in {\mathcal{R}}^n$ are the system state-space variables, $u(t)\in {\mathcal{R}}^m$ are the system control inputs, $y(t)\in {\mathcal{R}}^l$ are the system outputs, and $f(x(t),u(t))$ and $g(x(t),u(t))$ are, respectively, n-dimensional and l-dimensional vector functions.

Remark 1.

It should be emphasized that having unknown initial system conditions is the fundamental assumption of the observer design. If $x(t_0$) is known, then an observer for (14), assuming that value of the input vector is known, will be a computer program that solves the corresponding nonlinear differential Equation (14). Such an observer has no observation error at all times. In the case when the system’s initial condition, $x(t_0)=x_0$, is known in (14), then the output injection is not needed, since a perfect observer (with zero estimation error at all times, $e(t)=0,\forall \ge t_0$) can be simply obtained from the following (with known $u(t)$ or $u(x(t))=u(\widehat{x}(t))$:

$$\frac{d\widehat{x}(t)}{dt}=f(\widehat{x}(t),u(t)),\widehat{x}(t_0)=x(t_0)=x_0\forall t\ge t_0$$

(15)

This remark will be used when we address the observers designed so far for distributed-parameter systems.

A full-order nonlinear observer for (14) can be constructed as follows [16]:

$$\begin{array}{l}\frac{d\widehat{x}(t)}{dt}=f(\widehat{x}(t),u(t))+K^{n\times l}(\widehat{x},u)[y(t)-g(\widehat{x}(t),u(t)]\\ \widehat{y}(t)=g(\widehat{x}(t),u(t))\end{array}$$

(16)

where $K^{n\times l}(\widehat{x},u)$ is a matrix function that has to be chosen such that the observation error, $e(t)=x(t)-\widehat{x}(t)$, decays to zero. The term $y(t)-\widehat{y}(t)$ stands for the “output injection”. The dynamics of the observation error are given by the following:

$$\begin{array}{lll}\frac{de(t)}{dt}& =& f(x(t),u(t))-f(\widehat{x}(t),u(t))y(t)-K(x(t),u(t))(g(x(t),u(t))-g(\widehat{x}(t),u(t)))\\ & =& f(x(t),u(t))-f(x(t)-e(t),u(t))y(t)-K(x(t)-e(t),u(t))(g(x(t),u(t))-g(x(t)-e(t),u(t)))\\ & & e(t_0)=x(t_0)-\widehat{x}(t_0)=\mathrm{unknown}\end{array}$$

(17)

Note that the observation error differential equation has an unknown initial condition due to the fact that the nonlinear system’s initial condition is not known. It is important to emphasize, as previously indicated, that if the system’s initial condition, $x(t_0)$, is known, then an observer is not needed, since complete information about the state variables at all times can be exactly obtained by directly solving the corresponding nonlinear differential Equation (14), as presented in (15).

It follows from (17) that $e(t)=0$ is one of the steady-state points of (17). Several approaches to the shape dynamics of (17) are feasible at this point by appropriately choosing the observer gain $K(x(t),u(t))$ to achieve asymptotic or exponential stability [17]. To find $K(x(t),u(t))$, assuming that $e(t)=0$, the unique steady-state solutions of (17), the first and second stability methods of Lyapunov [18] can serve as starting points in analyzing the dynamics of (17). For example, using the first stability method of Lyapunov, we can linearize (17) at $e(t)=0$, which produces the Jacobian matrix:

$$\begin{array}{ll}{J}_e=& \frac{\partial }{\partial (x(t)-e(t))}\left\{-f(x(t)-e(t),u(t))+K(x(t),u(t))g(x(t)-e(t),u(t))\right\}\frac{d(x(t)-e(t))}{de(t)}\\ & =\frac{\partial }{\partial \widehat{x}(t)}{\left\{f(\widehat{x}(t),u(t))-K(x(t),u(t))g(\widehat{x}(t),u(t))\right\}}_{|\begin{array}{l}e(t)=0\\ x(t)=\widehat{x}(t)\\ u(t)=u(\widehat{x}(t))\end{array}}\end{array}$$

(18)

Hence, the main observer design goal is to find the matrix function, $K(x(t),u(t))$, that provides asymptotic stability for the Jacobin matrix derived in (18). However, the problem is much more complicated, since in general there are many steady-state points of (17) that are not necessarily equal to zero. Moreover, since $e(t_0)$ is not known and theoretically it can be anywhere in the state space, in general, it might not be in the region of attraction of the steady point, $e(t)=0$. Due to the potential existence of multiple steady-state points, the design of nonlinear observers is still the subject of research, as demonstrated in several relatively recent books written on this subject [19,20,21].

1.4. Design of Reduced-Order Nonlinear Observers

Reduced-order observers for nonlinear systems can be similarly designed, as demonstrated in [16]. For simplicity, assume that a portion of the state-space vector $x_1(t)\in R^{n_1},x_2(t)\in R^{n_2},x(t)\in R^n,n=n_1+n_2$ is directly and exactly measured:

$$\begin{array}{l}\frac{dx(t)}{dt}=\left[\begin{array}{c}\frac{d{x}_1(t)}{dt}\\ \frac{d{x}_2(t)}{dt}\end{array}\right]=f(x(t),u(t))=\left[\begin{array}{c}{f}_1(x(t),u(t))\\ f_2(x(t),u(t))\end{array}\right],x(t_0)\mathrm{unknown}\\ y(t)=x_1(t)\end{array}$$

(19)

In that case, only an observer of the reduced order is needed to estimate $x_2(t)$. Using Formula (19), we look for the estimate of $x_2(t)$ as a linear combination of the system’s actual output and the estimate obtained by the reduced-order observer of dimension $n_2$:

$${\widehat{x}}_2(t)=z(t)+K_2(x(t),u(t))y(t),{\widehat{x}}_1(t)=y(t)$$

(20)

where $K_2(x(t),u(t))$ represents the reduced-order observer gain, with the reduced-order observer dynamic equation given by [16]:

$$\frac{dz(t)}{dt}=f_2(x_2(t),y(t),u(t))-K_2(x(t),u(t))f_1(x_2(t),y(t),u(t))$$

(21)

where $f_1(x_2(t),y(t),u(t))$ and $f_2(x_2(t),y(t),u(t))$ are, respectively, $n_1$ and $n_2$ dimensional vector functions. The observation error from (19)–(21) satisfies the following differential equation:

$$\begin{array}{ll}\frac{d{e}_2(t)}{dt}=& f_2(x_2(t),u(t),y(t))-f_2(x_2(t)-e_2(t),u(t),y(t))\\ & +K_2(x(t),u(t))\left[f_1(x_2(t),u(t),y(t))-f_1(x_2(t)-e_2(t),u(t),y(t))\right]\\ & e_2(t_0)=x_2(t_0)-\widehat{x}(t_0)=\mathrm{unknown}\end{array}$$

(22)

Note that the reduced-order observation error differential equation has an unknown initial condition due to the fact that the nonlinear system initial condition is not known. Several approaches to shape the dynamics of the reduced-order observer estimation error (22) by appropriately choosing the reduced-order observer gain, $K_2(x(t),u(t))$, are feasible. Apparently, $e_2(t)=0$ is one of the steady-state points of (22). As indicated in the case of the full-order nonlinear observer design, the problem becomes complicated, since, in general, there are many steady-state points of (22) that are not necessarily equal to zero. Moreover, $e_2(t_0)$ is not known, and, theoretically, it can be anywhere in the state space; in general, it can happen that it does not lie in the region of attraction of the desired steady point, $e_2(t)=0$. The research on reduced-order observers for nonlinear systems remains an open area [18,19,20,21].

In addition to the basic fundamental idea used for the design of reduced-order observers presented in formulas (19)–(22), there are several other approaches for nonlinear system full- and reduced-order observer designs [22], including those based on the Lie algebra [23]. It is interesting to observe that for some classes of nonlinear systems, the observer design does obey the separation principle (in which the estimation (observation) task and the control task are fully independent) [24]. A reduced-order observer-based controller for a class of Lipschitz nonlinear systems was presented in [25]. A reduced-order observer for backstepping tracking control was considered in [26]. The design of high-gain observers was considered in [19].

In general, the following are important features that can be achieved with the use of nonlinear observers: (a) locally stabilizing observer gains exist in almost all cases; (b) observation error can decay to zero asymptotically, and in rare cases exponentially; (c) separation of estimation and control tasks (the separation principle) is valid only for some special cases; (d) reduced-order observers are feasible.

1.5. Observers for Distributed-Parameter Systems

In the case of distributed-parameter systems, described by partial differential equations (PDEs), the observer design problem is much more complex than in the case of linear and nonlinear concentrated-parameter systems. The most fundamental observer design question for distributed-parameter systems has not been answered so far in the control literature (nor in the applied mathematics literature): “Is it possible, based on the information of the system’s output and the system’s input, to observe the system’s state at all times?” When an observer is used for the purpose of only observing the system’s states (not for feedback control), then the input signal does not exist and observation has to be performed solely on the basis of information provided by the system’s output. No general set of conditions exist that guarantee that the state of the distributed-parameter system can be observed using an observer. This problem is system dependent, and it also depends on the system’s boundary conditions and the observer’s initial conditions (in general, it is assumed that the system’s initial conditions are not known). Moreover, measurements (sensors) for distributed-parameter systems mostly produce only limited information about the state of the system, since they are mostly placed on the domain boundaries. It is impractical and/or impossible to measure the system’s state at all times over the entire domain: linear, planar, or three-dimensional space. An important question to be answered is the following: “Are the given measurements sufficient to provide sufficiently accurate estimates of the space variables at all times?” Finally, nothing is known about the separation principle for this kind of system. Is it possible to independently design an observer and independently design a feedback controller using the observed states as feedback when the system and the observer are connected together.

Observers for distributed-parameter systems have been studied in the past decade by several researchers. Most notable are the results obtained by Krstic and his coworkers. Backstepping observers for a class of parabolic partial integro-differential equations were considered in [27,28] with application to a chemical tabular reactor. The authors considered observer designs with both anti-collocated and collocated sensors and actuators. Reference [29] designed a backstepping observer for a shear beam using the observer design strategy of [27,28]. It is important to observe that, in both papers, the system’s initial conditions were not introduced at all in the problem formulation. However, they were given in the simulation section (being different for the system and the observer). Moreover, closed-loop feedback control was obtained in [29] in terms of initial conditions, implying that both the system’s initial conditions and the observer’s initial conditions were known and were the part of the original problem formulation. Along the lines of [27,28], reference [30] designed an observer for the Schrodinger equation. An observer design was considered in [31] for a heat conduction system and for a viscous PDE in [32]. An observer for a battery’s state of charge was presented in [33]. An extended Luenberg-type observer for a PDE was presented in [34]. Specific to corresponding applications, PDE observers were considered in [35,36,37].

It is interesting to note that observer design for distributed-parameter systems has been considered only for full-order observers and that we have not yet seen an extension of the reduced-order observer design for this class of systems.

2. Linear and Nonlinear Observers for Energy Systems and Processes

In this section, we present an overview of concentrated (lumped)-parameter observers used for various types of renewable energy systems, such as fuel cells, wind turbines, solar cells, and batteries, as well as electric power systems. In the next section, distributed-parameter observers used for some renewable energy systems and electric power grids will be surveyed.

2.1. Observers for Fuel Cell Systems

In the context of optimal control, subject to optimization (minimization) of the performance criterion defined in (13), reduced- and full-order observers were designed for a polymer electrolyte (PEM, also known as a proton exchange membrane) fuel cell model in [14,38]. Two linear observers were designed in [39] to estimate water activity dynamics at both sides of a PEM fuel cell membrane, for both anode and cathode models. These variables were needed to facilitate a model predictive controller (MPC) design. It was shown via simulation results that such observers generate the required state-space variables rather quickly; roughly, the observation error was reduced to zero in half the sampling time of the MPC controller, which facilitated a very efficient controller performance.

A second-order adaptive nonlinear sliding-mode observer was designed in [40] for a PEM fuel cell air-feed system. The experimental results validate such an observer design, which can be used for real-world applications. A higher-order sliding-mode controller for the air excess ratio in a PEM fuel cell was designed in [41]. Similarly, a higher-order sliding-mode observer based on oxygen excess ratio control of a PEM fuel cell air-feed system was considered in [42], with experimental results showing the effectiveness of such a controller design.

A simple nonlinear observer was efficiently used in [43] for partial pressure hydrogen estimation in a PEM fuel cell. It was shown that the designed observer is robust to variations in the inlet partial pressure and insensitive to modeling errors. Reference [44] demonstrated the use of a class of nonlinear robust observers for a PEM fuel cell stack in the presence of disturbances in both system dynamics and system output.

A linear minimum-order functional observer for a solid-oxide fuel cell (SOFC) was considered in [45] with respect to the problem of connecting an SOFC to a power grid. The obtained observer has a simpler structure than the conventional observer. A fractional-order unknown-input nonlinear observer was used in [46] for proportional–integral–derivative (PID) control of a PEM fuel cell air supply system in order to regulate the cathode pressure and the oxygen excess ration. Such a controller was validated both via simulation and experimentally. A disturbance observer was designed for a PEM fuel cell for an air management system via the use of a second-order sliding-mode controller [47] with the goal of minimizing the fuel cell net generated power and avoiding fuel cell oxygen starvation by keeping the oxygen excess ratio at the desired value. The obtained controller was also validated experimentally in [47].

A Luenberger observer was implemented in [48] for switch fault diagnosis in a DC-DC interleaved boost converter for fuel cell applications. Simulation and experimental results showed good robustness of the proposed fault diagnosis method to false alarms. An Internet-based distributed test platform for a fuel cell electric vehicle was developed in [49] using an observer. It was shown that the observer has an impact on vehicle velocity, fuel cell output power, and electric motor output torque.

An extended Kalman filter that can be used as a stochastic observer for a PEM fuel cell was considered in [50]. Extended observers (ESOs) were also used in [51,52,53] for various processes and devices in fuel cells: flatness control [51] and control DC-DC converters [52,53].

2.2. Observers for Solar Cell Systems

Not very many papers on the use of observers for solar cell (photovoltaic) systems can be found in the engineering literature. Among the interesting ones are the papers [54,55,56,57,58,59,60,61,62,63]. One of the early papers on this topic [54] uses a sliding-mode observer to estimate the solar array-produced current in a grid-connected photovoltaic system. The designed sliding-mode observer has a simple structure, and it is robust to modeling uncertainties and parameter variations while providing robust tracking properties.

In [55], a linear Luenberger observer that provides frequency estimation was used to facilitate the disturbance rejection problem in a wind–solar integrated AC microgrid. An adaptive observer-based controller was designed in [56] for domestic single-household solar cell electric energy generation. Connected to the power grid, such an observer facilitates grid current balancing, unity power factor, harmonic compensation, and DC-like voltage regulation. The results obtained were also verified experimentally [56]. Based on the work of [57], ref. [58] designed a second-order sliding-mode observer for current estimation of a solar array. Such an observer is integrated into a single-phase control loop that contains a grid-connected invertor and a current sensor.

The mathematical model for a two-axis solar tracker (horizontal and vertical rotary axes), which has the state space defined in (1), was derived in [57] with the state-space matrices for horizontal rotation in the phase-variable canonical form:

$$A_h=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& -77.4920& -6250\end{array}\right],B_h=\left[\begin{array}{c}0\\ 0\\ 5.9356\times {10}^6\end{array}\right],C_h=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$$

(23a)

and for vertical rotation:

$$A_h=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& -40.775& -6250\end{array}\right],B_v=\left[\begin{array}{c}0\\ 0\\ 3.1232\times {10}^6\end{array}\right],C_v=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$$

(23b)

It is interesting to observe that the state-space models for the horizontal and vertical axes are quite similar. The Luenberger observer was used to estimate the angular velocities of the solar tracker in both the horizontal and vertical axes. An observer-based optimal controller for solar panel rotation (4) that minimizes performance criterion (13) was obtained in [59]. The weighted matrices in performance criterion (13) that produced the best system response were selected, after several attempts via a trial-and-error method, as $R=0.1$ and $Q=\mathrm{diag}\left\{\begin{array}{ccc}1000& 100& 10\end{array}\right\}$. The simulation results were excellent, indicating that such an observer-based controller design facilitates solar panels practically perfectly following the position of the sun.

In [60], a robust adaptive hybrid second-order integrator-based rotor flux observer was designed for self-sensing control of a solar array-fed permanent magnet synchronous motor drive used for a water pump. Reference [61] designed a sliding-mode observer for fault detection in a photovoltaic system connected to a microgrid. The method was validated both experimentally, using a corresponding microgrid benchmark system, and via simulation with MATLAB/SIMULINK. In [62], a nonlinear (extended) observer-driven sliding-mode controller was proposed for maximum power point tracking in a solar cell system. Estimates of a Luenberger-type observer were used in [63] to design a controller for a photovoltaic grid-connected energy conversion system. The controller was tested for various load (linear and nonlinear) conditions and various climatic conditions. The performance of such a controller was found to be satisfactory according to IEEE-519 standards. Reference [64] used a nonlinear observer for maximum power point tracking (MPPT) in a photovoltaic system in a corresponding fault identification system to detect short- and open-circuit faults. A proportional–derivative (PD)-based controller was designed in the same paper for voltage regulation at (or around) the maximum power point.

2.3. Observers for Wind Turbine Energy Systems

A reduced-order observer was designed in [65] whose estimates were used for control of a double-fed induction generator (DFIG) for variable-speed wind turbines. It was shown that such an observer design is robust to parameter variations. Simulation results demonstrated that the controller provided the desired behavior for the considered wind turbine. A full-order observer-based controller was proposed in [66] for effective control of a variable-speed wind turbine. Numerical simulation showed the efficacy of this controller as far as the optimal power was extracted. Reference [67] shows how an observer can be used for fault diagnosis in a wind power system. A nonlinear disturbance observer-based fuzzy sliding-mode controller was used in [68] to track the optimal power by regulating the rotor speed of the generator in a wind energy conversion system. Reference [55] designed a frequency estimation observer for the purpose of disturbance rejection in a hybrid wind–solar integrated microgrid.

Rotor speed and turbine torque were estimated in [69] using a nonlinear sliding-mode observer, and a corresponding controller was designed for a wind energy conversion system. The obtained observer and controller were verified both experimentally and via simulation and showed fairly accurate results. A fault-tolerant observer-driven LQ controller was considered in [70] for tracking optimal power in wind energy conversion systems. Reference [71] proposed the use of an adaptive observer-based controller in a wind energy conversion system. The obtained controller was verified both via simulation and experimentally. A sliding-mode observer was used in [72] for simultaneous detection of actuator and sensor faults. Since the separation principle holds during faulty conditions, a controller-driven observer is implemented in such cases.

An observer-based controller was considered in [73] for the situation of cyberspace security attacks (denial of service) in a wind power system. The role of the observer is, first of all, to monitor for fault detection in the case of attacks. The event-triggered controller has the role of decreasing occupation of the communication channel when attacks happen. Another paper on the use of observers to detect cyber attacks on wind energy systems is [74]. Reference [75] presents an observer designed for a wind–battery energy storage hybrid system. An observer-based controller for a permanent magnet synchronous generator of a wind energy conversion system was considered in [76].

Reference [77] designed a Luenberger observer-based controller with the goal of maximizing wind turbine aerodynamic power over the entire wind speed range. The observer, which had the form of (5), was used to estimate the aerodynamic torque and shaft speed. The second-order linear state-space dynamic model for aerodynamic torque, $T_a$, and turbine shaft speed, ${\omega }_m$, are presented in the paper as follows:

$$\begin{array}{l}\left[\begin{array}{c}\frac{d{\omega }_m(t)}{dt}\\ \frac{d{T}_a(t)}{dt}\end{array}\right]=\left[\begin{array}{cc}0& \frac{1}{J}\\ 0& 0\end{array}\right]\left[\begin{array}{c}{\omega }_m(t)\\ T_a(t)\end{array}\right]+\left[\begin{array}{c}-\frac{K_c}{J}\\ 0\end{array}\right]i_{sq}(t)\\ y(t)={\omega }_m(t)\end{array}$$

(24)

where $J$ is the total moment of inertia, $K_c$ is the torque constant, and $i_{sq}(t)$ is the current in the d-q reference system. The effectiveness of the designed controller was validated experimentally. Permanent magnet wind generator interturn short fault detection and location were performed in [78] using a Luenberger linear observer, with the interturn short fault dynamics represented by a fourth-order linear state-space model.

2.4. Observers for Batteries

There are quite a lot of papers on the use of observers for batteries [79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107], especially for battery state-of-charge (SoC) and state-of-health estimation (SoH). Reference [79] presents four mathematical models used for SoC estimation (utilizing state observers) and provides a definition of SoC most commonly used for lithium-ion batteries as follows:

$$\mathrm{SoC}=\frac{\mathrm{remaining}\mathrm{capacity}}{\mathrm{nominal}\mathrm{capacity}}$$

(25)

The SoC cannot be measured directly, and, most commonly, the ampere-hour method (AHM) is used for that purpose, which requires only the measured current signal. In that case, the SoC at time $t$ is given by the following formula [79]:

$${\mathrm{SoC}}_{\mathrm{t}}{=\mathrm{SoC}}_{{\mathrm{t}}_0}+{\displaystyle \underset{t_0}{\overset{t}{\int }}\frac{\eta I(t)}{C_n}dt}$$

(26)

where ${\mathrm{SoC}}_{\mathrm{t}}$ is the SoC at time $t$, ${\mathrm{SoC}}_{{\mathrm{t}}_0}$ is the initial SoC, $\eta$ is the current efficiency, $C_n$ is the nominal capacity of the battery, and $I(t)$ is the current (assumed to be positive when charging). Battery SoH estimation, first of all, stands for battery capacity fade and resistance deterioration estimation; however, some other battery parameters can be included in battery SoH characterization. Both the SoC and SoH of batteries can be estimated very accurately using observers in a relatively short period of time. We have selected some of the papers which, in our opinion, are the most relevant representatives of the problems studied. First, we review the papers dealing with various aspects of observers designed for batteries and then we review the use of observers for monitoring the state of charge and the state of health of different types of batteries.

Reference [80] presents the use of the extended Kalman filter (a nonlinear stochastic observer) for the observation of parameters in a lithium-ion lumped-parameter nonlinear dynamic model. The state estimation and fault detection of a lithium-ion battery model using the Luenberger observer were considered in [81]. The efficacy of the presented methodology was verified both experimentally and via simulation. Lithium-ion battery current estimation was performed in [82] using an observer. A linear simplified electrical circuit model was used in this process with an unknown input observer. A sliding-mode observer that estimates the lithium concentration in the anode and cathode simultaneously was presented in [83]. The Kalman filter and a nonlinear observer were used in [84] for entropy estimation for a lithium-ion battery.

Due to aging, battery capacity and power fade. An observer was designed in [85] to estimate the capacity for a lithium-ion battery. The obtained analytical results were verified experimentally, showing only up to 2% capacity estimation error. Reference [86] presents a functional observer [87] for lithium-ion batteries connected in parallel. The dynamics of such systems are described by differential-algebraic equations. The corresponding observer design is formulated in terms of linear matrix inequalities [88]. A prescribed degree-of-stability observer-based controller for a battery energy storage system was considered in [89]. An adaptive controller with a disturbance observer for a hybrid battery–supercapacitor energy source for electric vehicles was designed in [90].

It is important to emphasize that a large number of papers have considered the state of charge (SoC) and state of health (SoH) of different types of batteries and under different conditions [91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107]. For lead batteries, SoC and SoH were considered in [91,92], including the use of the extended Kalman filter and an electric vehicle application. The work of [91] was based on a second-order linear RC model, and it was established that its observability facilitated the observer design.

SoC was considered in [93] for lithium-ion nickel–manganese–cobalt batteries using a switched $H_{\infty }$ observer and a switched linear second-order battery model. The obtained results were validated experimentally, indicating high accuracy and robustness. The SoC was determined from obtained information about the battery current and the terminal voltage.

An observer was designed for SoC estimation for lithium iron phosphate batteries (LFPs) that are used for electric vehicles [94]. The paper used a fractional-order observer for that purpose, with the battery fractional model represented by the following:

$$\begin{array}{l}\frac{d^{{\alpha }_1}{v}_1(t)}{d{t}^{{\alpha }_1}}=-\frac{1}{R_1{C}_1}{v}_1(t)+\frac{1}{C_1}i(t)\\ \frac{d^{{\alpha }_2}{v}_1(t)}{d{t}^{{\alpha }_2}}=-\frac{1}{R_2{C}_2}{v}_2(t)+\frac{1}{C_2}i(t)\end{array}$$

(27)

where $0<{\alpha }_1<1$ and $0<{\alpha }_2<1$ and $v_1(t),v_2(t)$ are the voltages over the constant phase elements, whose connection in series, together with the internal resistor and the open voltage source, models the LFP battery. In terms of the LFP current, $i(t)$, the SoC is given by the following simple differential equation:

$$\frac{d(\mathrm{SoC})}{dt}=-\frac{\eta }{C_n}i(t)$$

(28)

where $\eta$ is the charging/discharging efficiency and $C_n$ is the nominal capacity of the LFP battery.

SoH in terms of capacity fade and resistance deterioration for lithium batteries was considered in [95] using a dual sliding-mode observer. A Lypunov function [18] was selected to show the stability and convergence of the observation error to zero. A proportional–integral observer was used in [96] to estimate a lithium-ion battery’s SoC, with the battery model represented by a simple first-order RC circuit. It was shown that the observer integral action compensates for the model’s errors and uncertainties. The proposed observer was validated experimentally. An evaluation of different circuit-based lithium battery models can be found in [97]. A discrete-time nonlinear observer was proposed in [98] to estimate the SoC of a lithium-ion battery. To that end, the second-order battery model was discretized using the Euler approximation. Lyapunov stability was employed to show convergence to zero of the observation error. In addition, the paper considered the relation of the SoC and the open battery voltage.

Flow batteries have attracted attention recently as potential substitutes for the lithium-ion batteries used in vehicular applications [99]. These batteries use a new kind of fluid known as “nanoelectrofuel”. According to [99], “Flow batteries are safe, stable, long lasting, and easily refiled, qualities that suit them well for balancing the grid, providing uninterrupted power, and backing up sources of electricity”. In comparison, lithium batteries have long charging times, the danger of fire, and a relatively short working life, and there are difficulties in acquiring the battery materials and recycling the batteries. Sliding-mode observers were used in [100,101] to estimate the SoC for vanadium redox-flow batteries. In addition, in [100], the flow battery capacity fading factor was also estimated.

References [102,103,104,105,106,107] deal with various aspects of SoC and SoH estimation using different types of observers, including system parameter identification [104,105,106] and hysteresis compensation [106], and consider vehicular applications [102,107].

3. Distributed-Parameter Observers for Energy Systems and Processes

In recent years, significant work has been carried out by numerous authors on the modeling and control of distributed-parameter energy systems governed by partial differential equations (PDEs), as has been reviewed in a recent paper by the authors [108]. Observers are often a necessary component of control systems when feedback information about system states cannot be directly measured. As PDEs model infinite system states, PDE-based controllers can sometimes rely on knowledge of infinite system states as feedback information, which typically cannot be measured directly, and in this circumstance such controllers cannot be implemented without an observer [109]. In other contexts, observers are needed to estimate immeasurable states for diagnostic and monitoring purposes [110]. Due to the complexity of PDE models, observers have traditionally been developed for reduced-order or discretized models for such systems. Lyapunov stability theory has proven to be a tremendously powerful tool for establishing the convergence of PDE estimators, as have other, more recently developed techniques, such as the PDE backstepping method [28] and the extended Luenberger method [34]. In this section, recent developments and the implementation of observers for energy systems governed by PDEs will be reviewed.

3.1. Lithium-Ion Battery Systems

For a comprehensive review of batteries and other electrical energy storage technologies, the reader is referred to the comprehensive review paper [111]. Considerable research has been carried out on rechargeable batteries due to their numerous applications in laptops, mobile phones, electric vehicles, and renewable energy storage, among others [112,113,114]. In particular, lithium-ion batteries have been the fastest-growing rechargeable battery technology in recent years [115]. Lithium-ion batteries use a lithium metal oxide cathode, a carbon anode, and a nonaqueous organic liquid electrolyte containing dissolved lithium salts which is not consumed in the chemical reaction in charge and discharge cycles. 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, with electron flow through an external circuit [111,116,117].

Observers have an important role in lithium-ion battery technology. Since they are rechargeable, it is important to be able to know the state of charge as well as the state of health of batteries. In addition to that, the organic electrolyte in lithium-ion batteries is flammable, and therefore battery monitoring is of critical importance [118,119].

A derivation of governing equations for the electrochemical performance of a lithium-ion cell is available in [120]. This distributed-parameter model includes three ordinary differential equations, two partial differential equations, and one algebraic equation, presented below:

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

(29)

The dependent variables, ${\mathsf{\Phi }}_1\left(x\right)$ and ${\mathsf{\Phi }}_2\left(x\right)$, are the potentials in the solid and the 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; $c_o$ is the solvent concentration; and $i_2\left(x\right)$ is the current. The subscripts $+$ and − refer to cation and anion species, respectively. The parameters are defined as follows: $\kappa$ is the conductivity corrected by the Bruggeman relation, $F$ is Faraday’s constant, R is the universal gas constant, $T$ is the temperature in Kelvin, $t_i^o$ is the transference number of species $i$ with respect to solvent velocity, $f_{\pm }$ is the electrolyte mean molar activity coefficient, $\sigma$ is the effective electronic conductivity of the electrode, $ϵ$ is the electrolyte volume fraction, $a$ is the surface area of the active electrode material per volume, and ${\nu }_i$, $z_i$, and $j_i$ are moles of ions produced when a mole of the salt dissociates, the charge of the ions, and the total flux due to the rection, respectively, for species $i$. $D_s$ is the difficusion coefficient of lithium in an insertion electrode; $r$ represents the radial position across a spherical particle; ${\alpha }_a$ and ${\alpha }_c$ are the anodic and cathodic transfer coefficients, respectively; $U$ is the thermodynamic potential measured with respect to a lithium reference electrode; $i_0$ is the exchange current density; and $i_n$ is the transfer current normal to the surface of the active material. Details about the parameters, assumptions that can be made, and boundary and initial conditions can be found in [120,121,122,123,124].

A single-particle model was developed in [125] and adapted for a lithium-ion battery, as described in paper [126]. An adaptive PDE-based observer was designed in [33] to estimate the state of health and 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, $c_s^{-}\left(r,t\right) \mathrm{and} c_s^{-}\left(r,t\right)$. The PDE model with the corresponding boundary conditions is given by the following:

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

(30)

which has the Neumann-type boundary conditions. $I\left(t\right)$ is the input current, $A$ is the cross-sectional area of the cell, $a^{+}$ and $a^{-}$ are the specific interfacial surface areas, and $L^{+}$ and $L^{-}$ are the thicknesses of the respective electrodes. Here, $D_s$ remains the diffision coefficient for the respective cation ($+$) and anion (−) in the solid phase. 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 of [33] use a backstepping technique to estimate the system states $c_s^{-}\left(r,t\right)$ and $c_s^{-}\left(r,t\right)$ by voltage measurement.

The authors of [127,128] developed a partial differential equation model to describe lithium-ion diffusion with intercalation stresses, given here with the corresponding boundary conditions:

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

(31)

where $\theta =\frac{\mathsf{\Omega }}{RT}\left[\frac{2\mathsf{\Omega }E}{9\left(1-\nu \right)}\right]$, with $\mathsf{\Omega }$ as the partial molar volume, $E$ as the Young’s modulus, and $\nu$ as Poisson’s ratio. In [128], the authors developed a state-of-charge observer for this model. This model was also used in [129] to develop a sensitivity-based interval observer for state-of-charge estimation; in [130], where the robustness of observers to parameter uncertainties in the model was also considered; and in [131], where an observer was developed for a simplified single-particle model. In [132], the authors used output terminal voltage information to develop a Luenberger-type observer. Kalman filter approaches to a discretized single-particle model PDE were used to develop state-of-charge observers in [133,134], while the authors of [135] used a similar approach to construct a globally stable circle-criterion observer and examined its effectiveness using the original PDE model. Similarly, the authors of [136] also used the single-particle PDE model to validate parameter estimations made using a simplified linear ODE-based estimator.

Some recent research has been dedicated to improving model performance in general and in high-current applications in particular by including the dynamics of the electrolyte, as, for example, in [137,138,139]. The problem of observability for such models was considered in [140,141], and estimators have been developed with the inclusion of electrolyte dynamics in [138,139,142,143,144]. Such advancements have the potential to significantly improve state-of-charge and other lithium-ion estimates in fast-charging contexts.

A recent review paper on lithium-ion battery models has an excellent section on online identification methods; the reader is directed to [145] for an overview of some of the other works on observers for lithium-ion batteries conducted by some of the authors cited above, and others, such as [146], which developed an observer for the full electrochemical model to estimate the parameters and system states; [147,148], which developed an observer for a reduced-complexity version of the single-particle model using an iterated extended Kalman filter; [149], which used a coupled pair of observers for an electrochemical–thermal model to determine both state of health and state of charge using Lyapunov stability analysis; and [150], which developed an observer that estimates not only state of charge but other important battery health conditions, such as temperature. Other valuable review papers that discuss observers for lithium-ion batteries, including ones based on both lumped and distributed-parameter models, are available in [151,152,153,154].

3.2. Power Grids

An interesting distributed PDE model was developed for the demand-side management of thermostatic loads in the power grid in [155]; building on existing diffusion and transport models, the authors developed the following model:

$$\begin{gathered} \frac{\partial X_{\mathrm{on}/\mathrm{off}}\left(t,T\right)}{\partial t}=-\frac{\partial \left[\left({\alpha }_{\mathrm{on}/\mathrm{off}}-{\dot{T}}_{sp}\right)X_{\mathrm{on}/\mathrm{off}}\left(t,T\right)\right]}{\partial T} \\ \left[({\alpha }_{\mathrm{on}}-{\dot{T}}_{sp}\right)X_{\mathrm{on}}\left(t,T\right){]}_{@T_{\max }^{-}}-\left[({\alpha }_{\mathrm{on}}-{\dot{T}}_{sp}\right)X_{\mathrm{on}}\left(t,T\right){]}_{@T_{\max }^{+}}+\left[({\alpha }_{\mathrm{off}}-{\dot{T}}_{sp}\right)X_{\mathrm{off}}\left(t,T\right){]}_{@T_{\max }}=0 \\ \left[({\alpha }_{\mathrm{off}}-{\dot{T}}_{sp}\right)X_{\mathrm{off}}\left(t,T\right){]}_{@T_{\min }^{+}}-\left[({\alpha }_{\mathrm{off}}-{\dot{T}}_{sp}\right)X_{\mathrm{off}}\left(t,T\right){]}_{@T_{\min }^{-}}+\left[({\alpha }_{\mathrm{on}}-{\dot{T}}_{sp}\right)X_{\mathrm{on}}\left(t,T\right){]}_{@T_{\min }}=0 \end{gathered}$$

(32)

where $T_{sp}$ represents a nonconstant setpoint temperature, $T$ represents the temperature relative to this setpoint, and $t$ represents time. $X_{\mathrm{on}}$ and $X_{\mathrm{off}}$ represent the distribution of loads (number of loads/$°$C) in the on and off states, and ${\alpha }_{\mathrm{on}}$ and ${\alpha }_{\mathrm{off}}$ represent local load transport rates. The bounds for the independent variable, $T$, are $T\in \left[T_{\min },T_H\right]$ for the equation governing $X_{\mathrm{on}}$ and $T\in \left[T_L, T_{\max }\right]$ for the equation governing $X_{\mathrm{off}}$, where $T_L$ and $T_H$ are the lowest and highest practical temperature extremes, and $T_{\min }$ and $T_{\max }$ are the upper and lower limits of the temperature deadband. This model is further developed in [156,157,158] to include other aspects, such as the effect of heterogeneity in populations of thermostatically controlled loads.

The authors of [158] developed a boundary observer for online estimation of the distribution of thermostatically controlled loads in the PDE model using the PDE backstepping method that only requires knowledge of times when a thermostatically controlled load in the grid switches between on and off states. This greatly reduces the feedback information that needs to be measured for a grid power management system, making control strategies such as those developed in [159,160,161] more practical. A recent review of lumped and distributed models for this problem, including a discussion of observability, is available in [162].

Opportunities for future PDE observer development include the inverse of the problem of demand-side grid management, namely, the supply-side problem. The authors of [163] developed a PDE model and a control strategy for wind farm management, where the number of individual wind generators is a parameter in the PDE and does not increase the complexity of the model. However, to the best of the authors’ knowledge, the problem of observation for this PDE model has not been addressed to date.

3.3. Proton Exchange Membrane Fuel Cells

Proton exchange membrane fuel cells (PEMFCs) require measured system states to operate properly, for reasons that include the durability, safety, and reliability of the system. While some states can be directly measured, others must be estimated due to the physically closed nature of the technology [110]. A finite-element discretization methodology was utilized in [110] to design a nonlinear observer for a PEMFC model described by a PDE. The state estimates generated by the observer were used in [110] for an efficient continuous-time sliding-mode controller design whose purpose was to reduce the observer estimation error to zero. In general, the complexity of the partial differential equations governing PEMFC processes leads to high computational costs, making observers based on these models difficult for control applications [164,165]. However, the authors of [166] developed a PDE model of a PEMFC from first principles and with it an observer-based controller. The observer requires only a handful of measurements, including the pressure in the middle of the gas channels, humidities, temperature, and pressures at the outlet of the anode and cathode gas channels, the temperature of the solid midpoint, and the cell voltage and current.

3.4. Solar Collectors

Solar collectors work by reflecting sunlight to generate heat energy. With parabolic trough solar collectors, mirrors of a parabolic shape reflect sunlight onto a heat-absorbing pipe that has a fluid flowing through it (typically oil). The thermal behavior of distributed solar collector fields has been modeled and summarized in [167,168,169,170] and in other works by partial differential equations. The energy balance equations are given as follows:

$${\rho }_m{c}_m{A}_m\frac{\partial T_m}{\partial t}\left(t,x\right)={\eta }_{0}GI\left(t\right)-P_{rc}-D_i\pi H_t\left(T_m\left(t,x\right)-T_f\left(t,x\right)\right)$$

(33a)

$${\rho }_f{c}_f{A}_f\frac{\partial T_f}{\partial t}\left(t,x\right)+{\rho }_f{c}_{f}q\left(t\right)\frac{\partial T_f}{\partial x}\left(t,x\right)=D_i\pi H_t\left(T_m\left(t,x\right)-T_f\left(t,x\right)\right)$$

(33b)

where the independent variables, $t\mathrm{and} x$, represent time and space, respectively; $T$ represents the temperature, with the subscripts $m$ for metal and $f$ for fluid, referring to the metal of the pipe and the fluid inside it. Parameters $\rho , c, \mathrm{and}A$ are the density, the specific heat, and the cross-sectional area, respectively; ${\eta }_0$ and $G$ are the mirror optical efficiency and the optical aperture, respectively; $D_i$ is the inner diameter of the pipe; $H_t$ is the coefficient of the metal fluid transmission; $P_{rc}$ is the sum of the radiative and conductive thermal losses; and the time-dependent functions $I\left(t\right)$ and $q\left(t\right)$ represent the solar radiation and the oil pump volumetric flow rate, respectively. A simplified model for the fluid temperature is also frequently used:

$$A\frac{\partial T}{\partial t}\left(t,x\right)+q\left(t\right)\frac{\partial T}{\partial x}\left(t,x\right)=\frac{{\eta }_{0}G}{\rho c}I\left(t\right)$$

(34)

which is a 1D hyperbolic PDE. The authors of [171] show the exact observability of the solar radiation (the “source”) using the temperature measurement at the boundary only and develop an adaptive boundary observer that guarantees $L_2$ convergence. In this paper, the authors reproduce a proof from the literature which establishes that the boundary of linear first-order hyperbolic systems are exactly observable, i.e., the boundary, $T(0,t)$, is exactly observable from a measurement of the opposite boundary, $T(1,t)$, for the domain $0<x<1$. The source term is in the governing equation, $I(t)$, however, and the challenge in [171] of obtaining this exact estimation from the boundary is achieved by the authors’ use of prior work on the concept of persistent excitation for PDEs. In [172], the authors develop an observer for the system states $T\left(t,x\right)$ using Lyapunov theory in the development of a controller for the fluid flow rate to achieve a desired temperature profile. Other Lyapunov-based observers were developed in [173] for the system states using fuzzy logic and [174] for uncertain parameters. A high-gain observer was developed for a class of hyperbolic PDEs in [173], and the authors used the solar collector governing equation as a test case. The authors of [173] make the claim that their observer can also be applied to bioreactors and chemical tubular reactors; however, to date, it does not appear that the authors have developed observers for these specific contexts. This might be an interesting future research topic. The authors of [175,176] also developed Lyapunov-based observers in the context of identifying soiling rates, so that cleaning schedules for a plant can be determined for increased efficiency and reduced maintenance costs.

3.5. Wind Turbines

The majority of wind turbine generators are of the horizontal-axis type, which consist of a tower with a nacelle on top containing an electrical generator and mechanical components which connect to a turbine with one or more blades. Wind forces in the normal direction to the turbine generate rotation around an axis parallel to the wind. This rotation is transferred through a shaft and typically through a gearbox to another shaft which powers a generator to produce electrical energy. Designs for such generators vary significantly; variable factors include the number and the geometry of the blades, whether they are powered by wind approaching from the front or behind, their ability to rotate to face the wind, etc. [177]. In any case, the slender geometry and flexible nature of both the tower and the blades, and the fact that they are exposed to consistent and repeated wind forces, make them vulnerable to vibration and fatigue. Vibration could cause inefficient operation or catastrophic failure of parts, and fatigue leads to failure in long-term operation [178,179,180]. Observation of stress or vibration in these components is necessary, either as part of a control algorithm to reduce vibration or as a health-monitoring tool to minimize operation and maintenance costs. This can have a major effect on the economic potential of energy generation via wind turbines [181,182,183].

The flexible vibrations of a beam tower or blade can be modeled using the familiar linear Euler–Bernouuli model:

$$\frac{{\partial }^2}{{\partial }^{2}x}\left(\frac{EI(x)\partial w(x,t)}{{\partial }^{2}x}\right)+\frac{n_b\partial w(x,t)}{\partial t}+\frac{\rho A_t(x){\partial }^{2}w(x,t)}{\partial t^2}=f(x,t)$$

(35)

where $w(x,t)$ is the displacement of the beam as a function of the location along the axis, $x,$ and time, $t$. This model is sufficient if one assumes that the flexible structure is made of a single material with a nonuniform cross section, where $EI(x)$ is the modulus of rigidity multiplied by the area moment of inertia, $\rho$ is the mass density of the material, $A_t$ is the cross sectional area, and $nb$ is an internal damping constant. The external force applied to the flexible body is $f(x,t)$. While this model has been used and studied by generations of mathematicians, control and observation of this and other PDE models is typically approached using discretization of the infinite-dimensional model, with some exceptions, such as [184], which used a Lyapunov method for the semi-linear beam equation; [109], which used a backstepping approach for estimation of an undamped Euler–Bernoulli beam; [29], which used a backstepping approach for the related shear beam model; and [185,186,187,188,189], which developed observers for other related PDE beam models.

Turbine towers have been modeled as flexible beams for the purpose of control in [190], for example, and in [179], the authors modeled the nonuniform tower of a horizontal-axis type turbine using the damped Euler–Bernoulli model in Equation (35) and developed an on-observer-dependent vibration control technique to stabilize the system using a Lyapunov-based approach. The controller is dependent on a disturbance observer designed to estimate the time-dependent disturbance at the top boundary of the tower. In [191], the authors used a flexible beam model for the blades of a horizontal axis turbine and developed a boundary controller which is similarly dependent on a disturbance observer at the free boundary of the blade.

In the concluding part of this section, we outline some of the most significant research challenges for the observer design of distributed-parameter systems. Compared to the concentrated-parameter observer design techniques, the study of observers for distributed-parameter systems is still in its infancy. A relatively small number of papers have been published in the past decade on this topic, and many important issues have not yet been resolved. For example, the potential validity of the observer design separation principle has not been discussed and confirmed even for special or simple PDEs. What happens when the system PDE driven by feedback control based on state estimates coming from the PDE representing the observer dynamics (driven by the system measurements and the same control that drives the system) are connected together? Will we be able to design the system feedback gain (controller) and the observer feedback gain (observer) independently and keep estimation (observation) errors within acceptable limits? It is important to emphasize that both gains will be functions of the estimated states. This difficult question and its solution will heavily depend on the class of the PDE system, the type of boundary conditions, the choice of sensors (system measurements), and the choice of the observer’s initial conditions. Even if it is possible to find only one distributed-parameter system and identify very restrictive conditions under which the separation principle holds, this will be considered a very successful result. The design of sliding-mode observers for PDEs will also be a challenging and interesting future research topic. It is important to emphasize that observer design for PDEs has been considered only for full-order observers and that we have not seen yet an extension of the reduced-order observer design to this class of systems. This will be a significant future research task.

4. Observers for Electric Power Systems

Since their invention in the 1960s, observers have been used for electric power systems. We can find a large number of papers on observers for electric power systems, either for monitoring, measuring, observing, estimating electric power system state variables or for the purpose of feedback control. It seems that this research and development field has been saturated, so there have not been very many results published on observers for electric power systems in the past decade. The recent papers are mostly on power electronics and components of electric power systems. For example, papers on observers [192,193] have focused on supercapacitors. Reference [194] focuses on observers for a microgrid. Reference [195] addresses balanced connection to the power grid for a combination of a wind turbine, a synchronous generator, and a photovoltaic array. Reference [196] presents the use of observers for DC-DC converters. Several recent papers have dealt with the emerging issues in observer design: observers were used to solve various aspects of grid-connected inverters [197,198,199,200], in a neutral-point clamped (NPC) converter that acts as an interface between the grid and many energy systems [201], as well as in microgrids, as demonstrated in papers [202,203], where linear discrete-time (digital) Luenberger observers were used. The most recent advances and potential challenges in the use of nonlinear observers for energy systems were nicely discussed in a very recent dissertation [204] published in 2023.

5. Conclusions

The authors have presented an overview of research results and applications of observers for (renewable) energy systems. Since there is a very large number of papers on this topic, the overview is limited in scope and based on the research and practical experience of the authors in dealing with energy systems and their theoretical research experience with observers. For this reason, the overview is mostly based on papers published in high-quality journals within the past twenty years.

It is the authors’ opinion that observers are very powerful tools for all dynamic systems, especially energy systems, where there are many state variables and parameters that need to be estimated (observed, monitored). It is our opinion that the work performed so far is fairly extensive as far as full-order observers are concerned with respect to the estimation of state variables and parameters in most energy systems. What is missing in this area is more utilization of reduced-order observers, since they are simpler to design and more accurate than full-order observers. Only a few studies on the use of reduced-order observers for energy systems are available in the literature.

There are quite a lot of open problems regarding the use of full- and reduced-order observers for feedback control of energy systems. In particular, the renewable energy systems areas that can benefit more from the use of observers are flow batteries, photovoltaic systems, and wind energy systems. Presently, observers are powerful tools for monitoring the dynamics of energy systems and for diagnostic purposes, replacing sensors where they are either not available or not physically realizable, or too expensive to be produced and installed. By providing state estimates at all times, observers also facilitate the design of full-state feedback controllers. The main challenge in such cases is the availability of reliable and very accurate mathematical models for corresponding dynamic processes, since the design of observers is based on these models and the accuracy of observers (estimated state trajectories in time) depends on the accuracy of the mathematical models.

As far as classes of systems related to this overview that are not very well represented are concerned, definitely, distributed-parameter systems (systems whose dynamics are represented by partial differential equations) deserve much more attention from researchers and practitioners working on energy systems. Designing observers for this class of systems is much more challenging than for the lumped-parameter systems, and it seems, in general, that the research on observers for distributed-parameter systems is still in development (it has not reached maturity), so it will take some time before the corresponding observers become routinely implemented in energy systems. The challenges of the reduced-order observer design for PDEs and eventual proofs of the validity of the separation principle in some special cases seem to be the most significant future research tasks.