In this paper, we study a solid oxide fuel cell (SOFC) controlled by a multi-input multi-output (MIMO) compensator, which uses the blower/fan power and cathode inlet temperature as actuators. The usable power of the fuel cell (FC) is maximized by limiting the air flow rate deliberately when an increase in power is demanded. Possible rate bounds on the cathode inlet temperature are also modeled. These bounds could represent the physical limitations (due to slow dynamics of heat exchangers) and/or a control concept for accommodating the power saving objective. Applying proper limits to the amplitude and rate of the actuator signals, and incorporating antiwindup (AW) techniques, can raise the net power of the FC by 16% with negligible effects on the spatial temperature profile.

## Introduction

Conventional electricity generation using fossil fuels is neither efficient nor suitable in terms of the high pollution it produces. Increasingly, fuel cells (FCs) are used in a variety of applications for electricity generation [1,2]. The ultralow emission, zero noise pollution of fuel cells, and their high efficiency help fuel cells as strong candidates for the new generation of power plants. Fuel cells are classified based on their efficiency, operating temperature, and type of electrolyte. A popular class of fuel cells with efficient long-term operation and fuel flexibility is solid oxide fuel cells (SOFC), which is characterized by its solid electrolyte. SOFCs, operating at high temperature, are being used in a wide range of mobile and stationary applications taking advantage of the high quality heat byproduct, useful for reformation [3,4]. One of the main challenges in commercialization of SOFCs is to be able to safely control the system in the presence of disturbances such as power demand variation, and/or nonlinearities such as actuator saturation [5,6]. One of the objectives of this technical note is to ensure power following while controlling the resulting changes in the temperature profile, in transient operation of solid oxide fuel cells. This would require a high performance control technique for minimizing the temperature gradients and, thus, thermal fatigue and material damage.

For high efficiency and low degradation of the fuel cell due to thermal cycling, the fuel cell temperature should remain fairly constant during operation. This has led to increased attention to SOFC thermal control [7,8]. To meet the demands of developing control strategies, in Ref. [9], a control-oriented multi-input multi-output (MIMO) nonlinear thermal model of the SOFC is developed and a temperature controller is proposed. In Ref. [10], a systematic approach to the multivariable robust control of a hybrid fuel cell gas turbine plant is presented, and the transient operation of the system is studied using an empirical model. In a recent work [11], a model of the SOFC in frequency domain is developed for system control design and stability. Such models are useful to understand the fuel cell interface with remote power systems for applications such as those found in aviation, terrestrial vehicles, and naval transport industry. In Ref. [12], a dynamic states estimator is designed to track and predict the behaviors of unmeasurable states inside SOFC using stochastic filtering algorithms. In Ref. [13], dynamic programming strategies are proposed for a hybrid system to achieve an optimal schedule while minimizing fuel consumption. Chaudhari et al. [14] investigate the transient behavior of a 25 kW SOFC and an internal combustion engine hybrid power system and apply model predictive control techniques in order to determine control parameters and set-points. Ansari et al. [15] propose a neural network model for prediction of SOFC performance based on the Levenberg-Marquardt back propagation algorithm.

The use of compressor/turbine generators with fuel cells in a hybrid format is common [5,16]. Here, for a 5 kW SOFC, we use a variable speed blower for cooling purposes and assume that there exists an external reformer. The complex dynamics of the SOFC system is modeled and advanced control techniques are applied in order to achieve stable and safe operation while maintaining system performance. The objective is to maximize usable power of a 5 kW SOFC while minimizing the spatial temperature variations. We start by assuming that a controller is designed for SOFC to primarily meet this thermal control objective. By relying on a high-performance MIMO controller, proposed in Ref. [17], the nonminimum phase like behavior caused by the fan can be addressed; however, a temporary surge in blower power due to the inertia of the blade will result (overshoot in blower power demand). Large overshoots in the blower power lowers the net power of the FC available for external use. That would necessitate conservative operation of the fuel cell. In order to address this issue, we propose to limit the power sent to the blower with a fictitious saturation bound. While all actuators have a limited range of operation due to physical characteristics and constraints, the approach here creates a limit that is lower than the physical limitations, and attempts to leverage the thermal mass of the fuel cell. By preventing the blower from using too much power, more of the power of the fuel cell can be made available for external demand, while keeping a high-performance controller, for tracking smaller changes or when the power demand is reduced.

As in any other physical system, actuation limitations can cause severe performance degradation (even system failure) [18]. Of course, actuators can be saturated both in terms of the size of the input generated and the rate at which the input can change, i.e., the magnitude and rate actuator saturation, respectively. In this paper, for the first time, the usable power of the fuel cell is maximized by artificially bounding the inlet flow rate, thus, avoiding the overshoots in the blower power. Antiwindup (AW) control techniques are then incorporated in order to guarantee stability and provide a satisfactory performance. The effect of AW on cathode inlet temperature, as the second actuator signal, is studied and possible rate bounds are modeled and compensated using antiwindup magnitude and rate augmentations.

The paper outline is as follows: In Sec. 2, the nonlinear model of a solid oxide fuel cell in coflow configuration is presented followed by the controller, which is designed assuming ideal actuation (Sec. 3). The issue of maximizing usable power of a SOFC by scheduling the actuator's amplitude bounds is discussed in Secs. 4. In Sec. 5, the concept of actuator rate saturation to accommodate the maximized power with acceptable performance is discussed. Section 6 combines both magnitude and rate limitation on actuators and proposes an antiwindup design using the peak-to-peak bound minimization approach to improve the performance of the system constrained by saturation nonlinearities. Simulation results are presented in Sec. 6.2, evaluating the effect of the proposed AW augmentations on system performance. Section 7 concludes the paper.

## Solid Oxide Fuel Cell Model

Solid oxide fuel cells are composed of a solid electrolyte separating the two electrodes, cathode and anode, which are fed by the air and fuel, respectively. In contrast to the typical heat engines running due to a temperature difference, the chemical potential difference is the driving force of the fuel cells. As shown in Fig. 1, the chemical potential difference requires the fuel and air to be apart in anode and cathode, respectively. Under proper operating conditions, negatively charged oxygen ions from the cathode chamber are transferred to the anode chamber, through the separating layer of electrolyte. The release of electrons in cathode then leads to the current and formation of water.

The main product of the fuel cell is the clean desirable electricity directly converted from the chemical energy avoiding the thermal cycle. The voltage available from a single fuel cell is approximately 1 V; thus, a stack of a large number of fuel cells is always of interest.

An integrated nonlinear dynamic model of a SOFC system developed with multiple subsystems is studied here. The model is the same coflow SOFC investigated in several previous efforts including [17,19] and has been evaluated with experimental data. The system is a typical planar coflow SOFC in the 5 kW scale integrated with a variable speed blower. The model has sufficient spatial resolution (i.e., along the flow channels) to capture the effects of power demand variations. A schematic of the SOFC model is shown in Fig. 2 representing four control volumes: cathode and anode gas channels, positive-electrode electrolyte negative-electrode (PEN), and the top and bottom interconnecting plates, which are the same due to periodic boundary condition assumption.

For brevity, only the key features of the physical modeling are presented here and detailed discussions are provided in the references (e.g., in Refs. [17] and [19]). Conservation of mass, energy, and species, together with the convective and conductive heat transfer, form the basis of the SOFC dynamic model. The flow throughout the system is solved for temperature, pressure, molar flow rate, and species concentration (for $CH4,CO,CO2,H2,H2O,N2,O2$). For gas channel control volumes, the energy conservation equation is given by
$NCvdTdt=N˙inhin−N˙outhout+ΣQ˙in−ΣW˙out$
(1)
and species conservation
$NdXidt=N˙inXi,in−N˙outXi,out+R˙i$
(2)
where the exit molar flow rate $N˙out$ is given by
$N˙out=N˙in+ΣR˙i$
(3)
and the description of other variables is provided in the nomenclature. The solid-state PEN energy conservation equation for each control volume is also given by
$ρVCdTdt=ΣQ˙in−ΣW˙out$
(4)
which is used to solve for the temperature of the solid control volumes (PEN) based on the heat transfer through the control volumes. The parameter values are specified in Table 1.
Convection heat transfer between each stream and the plate is modeled using Newton's law of cooling. Also, Fourier's law is used to model conduction heat transfer along the heat exchanger plate. The governing equation of the blower is the shaft torque balance presented in the state-space form:
$Jwdwdt=Pblower+Pimpeller$
(5)
where Pblower is the motor power supplied to the blower and Pimpeller is the loss associated with the impeller, which is given by
$Pimpeller=1ηγRTambγ−1[(PoutPamb)γ−1/γ−1]$
(6)

where η is the blower isentropic efficiency (85%), γ is the air-specific heat ratio, R is the universal gas constant, Tamb and Pamb are the ambient temperature and pressure, respectively. The dynamic nonlinear model of a coflow SOFC studied here is spatially discretized into five nodes along the flow direction (Fig. 2). Each node includes four control volumes: cathode and anode gas channels, PEN, and the interconnecting plate, which adds up to 20 control volumes. A list of the 60 states of the SOFC model is given in Table 2, having N2 and O2 as anode flow species and $CH4,CO,CO2,H2,H2O,N2$ as cathode flow species.

## Controller Design

One of the main challenges in transient operation of the solid oxide fuel cells is achieving load following with minimum risk of damage. Power variations lead to large changes in temperature profile of the SOFC, which result in undesirable thermal fatigue and consequently serious damage to the structure of the fuel cell [20]. Thermal control design aims at minimizing the spatial temperature variation throughout the SOFC, while maintaining a reasonable performance in load following.

For controller design, the detailed nonlinear model of the SOFC, integrated with a variable speed blower, is first linearized around the nominal conditions, and its order is reduced by removing the unobservable and uncontrollable states, and those with small Hankel singular values, along standard techniques [21]. Compared to the performance of the original nonlinear system, the linearization error observed for a 15% change in power demand is around $1 K$, which is negligible compared to the operating temperature of the SOFC being around $1000 K$ [19]. For simplicity, we assume there is no feed-through term for u and w. The linearized reduced order plant is then given by
$δx˙p=Apδxp+B1δw+B2δup,δz=C1δxp,δy=C2δxp$
(7)

The δ variables represent the variations from the nominal operating conditions. The state vector $δxp∈ℝnp$ is the difference between the current state of the system from those at nominal condition, while $δup∈ℝnu$ is the control commands that would be added to the nominal values of the input to obtain the commands to the actuators. Other variables, i.e., the measurement outputs (sensors) $δy∈ℝny$, and the performance outputs $δz∈ℝnz$, similarly denote variations from the nominal conditions. The exogenous input $δw∈ℝnw$ is the reference signal, representing the change in power demand.

The block diagram of the SOFC control system is shown in Fig. 3. Table 3 summarizes the input and output signals associated with the SOFC model. According to the table and figure, cathode inlet temperature and blower power are inputs to the plant $up1$ and $up2, up=[up1 up2]⊤$, respectively. The anode outlet temperature, plate temperatures at the first, middle, and the last nodes, and the blower shaft speed are the measurement outputs, y. The electrolyte temperature at each of the five nodes are the performance outputs, z. An outer proportional–integral–derivative loop uses the power tracking signal to obtain the voltage needed and the nominal voltage is then used as the disturbance w.

The unconstrained MIMO controller is designed for spatial temperature control assuming ideal actuation using $L2$-gain (or $H∞$) approach [17]. The controller aims at controlling the actuators to minimize temperature deviations from nominal conditions whenever a power demand disturbance is encountered. The stable linear controller C, with same order as of the plant, is presented by
$x˙c=Acxc+Bcyδy,δu=Ccxc+Dcyδy$
(8)

## Actuator Magnitude Limitations

The fuel cell power is prone to large variations often due to the grid (or microgrid, for example) demand. Therefore, actuator saturation is likely, particularly for a high performance controller. Addressing the problems created by actuation saturation is one of the challenges in FC controller design. As discussed in Ref. [22], fast load following is possible under the assumption of ideal actuation. Under realistic actuation, however, power following degrades and the stability and performance are no longer guaranteed. The main actuation problems are the power needed by the blower to deliver the required cathode air flow, and the proper temperature needed at the cathode inlet. If the power demand change is large, it might lead to saturation problem for the blower power. On other hand, due to the inertia of the blower, the MIMO controller designed for the integrated SOFC and the blower model can cause relatively large overshoots in the blower power and consequently limit the available net power of the FC. Since the power of the blower is provided by the fuel cell, any power not used by the blower can be added to net power supplied by the FC. While we can reduce the overshoot, by using a less aggressive controller, we would like to avoid this solution particularly when power demand is lowered and the FC temperature is prone to a significant drop. This might lead to lower efficiency in ion transport and large temperature variations.

Alternatively here, in order to avoid the overshoot and increase the range of power available, we use an aggressive controller and rely on artificial saturation (enforced by software) that limits the power sent to the blower. This acts on the controller as a standard saturation bound, which can result in performance degradation or instability. The next step is then to develop antiwindup protection schemes, which help maintaining the stability as well as an acceptable performance for the fuel cell under such actuator saturation.

The power demand profile used in simulations in this paper corresponds to ±15% variations in power demand. Figure 4 shows the demand profile composed of a 15% decrease in the power at time t = 11,000 s, and then 30% of the nominal power increase at time t = 12,000 s. The times chosen for changes in power are far apart from each other and the start-up time, to focus on the dynamics of power set-point change. The simulation results are presented for the nominal conditions, given in Table 4, satisfying the temperature range often used as the most common electrolyte average temperature.

### Blower Power Enforced Saturation.

The idea of the enforced saturation, presented here, is to intentionally limit the air flow feeding the FC by putting an artificial bound on the power of the blower, which is lower than the actual physical constraints. The blower power command is thus subjected to a magnitude limit $m$ and the actuator commands would not exceed this bound. Therefore, for the second plant input (blower power), we require
$0≤up2≤ulim$
(9)
with ulim as a known positive constant, and $up2$ to be the blower power. The inputs to the plant up are thus modeled as
$up=[δu1+u1,nomsatm(δu2+u2,nom)]$
(10)
When saturation is not expected to happen frequently, it is possible to design a controller for the unconstrained system, to obtain a high performance controller. Then, augmentations can be introduced to the controller, known as AW compensation, which is responsible for maintaining the characteristics of the linear system in the absence of saturation, and guaranteeing stability together with an improved performance once saturation occurs. As a result, the small signal performance provided by the aggressive unconstrained controller is not compromised in order to achieve an acceptable large signal performance. Figure 5 shows the AW augmentations for the SOFC integrated model with blower power saturation. According to the figure, using the saturation block, $δu2+u2,nom$ is kept within the interval $[0 ulim]$; thus, the actuator signal is
$up2= 0 if δu2+u2,nom<0, δu2+u2,nom if 0≤δu2+u2,nom≤ulim, ulim if δu2+u2,nom>ulim$
(11)
The position of the saturation element is consistent with actuators with physical limits. To be more consistent with AW models, this element could be moved to the left of where $u2,nom$ is added to $δu2$. In that case, the saturation limit would be $−ulim−u2,nom$ and $ulim−u2,nom$. The asymmetry of the bound does not cause any difficulties in the AW development since that requires only the resulting dead-zone function q
$q=δu2+u2,nom−up2$
(12)
to have the same sign as the corresponding $δu2$, which holds as long as $ulim−u2,nom>0$. As a result, the antiwindup gains would be the same in either arrangement. Figure 5 shows how this antiwindup augmentation would be implemented, and we use the same setup for our simulations.

### Antiwindup Design With Magnitude Bound on Blower Power.

As depicted in Fig. 5, the objective here is to design an antiwindup augmentation that introduces suitable additive modification signals $ν1∈Rnc$ and $ν2∈Rnu2$ and augment the unconstrained compensator (8) as
$x˙c=Acxc+Bcyδy+ν1,δu=Ccxc+Dcyδy+[0ν2]$
(13)
These modifications should make the closed-loop system internally stable with a guaranteed input–output performance level in the presence of saturation nonlinearities in the control loop. In Fig. 5, since only the second output of the controller (blower power actuator) is subject to saturation, AW term ν2 is added only to the second row of the input vector u, which corresponds to the blower power as from Eq. (10)
$x˙p=Apxp+B1w+[B21 B22][up1up2−q]$
(14)
The static antiwindup block containing matrix gain
$AW(q)=−Λq$
(15)
is then applied to the dead-zone function defined in Eq. (12).
Following standard techniques in e.g., Refs. [23] and [24], and recalling the fact that here only one of the actuators is subject to saturation, it is straight forward to build the augmented system with state vector $x=[δxp⊤ xc⊤]⊤∈Rn$ and w and q as input signals. Given the linearized SOFC state-space model (7) and the unconstrained controller (8), the closed-loop system with magnitude antiwindup gains can be written as
$x˙=Ax+Bwδw+(Bq−BηΛ)q,δz=Czx+Dzwδw+(Dzq−DzηΛ)q,δu=Cux+Duwδw+(Duq−DuηΛ)q$
(16)
with further details omitted for brevity.

In order to guarantee the stability of the closed-loop system (16) and establish a performance bound for the AW design, an upper bound for the $L2$ (or Energy) gain γ, from the disturbance signal w to the performance output z, is minimized and the stabilizing AW gain Λ is obtained. The minimization is constrained by a linear matrix inequality ensuring the stability and performance bound γ (details omitted due to space limitation). While the algorithm presented is to bound the energy of the performance output (by a fix multiple of the energy of the reference input), other techniques that bound the peak norm (i.e., peak-to-peak or energy-to-peak) are straightforward and can be implemented in a similar fashion with ease [24]. Indeed, we use minimization of the upper bound for peak-to-peak gain in Sec. 6.

Setting aside a few percentage of power for the balance of plant (BoP) and other parasitic losses, the net power of the fuel cell available for external use is the difference between the total power and the power required for the blower
$Pnet=PFC−Pblower$
(17)

While the blower power for the nominal condition is 0.288 kW (from Table 4), increasing the overall power level requires higher air flow rates and thus higher blower power. Furthermore, the overshoot of the blower should also be taken into account. As shown in Fig. 6, the peak blower power (for both overshoot and steady operation at higher power levels) reaches close to 1 kW. Therefore, the maximum net power, which can be achieved form the fuel cell, is limited to 4 kW. Assuming a ±15% variation in the nominal power demand, the nominal value of the fuel cell net power should then be approximately 3.5 kW, in order to respect the 5 kW upper bound of the fuel cell capacity, i.e., $Pnet,max=3.5$ kW. In order to increase the maximum net power available by the fuel cell, the idea of the enforced saturation (bounding the blower power intentionally) is applied, and AW techniques are used to ensure stability and performance. The steady-state value of the blower power (0.66 kW) could be selected as the magnitude bound in order to avoid the excess power in overshoot. Due to the thermal mass of the FC, cutting the power for the short duration of the overshoot would not change temperature profiles significantly, while a %7 increase in the net power could be achieved.

A lower power bound results in more power saving, particularly if the higher power demand is not permanent and a decrease in tracking profile is expected. Therefore, in Fig. 6, the limit of 0.4 kW is selected for the blower power saturation element (recall the nominal value is 0.288 kW).

Figure 7 shows the total power of the fuel cell with or without limiting the blower power. Avoidance of the overshoot shown in Fig. 7 allows the nominal power of the fuel cell to be at 0.57 kW higher level. Adding this value to the maximum nominal net power of the FC, i.e., 3.5 kW, gives
$Pnet,max=4.07kW$
(18)
having ulim = 0.4 kW on Pblower. This level implies a 16% increase in the FC available net power. Therefore, by putting an aggressive limit on the blower power and taking advantage of the AW techniques, the same SOFC can be used for 16% higher power demands.

Without antiwindup, as shown in Fig. 8, the 16% additional power made available is at the cost of a large temperature increase while the signal is saturated. By limiting the blower power, not enough air is sent through, causing a large temperature rise along the cell, especially at the outlet. These changes in the temperature profile can have negative impact on the FC and may lead to degradation and thermal fatigue. However, taking advantage of the proposed AW techniques, the error can be reduced to the negligible amount of less than 2 K.

Figure 9 shows the variation of the cathode inlet temperature, the other actuator of the fuel cell system, when it is not subject to any saturation constraint. This figure shows that in order to address the blower power saturation, and reduce the error in the outlet temperature, the augmented controller avoids further heating up of the inlet using the other actuator signal ($Tcath,in$). As shown in Ref. [17], in coflow FC, without saturation constraint, both the air flow rate and cathode inlet temperature are increased in response to higher power demand. The elevated airflow rate is aimed at reducing the average temperature while the higher inlet temperature ameliorates potential thermal gradients along the cell.

In order to accommodate the instantaneous drop in inlet temperature observed in Fig. 9, an actuator with high rate of change is required. However, in the absence of ideal actuation assumption, the rate at which $Tcath,in$ can change is limited by the mechanism used (e.g., heat exchanger, mixing chamber). Under operational conditions, this may lead to actuator rate saturation, causing severe performance degradation. The rate antiwindup design introduced in Sec. 6.1 is used as the remedy, since an artificial rate bound can address this sudden drop in cathode inlet temperature, thus avoiding large temperature gradients. The objective is to extend AW protection schemes, which help maintaining the stability as well as an acceptable performance for the fuel cell under actuator both magnitude and rate saturations.

## Actuator Rate Saturation

The inlet air temperature is controlled by bypassing the air through a heat exchanger or a mixing chamber with possibly slow dynamics. As the SOFC is a high temperature fuel cell, there might not be a real magnitude bound on the temperature control signal for the cathode inlet. However, the rate at which this temperature can change, in order to satisfy the control commands, can be limited by the heat exchange mechanism or transport delays.

### Rate Model.

In order to study the effects of actuator rate saturation on the fuel cell performance, we first need to have access to the rate signal (not typically available) and apply rate limits. One common approach to model, the rate signal is to insert a first-order filter with gain K in the forward loop connecting the controller's output to the plant's input. As shown in Fig. 10, a first-order circuit with a saturation element is added before the actuator. As a result, the actuator signal is guaranteed to be rate bounded.

The gain K needs to be selected large enough in order not to affect the dynamics of the original system. Figures 11 and 12 illustrate the effect of rate model with different K values on inlet and outlet temperatures, respectively. In fact, gain K controls the delay in system response as this is a first-order filter that may influence the FC behavior. Higher K corresponds to smaller delay and thus faster response compared to the time constants of the original system. For relatively small values of gain K, as shown in Fig. 11, the delay is observed in the inlet of the fuel cell. For $K≥0.1$ temperature profiles throughout the cell remain unchanged after adding the rate model. Therefore, for the rest of the simulations in this paper, K = 0.1 is selected for the gain of the rate model. The signals are available for the antiwindup loop since the loop in Fig. 10 is a part of the compensator.

### Cathode Inlet Temperature Rate Saturation.

Figure 13 shows the block diagram of the SOFC control system with the first-order model added, having the first actuator subject to rate saturation. For the first plant input ($Tcath,in$), we require $|u˙p1|≤r$ with $r$ as a known positive constant with unit $K/s$. This model guarantees the cathode inlet temperature signal to be rate bounded.

Figure 14 shows the effect of different rate saturation bounds on the profile of the cathode inlet temperature. When an increase in power demand is requested, the unconstrained controller increases both inlet temperature and the airflow rate in order to minimize the spatial temperature gradients along the cell (dashed curve). However, as shown in the figure, a limited rate of change of $Tcath,in$ causes a lower slope in the inlet temperature rise.

Figure 15 shows the effect of different rate saturation levels on profile of the anode outlet flow temperature. As the rate bound gets tighter, inlet temperature rises slower; thus, a drop in temperature at the outlet is expected. Compared to the cathode inlet temperature (Fig. 14), the anode outlet temperature, Fig. 15, shows less sensitivity to rate saturation since it is located far from the point of restriction and the effects of the bound become less profound toward the end of the cell. The decrease in inlet temperature and the increase in outlet temperature may cause a minor increase in spatial temperature gradients from nominal conditions, particularly at the nodes near the inlet. To address this, we add a rate limited AW with details discussed in Sec. 6.

## Actuators Magnitude and Rate Saturation

The block diagram of FC control system with a magnitude bound on blower power and rate bound on cathode inlet temperature is shown in Fig. 16, which is a combination of Figs. 5 and 13. Rate-bounded cathode inlet temperature and magnitude-bounded blower power are modeled as
$xI˙=satr(K(u1−xI)),up2=satm(u2)$
(19)

where $K=diag(K1,…,Knu1)∈ℝnu1×nu1$ and xI is the integrator's state. In order to minimize the negative effects of rate saturation on fuel cell performance and guarantee the system stability, we rely on antiwindup design for both magnitude- and rate-bounded actuators.

### Antiwindup Design.

The objective here is to design an antiwindup augmentation $v=[v1⊤ v2⊤]⊤$ that introduces suitable additive modification signals $v1∈ℝnc$ and $v2∈ℝnu1$ to the unconstrained controller (8). According to Fig. 17, since the first output of the controller ($Tcath,in$) is subject to rate saturation, and the second output (Pblower) is magnitude bounded, the antiwindup term ν2 is added to the entire input vector u (see Eq. (13) for comparison). Therefore
$x˙c=Acxc+Bcyy+ν1,u=Ccxc+Dcyy+ν2$
(20)
The static antiwindup block containing matrix gains
$AWr(q)=−Λrqr,AWm(q)=−Λmqm,AW(q)=−Λmqm−Λrqr$
(21)
is applied to dead-zone function $q=η−satr(η)=η−xI˙$, and signal xI is defined as a new state variable with dynamics
$xI˙=K(u1−xI)−qr$
(22)
in which signal u1 is the first row of u in Eq. (20). The augmented closed-loop system with state vector $x=[δxp⊤ xc⊤ xI⊤]⊤$, and w and q as input signals in then presented as
$x˙=Ax+Bwδw+(Bqm−BηΛm)qm+(Bqr−BηΛr)qr,δz=Czx,δup=Cux+Duwδw+(Duq−DuηΛm)qm+(Duq−DuηΛr)qr$
(23)
with system matrices presented in Appendix  B. Antiwindup gains Λm and Λr are then obtained by solving a convex optimization problem subjected to performance and stability constraints (inspired by the result presented in Ref. [25]). Further details are provided in Appendix  B.

### Results.

In this section, the behavior of an SOFC under the magnitude and rate actuator bounds discussed in Sec. 6 is studied through simulations. The objective is to achieve a %16 rise in the FC net power by artificially bounding the inlet flow rate, considering the possible physical bounds on the rate of change of the inlet temperature due to slow dynamics of heat exchanger/mixing chamber. The reasonable bound of $0.1 K/s$ is selected as the rate limit with gain K = 0.1 based on the discussions in Sec. 5.1.

Figures 18 and 19 show the performance of the optimized system with maximum net power, compared with the original system. Figure 19 shows the temperature gradients from the nominal conditions at each node along the cell. As discussed in Sec. 2, the cell is discretized into five nodes along the flow direction: node 1 located at the inlet toward node 5 placed at the outlet. The required magnitude and rate bounds are applied to the actuators in order to achieve a %16 increase in the net power. The maximized power is available at the cost of at most $5 K$ increase in temperature gradients, from nominal conditions, along the cell.

As an alternative approach, in Ref. [26], a magnitude lower bound was used for cathode inlet temperature, avoiding it to drop significantly due to AW compensation. Here, the rate limit on the inlet temperature, which could be due to physical constraints and/or control design purposes, addresses this issue and keeps the temperature gradients within an acceptable interval. The overall performance here is superior to the one achieved in Ref. [26]. As an example, the rate bound eliminates excessive temperature gradients along the cell.

## Conclusions

The power of the cooling fan of an SOFC is intentionally limited, in order to increase the fuel cell net power available for external use. Possible rate limits on cathode inlet temperature, as the second actuator signal, are also simulated and their effects on the performance of FC are studied. The existing MIMO controller used for spatial temperature control is then augmented using antiwindup techniques to recover the performance of the unconstrained system. A 16% increase in the FC usable power is achieved by applying suitable actuator limits and incorporating antiwindup techniques, only at the cost of less than $5 K$ variations in temperature gradients.

## Funding Data

• NSF Grant No. CMMI-1461583.

## Nomenclature

• C =

controller

•
• N =

molar capacity

•
• $N˙$ =

molar flow rate

•
• P =

plant

•
• Pblower =

blower power

•
• Pimpeller =

impeller loss

•
• t =

time

•
• $Tcath,in$ =

cathode inlet temperature

•
• $X˙$ =

concentration of species

•
• η =

blower isentropic efficiency

### Appendix A: Antiwindup Design for Multi-Input Multi-Output Systems With a Magnitude Bounded Actuator

The system matrices in Eq. (16) are given by
(A1)

where B22 and D122 are the second columns of B2 and D12, and $Dcy2$ is the second row of Dcy. The system matrices are slightly different from the general magnitude AW design due to the fact that here only a subset of input signals are subject to saturation (only the blower power, u2, and not the cathode inlet temperature, u1).

Algorithm. [Magnitude AW for MIMO systems with partially bounded actuators: Energy gain approach] [22,26]. Consider the plant and the controller introduced in Eqs. (7), (8), and (10), as well as the magnitude limit $m$. Given any solution to the optimization problem
$minQ,M,X,γ γ$
(A2)
subject to the linear matrix inequality constraints
$(QA⊤+AQ***Bw⊤−γI**CzQDzw−γI*Φ4,1Duw2MmDzq⊤−X⊤Dzη⊤Φ4,4)<0,Q>0, M=W−1, X=ΛM,Φ4,1=MBq⊤−X⊤Bη⊤+Cu2Q,Φ4,4=−2M+Duq2M+MDuq2⊤−Duη2X−X⊤Duη⊤$
(A3)
with the AW gain $Λ=XM−1$, the augmented closed-loop system (16), has a guaranteed $L2$ gain of γ from w to z.

Inequality (A3) provides the stability condition for the saturated system with antiwindup gain Λ using the standard Lyapunov stability approach.

### Appendix B: Antiwindup Design for Multi-Input Multi-Output Systems With a Magnitude and a Rate-Bounded Actuator

The system matrices in Eq. (23) are given by
(B1)

(B2)

Here, we assume that a possibly conservative estimate of disturbance signal w(t) is known. Thus, peak-to-peak approach is applied to the current fuel cell saturation problem.

Algorithm. [Magnitude and Rate AW for MIMO systems with partially bounded actuators: Peak-to-peak approach] Assume that this system is only exposed to peak-bounded disturbances with known upper bound $wmax∈ℝ$, i.e., $w⊤(t)w(t)≤wmax2$. Given the saturation model (19) with the magnitude and rate limits $m$ and $r$, let us assume that for a given $0<α<(|Re(λmin(A))|/2)$, there exists a solution for
$minQ,Mm,Mr,Xm,Xr,Ym,Yr,γ2 γ2(QA⊤+AQ+Qα***Bw⊤−αI**Φ¯3,10−2Mm*Φ¯4,1KDuw1Φ¯4,3Φ¯4,4)<0$
(B3)

$(QQCz⊤CzQ γ2/wmax2)>0, (r2/wmax2YrYr⊤Q)>0,$
(B4)

$Φ¯3,1=MmBqm⊤−Xm⊤Bη⊤+[0 0 I]Q,Φ¯4,1=MrBqr⊤−Xr⊤Bη⊤+K[Cu1−[0 0 I]]Q−Yr,Φ¯4,3=DuqMmK−KDuη1Xm,Φ¯4,4=−2Mr−KDuη1Xr−KXr⊤Duη1⊤$

Parameter α can be selected by performing a typical line search. The antiwindup gains satisfying the stability and performance are then given by $Λm=XmMm−1,Λr=XrMr−1$.

Using the peak-to-peak Lyapunov approach, the stability of the magnitude and rate limited system is guaranteed by inequalities (B3) and (B4) right, while the left inequality in Eq. (B4) provides the optimized performance measure γ. Further technical details could be found in Ref. [25].

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