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Cell voltage and power density as a function of current density. Solid lines: ˙ n fuel,in = 10 −3 mol sec ; Dot-dashed lines: ˙ n fuel,in = 1.2 × 10 −3 mol sec .
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On-line control and optimization can improve the efficiency of fuel cell systems whilst simultaneously ensuring that the operation remains within a safe region. Also, fuel cells are subject to frequent variations in their power demand. This paper investigates the real-time optimization (RTO) of a solid oxide fuel cell (SOFC) stack. An optimization pr...
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... plot of cell voltage and power density as a function of the current density (I-V curve) is shown in Figure 2 for fuel inlet flow rates of 10 −3 mol sec (or mass flow density of 6 ml min cm 2 ) and 1.2 × 10 −3 mol sec (7.2 ml min cm 2 ), and an excess air ratio λ air = 3. further, would result in a sharp dip in power due to increase in the overpotential losses. To deliver a higher power, it is necessary to increase the fuel inlet flow rate. ...
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Carbon-based solid oxide fuel cell (C-SOFC) is an effective approach for electric power generation because of its simplicity. The basic principle of the carbon-based fuel cell is electrochemical oxidization of solid carbon to CO 2 on the anode. CO 2 produced can further react with carbon fuel to generate CO via Boudouard reactions (C+CO 2 → 2CO). T...
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... On the other hand, our proposed algorithm handles this dual problem as part of an online optimization solution requiring no pre-or post-processing steps. Similarly, while others have proposed robustified RTO algorithms, these have primarily emphasised robustness against modeling errors and uncertainty in the steady-state input-output map [27][28][29]. However, these approaches have overlooked the critical aspect of robustness to implementation errors at the control layer. ...
... As mentioned, the biomass productivity of the reactor, subject to steady-state conditions, defines the objective function for the RTO problem where the decision variable is the biomass concentration set-point which will be sent to an underlying PI controller. The nominal problem is given as: (27) at steady-state (29) However, as noted, we assume no knowledge of the steady-state plant. Instead, we assume that steady-state measurements of the dilution rate and biomass concentration exist which can then be used to calculate the biomass productivity of the reactor by way of equation (28). ...
... Table of parameter values used for equation(27). ...
Real-Time Optimization (RTO) plays a crucial role in the process operation hierarchy by determining optimal set-points for the lower-level controllers. However, these optimal set-points can become inoperable due to implementation errors, such as disturbances and noise, at the control layers. To address this challenge, in this paper, we present the Adversarially Robust Real-Time Optimization and Control (ARRTOC) algorithm. ARRTOC draws inspiration from adversarial machine learning, offering an online constrained Adversarially Robust Optimization (ARO) solution applied to the RTO layer. This approach identifies set-points that are both optimal and inherently robust to control layer perturbations. By integrating controller design with RTO, ARRTOC enhances overall system performance and robustness. Importantly, ARRTOC maintains versatility through a loose coupling between the RTO and control layers, ensuring compatibility with various controller architectures and RTO algorithms. To validate our claims, we present three case studies: an illustrative example, a bioreactor case study, and a multi-loop evaporator process. Our results demonstrate the effectiveness of ARRTOC in achieving the delicate balance between optimality and operability in RTO and control.
... CA is built upon iterative updates of the model constraints on the basis of plant measurements for the purpose of tracking the active plant constraints [12,17,32]. If the set of active constraints were known, a simple multivariable feedback controller would enforce them. ...
... Since the optimization problem (3.3) considers the plant at steady state, it was originally proposed in Refs. [12,17,32] to compute the modifiers as ...
This paper describes an optimization strategy for operating solid-oxide fuel-cell systems at optimal efficiency. Specifically, we present the experimental validation of a real-time optimization (RTO) strategy applied to a commercial solid-oxide fuel-cell system. The proposed RTO scheme effectively pushes the system to higher levels of efficiency and maintains the system there despite perturbations by tracking active constraints. The optimization approach uses either steady-state measurements, or transient measurements in combination with a dynamic model, and can deal effectively with plant-model mismatch. In the reported experiments, the approach drives the system to the desired power demand at optimal efficiency. The experimental fuel-cell system reached 65% DC electrical efficiency. As such, the proposed RTO scheme is a promising candidate for enforcing optimal micro-CHP operation. In addition, the approach can deal with slow drifts such as degradation without compromising on efficiency. Finally, and important from a practical point of view, we suggest guidelines for safe and optimal operation.
... Furthermore, as these systems are often affected by slow drifts associated with component degradation, the optimal steady-state operating point changes with time. This motivates the use of real-time optimization (RTO) for optimizing and controlling the operation of SOFC systems [3]. ...
... As an alternative, modifier adaptation (MA) has been developed to enforce convergence to the plant optimum even in presence of structural plant-model mismatch [18]. Note that, if the plant optimum lies on active constraints, it is possible to use a simpler version of MA, called constraint adaptation (CA), that simply corrects the constraints in the model-based optimization problem by means of bias terms computed from the differences between the measured and predicted values of the constraints [3,10,19]. ...
There still exists a large gap between simulation work and industrial applications in the context of control and optimization of solid-oxide fuel-cell (SOFC) systems. In an effort to bridge this gap, this study describes the experimental implementation of steady-state real-time optimization (RTO) to an SOFC system that consists of both hardware and software components. The proposed adaptive optimization scheme uses an approximate steady-state model of the fuel-cell system and corrects it “appropriately” so that it becomes “excellent” for optimization. This way, the plant can be steered efficiently toward optimality, while meeting the varying electric power demand. In these experiments, the plant efficiency was increased from 55% to 62% through application of RTO. Furthermore, although the SOFC system is characterized by slow thermal dynamics that may take a few hours to settle to steady state, it has been possible to reduce the time necessary to reach the power setpoint from 1 h to about 5 min thanks to the use of transient measurements and a dynamic model. This experimental work has shown that it is possible, not only to control the SOFC system at a desired operating point, but also to operate it near optimality despite changes in power demand.
... Other examples of local control on fuel cells and in combination with gas turbines or magnetic energy storage systems are shown in [44][45]; while the system configurations are different, the holistic approach is similar to what is adopted in this study. Marchetti et al. [46] has simulated a robust real-time optimization on a solid oxide fuel cell stack, and followed through with a constrained model predictive controller. However, the optimization work is restricted to the fuel cell stack and does not consider other connected components. ...
While macroscopic modelling and dynamic analyses are topics approached by many to improve the efficiency of hybrid energy systems, the optimization and control of operational strategies need to be studied to allow practicality of such configurations in an industrial setting. Simple implementation and reliable operability of any critical sub-system component of a hybrid energy module are critical requirements for the development of their control algorithms. In this paper, a state-of-the-art hybrid renewable energy system consist of different sub-systems is presented with resolved economically optimal dispatch strategies. The local control of one of its key exemplary sub-system, a Solid Oxide Fuel Cell (SOFC), is realized via two methods i.e. a model-based approach of shrinking-horizon Model Predictive Control (MPC) with constraints and a data-driven approach using Virtual Reference Feedback Tuning (VRFT) to identify controller parameters. Fixed current load and fixed conversion efficiency operations are assessed respectively for 1-h and 24-h spans. For both simulation cases, results show that both control methods provide robust power tracking performance within constrained boundaries with marginal to no overshoot and the VRFT method achieving stability at least 10 times faster than the MPC method. While the two juxtaposed methods are diametrically different, they are both technically viable and efficient under industrial power loads. The main contribution of this work is to demonstrate the feasibility of the hybrid energy system by providing the practitioners with a broad range of local control options, depending on operational realities and constraints.
... Recently, Ramadhani et al. reviewed many optimization studies on SOFC, and summarized common decision variables, objective functions, constraints, and methods [3]. Marchetti et al. have performed realtime optimization of SOFC stack [4]. Hajabdollahi and Fu optimized the configuration of SOFC-GT cogeneration plant using MOO approach [5]. ...
Chemical process optimization problems often have multiple and conflicting objectives, such as capital cost, operating cost, production cost, profit, energy consumptions, and environmental impacts. In such cases, multi-objective optimization (MOO) is suitable in finding many Pareto optimal solutions, to understand the quantitative tradeoffs among the objectives, and also to obtain the optimal values of decision variables. Gaseous fuel can be converted into heat, power, and electricity, using combustion engine, gas turbine (GT), or solid oxide fuel cell (SOFC). Of these, SOFC with GT has shown higher thermodynamic performance. This hybrid conversion system leads to a better utilization of natural resource, reduced environmental impacts, and more profit. This study optimizes performance of SOFC-GT system for maximization of annual profit and minimization of annualized capital cost, simultaneously. For optimal SOFC-GT designs, the composite curves for maximum amount of possible heat recovery indicate good performance of the hybrid system. Further, first law energy and exergy efficiencies of optimal SOFC-GT designs are significantly better compared to traditional conversion systems. In order to obtain flexible design in the presence of uncertain parameters, robust MOO of SOFC-GT system was also performed. Finally, Pareto solutions obtained via normal and robust MOO approaches are considered for parametric uncertainty analysis with respect to market and operating conditions, and solution obtained via robust MOO found to be less sensitive.
... Based on the results related to the efficiency optimization, a new stack control strategy was proposed to achieve temperature safety and high efficiency in steady state and with power switching transients. In Ref. [13], a real-time optimization strategy which combines the real-time optimization of electrical efficiency and model predictive control was verified on a dynamic model of the SOFC power system. Simulation results showed that near-optimality can be obtained and constraints can be respected despite model inaccuracies and large variations in load demand. ...
... Simulation results showed that near-optimality can be obtained and constraints can be respected despite model inaccuracies and large variations in load demand. In Ref. [14], the experimental validation of the real-time optimization strategy proposed in [13] was presented. It was shown that the real time optimizer, which applies the static model of the system, was able to converge quickly and safely towards the optimum efficiency. ...
... Experimental validation of the model is provided in [20]. The model was already used for testing the real-time optimization strategies for SOFC power system [13,14]. The important aspects of the model are briefly described in the sequel. ...
One of the main challenges for wide-spread utilization of the solid oxide fuel cell (SOFC) power systems is how to achieve high electrical efficiency without increasing the degradation rate of the fuel cells. To run the SOFC power system at high efficiency over a long period of time, properly designed controllers are indispensable.
Although a number of various approaches to control SOFC have been proposed so far, it seems that the design of control system, along with simple tuning procedure, has not been treated in a consistent manner. This issue is addressed in the present paper resulting in a feedforward-feedback control structure. The feedforward part is based on the stoichiometry of electro-oxidation, reforming and combustion reactions, which allow immediate response to variable current demand. The feedback part performs additional fine adjustment of fuel and air supply in order to minimize the undesired system temperatures variations. The selection of pairings of manipulated and controlled variables for control is based on physical knowledge of the system. Input/output pairing for single-loop feedback control is assessed by the relative gain analysis. An efficient procedure for tuning the parameters of the feedback controllers is suggested, relying on simple open-loop step responses of the system.
The proposed low-level control is assessed on a detailed physical model of a 2.5 kW SOFC power system by simulating two nonstationary load regimes. Simulations show that the control provides a robust operation under large load variations while meeting the operating constraints. Due to its simplicity, the control is feasible for implementation on a real SOFC system.
... Improvements of SOFCs have been conducted based on modeling and simulation studies with the aim to achieve better performance [17][18][19][20][21][22][23][24][25]. Research in this topic has been developed in several areas involving improved model and observer studies [26][27][28][29][30][31], advanced estimation and identification [32,33], precision of control and management [34][35][36][37][38][39][40], and optimization of design and operation [41][42][43][44][45][46][47][48][49][50][51][52][53][54]. These previous studies frequently mentioned optimization of SOFC as a recent research topic compared to other fuel cell types such as PEMFC and MCFC [55]. ...
... Can solve non-linear problem Often the solution found is only a local optimum [51,177,181] Sequential Quadratic Programming (SQP) Used for solving non-linear constrained optimization problem ...
... Most of the studies in combined approach are related to the hybrid between optimization and control as well as optimization with management applied for operating the SOFC. Also, several works in the literature have shown these hybrid approaches between optimization and prediction in SOFC microstructure and stack models [51,103,107]. An interesting topic regarding the hybrid approaches has been performed by optimizing the control and prediction parameters [55,182,194]. ...
Solid Oxide Fuel Cells (SOFCs) have become the promising energy source for stationary and portable applications. The design of SOFCs including its structure and its operation are an important part to be properly optimized in order to achieve the best performance. The topic on optimization of SOFC has gained tremendous attention recently in line with the growing applications of fuel cell. This paper involves a literature survey of the application of SOFC in particular the optimization strategies. The strategies are reviewed in detail based on five features, which are the decision variable, objective analysis, constraint, method and tools. The future trends of research related to the optimization for SOFC are also discussed to provide better insight for future research work in this area.
... This means that the robust solution calculated will either remain feasible for all possible realizations of the uncertain parameters, or the feasibility can be recovered with a low cost. With this advantage, RO has gained an increasing interest in engineering optimization applications during the past few years [1,7,13]. Some previous works have been devoted to find robust solutions to the blending problem. ...
The blending operation is a key process in alumina production. The real-time optimization (RTO) of finding an optimal raw material propor tioning is crucially important for achieving the desired quality of the product. However, the presence of uncertainty is unavoidable in a real process, lead ing to much difficulty for making decision in real-time. This paper presents a novel robust real-time optimization (RRTO) method for alumina blending operation, where no prior knowledge of uncertainties is needed to be utilized. The robust solution obtained is applied to the real plant and the two-stage operation is repeated. When compared with the previous intelligent optimiza tion (IRTO) method, the proposed two-stage optimization method can better address the uncertainty nature of the real plant and the computational cost is much lower. From practical industrial experiments, the results obtained show that the proposed optimization method can guarantee that the desired quality of the product quality is achieved in the presence of uncertainty on the plant behavior and the qualities of the raw materials. This outcome suggests that the proposed two-stage optimization method is a practically significant approach for the control of alumina blending operation.
... Sometimes, experimental relationships are known to be either convex or concave in certain decision variables -i.e., if one were to keep all of the variables except a particular u i constant, the experimental relationship between the function and u i would either be convex or concave. For example, the relationship between power and current in a fuel-cell system, whose nature is almost always concave (Marchetti et al., 2011), is precisely such a case. ...
... While this particular example is somewhat trivial, there are others that are less so. For example, in the solid oxide fuel-cell system where one manipulates the current while needing to respect a lower-limit constraint on the cell voltage (Marchetti et al., 2011), it is known that the relationship between voltage and current is always inverse proportional, thus allowing one to set the corresponding κ as 0. And although the physical laws could not tell us the value of κ in this case, one could nevertheless rely on the multitude of available experimental data to come up with a good estimate. For the polymerization example of Section 2.4, one may know that both extended heating (increased u 1 ), together with shortened cooling (decreased u 2 ) and thus increased temperature, will generally lead to a favorization of the termination reactions and shorter polymer chains, thus resulting in a final product with a lower molecular weight. ...
Given a certain response surface function, its Lipschitz constants may be defined as the limits on its largest derivatives (sensitivities), and as such provide important information about the function's behavior. In this paper, it is argued that the explicit use of these constants may prove useful in the domain of experimental optimization. Most notably, it is shown that they offer a simple and robust way of rigorously guaranteeing constraint satisfaction, but may also be exploited to remove suboptimal portions of the decision-variable space, to filter out noise/error from the obtained experimental measurements, and to serve as regularizing bounds in the experimental derivative estimation problem. Because the constants are often conservative, a number of performance-improving refinements are outlined. Finally, some techniques for estimating the constants in practice are proposed.
... Even if a feasible operating point is guaranteed to be reached upon convergence, the iterates may follow an infeasible path, as seen in the simulated example. One can combine modifier adaptation with on-line control to enforce constraint satisfaction at all times (Marchetti et al., 2011). More work in that direction is needed. ...
The steady advance of computational methods makes model-based optimization an increasingly attractive method for process improvement. Unfortunately, the available models are often inaccurate. An iterative optimization method called "modifier adaptation" overcomes this obstacle by incorporating process information into the optimization framework. This paper extends this technique to constrained optimization problems, where the plant consists of a closed-loop system but only a model of the open-loop system is available. The degrees of freedom of the closed-loop system are the setpoints provided to the controller, whereas the model degrees of freedom are the inputs of the open-loop plant. Using this open-loop model and process measurements, the proposed algorithm guarantees both optimality and constraint satisfaction for the closed-loop system upon convergence. A simulated CSTR example with constraints illustrates the method.
