Networked Control of Distributed Energy Resources: Application to

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Ind. Eng. Chem. Res. 2009, 48, 9590–9602

Networked Control of Distributed Energy Resources: Application to Solid Oxide Fuel Cells Yulei Sun, Sathyendra Ghantasala, and Nael H. El-Farra* Department of Chemical Engineering and Materials Science UniVersity of California, DaVis, One Shields AVenue, DaVis, California 95616-5294

This paper presents a model-based networked control approach for managing distributed energy resources (DERs) over communication networks. As a model system, we consider a solid oxide fuel cell (SOFC) plant that communicates with the central controller over a bandwidth-constrained communication network that is shared by several other DERs. The objective is to regulate the power output of the fuel cell while keeping the communication requirements with the controller to a minimum in order to reduce unnecessary network utilization and minimize the susceptibility of the SOFC plant to possible communication disruptions in the network. Initially, a feedback control law is designed to regulate the power output of the SOFC plant at a desired set-point by manipulating the inlet fuel flow rate. Network utilization is then reduced by minimizing the rate of transfer of information between the fuel cell sensors and the central controller without sacrificing the desired stability or performance properties. To this end, a dynamic model of the SOFC plant is embedded in the controller to approximate the dynamics of the plant when measurements are not transmitted by the sensors, and the state of the model is updated using the actual state that is provided by the SOFC plant sensors at discrete time instances. When full-state measurements are not available, an appropriate state observer is included in the control structure to generate state estimates from the measured outputs, which are then used to update the model states. An explicit characterization of the maximum allowable transfer time between the sensor suite of the SOFC plant and the controller (i.e., the minimum allowable communication rate) is obtained under both state and output feedback control in terms of plant-model mismatch and the choice of control law. The characterization accounts for both stability and performance considerations. Finally, numerical simulations that demonstrate the implementation of the networked control architecture and its disturbance handling capabilities are presented. 1. Introduction With progressing utility deregulation and the increasing need to eliminate unnecessary transmission and distribution costs, reduce greenhouse gas emissions, and improve the availability and reliability of electrical networks, a transformation in energy production is emerging from traditional, central generation to new, distributed generation. Distributed generation refers to the integrated or stand-alone use of small, modular electric generation deployed close to the point of consumption. Examples of power sources that fit the definition of distributed generation include internal combustion engines coupled with generators, microturbines, fuel cells, and renewable systems such as photovoltaic arrays and wind turbines. In many applications, distributed generation technology can provide valuable benefits for both the consumers and the electric distribution systems.1 The small size and modularity of distributed energy resources (DERs) encourage their utilization in a broad range of applications, including generating the base-load power, as in the case of variableenergy DERs, providing additional reserve power at peakload intervals, providing emergency or back-up power to increase the stability and reliability of important loads, supplying remote loads separated from the main-grid system, and supporting the voltage and reliability by providing power services to the grid. The downstream location of DERs in distribution systems reduces energy losses and allows utilities to postpone upgrades to transmission and distribution facilities. Approximately 60000 megawatt (MW) of small-scale * To whom correspondence should be addressed. E-mail: nhelfarra@ ucdavis.edu. Tel.: (530) 754-6919. Fax: (530) 752-1031.

distributed generators, defined as under 10 MW, are on line in North America.2 They are mostly diesel generators and reciprocating engines, while fuel cells, microturbines and solar arrays are finding increasing markets. As the number and diversity of DERs on the grid increases, dispatching these resources at the right time and accounting for the flow of energy correctly become complex problems that require reliable monitoring and telemetering equipment, as well as reliable control and communication between DERs and loads. A distributed power network with thousands of small generators requires far more sophisticated communications and control systems than a radial grid focused on a few big plants. Traditional supervisory control and data acquisition systems with centralized control rooms, dedicated phone lines, and specialized operators, are not cost-effective to handle a large number of DERs spread over the grid. Advanced communication and control technologies are needed to enable the integration and interoperability functions of a broad range of DERs. These technologies offer a digitally controlled, “smart” electricity network with broadband communication capabilities. According to some estimates,3 the market potential for advanced control and communications technologies in managing DERs (based on 5-10% energy savings achieved) is between $3.75 billion and $7.5 billion domestically, and between $15 billion and $30 billion worldwide. From an environmental standpoint, each kilowatt-hour saved through real-time metering and demandside management is expected to offset about 590 g of carbon dioxide. The same offset applies to renewable resources. While managing DERs over communication networks offers an appealing modern solution to the problem of controlling distributed energy generation, it also poses a number of

10.1021/ie9008869 CCC: $40.75  2009 American Chemical Society Published on Web 09/16/2009

Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009

challenges that must be addressed before the full economic and environmental potential of DERs can be realized. These challenges stem in part from the inherent limitations on the information transmission and processing capabilities of communication networks, such as bandwidth limitations, networkinduced delays, data losses, signal quantization, and real-time scheduling constraints, which can interrupt the connection between the central control authority, the distributed generation units and the loads, and consequently degrade the overall control quality if not properly accounted for in the control system design. Despite the availability of fast and reliable communication networks, the fact that the distributed power market is primarily driven by the need for super-reliable, high-quality power implies that the impact of even a brief communication disruption (e.g., due to local network congestion or server outage) can be substantial. In sites such as hospitals, police stations, data centers, and high-tech plants which cannot afford blackouts, millisecond outages that merely cause lights to flicker will cause costly computer crashes. Such high-stakes risks provide a strong incentive for the development of robust control and communication strategies that ensure the desired levels of power supply and quality from DERs while minimizing the susceptibility of the overall system to data losses and communication outages. Over the past decade, several efforts have been made toward the development and implementation of control strategies for DERs.4-7 Important contributions in this direction include the use of conventional and model-based feedback control algorithms to regulate various types of grid-connected DERs in order to enhance power system stability,8,9 mitigate power quality problems10 and improve the continuity of electricity supply,11 as well as the development of various distributed control and coordination architectures using multiagent based control approaches.12,13 While the focus of these studies has been mainly on demonstrating the feasibility of the developed control algorithms, an important issue that has not been addressedsboth in the formulation and the solution of DER control problemssis the explicit characterization and management of communication constraints which are tied to the inherent limitations on the information transmission and processing capabilities of the communication media used to manage the information flow to and from the DERs. As the need for large-scale deployment and integration of DERs into power networks continues to drive the transition from dedicated, point-to-point connections to multipurpose shared communication networks, there is a greater need to understand the fundamental issues introduced by this technology and to develop systematic methods for their integration and handling in the controller design framework. A natural framework within which these problems can be addressed is the emerging paradigm of networked control systems. These systems differ from classical control systems in that their control loops are closed around communication networks. The flexible architectures and reduced installation and maintenance costs associated with these systems have motivated their use in a broad range of applications such as mobile sensor networks, remote surgery, automated highway systems, unmanned aerial vehicles, and process control.14,15 Not surprisingly, the study of control over communication networks has attracted considerable attention in the literature including, for example, studies on the stability of networked control systems,16-18 the development of scheduling algorithms,19-21 the analysis and compensation of data losses using asynchronous22-24 and hybrid25 system formulations, the design of networked control systems tolerant to faults,26,27 and the design of quasi-decentralized networked control systems for large-scale

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multiunit process systems (see also for other examples of recent works on control of process networks). At this stage, however, a comprehensive framework for the analysis and handling of communication constraints in the control of DERs over communication networks remains lacking. Motivated by these considerations, we focus in this work on the problem of controlling DERs over communication networks. As a model system, we consider a solid oxide fuel cell (SOFC) plant that communicates with the central controller over a resource-constrained communication network that is shared by many other DERs. A model-based networked control approach is presented to regulate the power output of the SOFC plant while keeping the communication requirements with the central controller to a minimum. The rest of the paper is organized as follows. Following an overview of the problem formulation and solution methodology in section 2, a SOFC model is presented in section 3 and used in sections 4 and 5 to design the networked control structure under full-state and output feedback control formulations, respectively. An explicit characterization of the minimum allowable rate at which the sensors of the SOFC plant must communicate with controller to maintain the desired closed-loop stability and performance properties is obtained for each case, and numerical simulations that demonstrate the efficacy of the networked control architectures are presented. Finally, concluding remarks are given in section 6. 2. Problem Formulation and Solution Methodology We consider an array of DERs managed by a higher-level supervisor over a resource-limited communication network as depicted in Figure 1. Each DER is modeled by a continuous-time system with the following general state-space description: x˙i(t) ) fi(xi(t)) + Gi(xi(t)) ui(t) yi(t) ) hi(xi(t), ui(t)) where xi(t) ∈ IRni denotes the vector of state variables associated with the ith DER (e.g., exhaust temperatures and rotation speed in turbines and internal combustion engines, operating temperature, and pressure in fuel cells), yi(t) ∈ IRqi is the vector of measured and/or controlled outputs (e.g., output power, voltage, and frequency), ui(t) ∈ IRmi denotes the vector of manipulated inputs associated with the ith DER (e.g., inlet fuel flow rate in fuel cells, shaft speed in turbines), and fi( · ), Gi( · ) and hi( · ), are smooth nonlinear functions of their arguments. In the hierarchical structure of Figure 1, each DER has local (on-board) sensors and actuators with some limited built-in intelligence that gives the DER the ability to run autonomously for periods of time when no communication exists with the remote software controller (the supervisor). The local sensors in each DER transmit their data over a shared communication channel to the supervisor where the necessary control calculations are carried out and the control commands are sent back to each DER over the communication network. On the basis of load changes, variations in utility grid power prices, and the state and capacity of each DER, the supervisor coordinates local power generation. The supervisor is also responsible for monitoring the operational health of DERs, issuing alarms and shutting down DERs if alarms go unheeded by operators, and automatically scheduling and dispatching DERs in an economically optimal manner.

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Figure 1. Management of DERs over a shared communication network.

One of the main problems to be addressed when managing a large number of DERs over a communication network is the large amount of bandwidth required by the different subsystems sharing the communication channel. Optimal control and coordination between the different DERs to meet changes in power demand is best achieved when information (e.g., measurements, control commands) flow continuously between each DER and the supervisor. In traditional control architectures, the dedicated point-to-point connections make the information available continuously. In a networked control system, however, the feedback path is a digital network which typically has limited bandwidth and transfers information in a discrete fashion. This gives rise to a trade-off where attaining maximal control performance requires frequent communication on the one hand, while minimizing unnecessary network resource utilization (e.g., to reduce network congestion and communication outages) favors limited communication, on the other. A proper characterization and management of this trade-off is an essential first step to the design of resource-aware networked control and communication strategies that ensure the desired performance while respecting the inherent constraints on the resources of the communication medium. To address this problem, we will focus in this work on minimizing the sensor-controller communication costs under the assumption that the actuators and the controller are collocated (i.e., the network exists between the sensors and the controller; generalizations to account for actuator-controller communication constraints are possible and the subject of other research work). To this end, we will consider the following approach: • Initially design for each DER an appropriate feedback control law that regulates its output (in the absence of communication constraints) at the desired set-point decided by the supervisor. • To reduce the unnecessary utilization of the shared communication network, reduce the collection and transfer of information between each DER and the supervisor as much as possible to limit the bandwidth required from the network and free it for other tasks (e.g., other control loops using the network and/or noncontrol information exchange processes) without sacrificing the desired stability and performance properties of the individual DERs and the overall system.

When communication is suspended, a model of the DER is used to generate the necessary control action, and when communication is re-established, the model state is updated using the actual measurements. • Obtain an explicit characterization of the maximum allowable transfer time between the sensor suite of each DER and the controller, which is the time between successive information exchanges. In general this time is different for each DER and depends on the degree of mismatch between the dynamics of each unit and the model used to describe it. The following sections demonstrate the application of this methodology on a solid oxide fuel cell plant example. 3. Application to a Solid Oxide Fuel Cell Fuel cells are important distributed resources owing to their high efficiency, low levels of noise and environmental pollution, and flexible modular designs that match versatile demands of customers. As an illustrative example, we consider in this work a stack of solid oxide fuel cells (SOFC) as a representative DER in a power distribution system. 3.1. SOFC Model. A solid oxide fuel cell is an electrochemical device that generates electrical energy from chemical reactions. It consists of two porous electrodes, an anode and a cathode in contact with a solid metal oxide electrolyte between them. Hydrogen-rich fuel is fed along the surface of the anode where it releases electrons that migrate externally toward the cathode. The electrons combine with oxygen in air that is fed along the surface of the cathode to form oxide ions. These ions diffuse through the electrolyte toward the anode where they combine with the H+ ions to form water. The Nernst equation describes the potential difference between the electrodes that drives the reaction and the movement of electrons, and is given by ∆E )

[

(0.5) RTs pH2 pO2 ln ∆E0 + 2F pH2O

]

(1)

where ∆E0 is the standard cell potential, F is Faraday’s constant, and pH2, pO2, pH2O are the partial pressures of hydrogen, oxygen, and steam, respectively. Typically, a


number of these cells are connected in series to form a stack, which can be used as a stand alone DER. The overall stack voltage is then given by Vs

[(

)]

1 1 ) N0∆E - r0 exp R T - T I s 0

(2)

where N0 is the number of cells in the stack, r0 is the internal resistance at T0, R is the resistance slope, and I is the load current. In eq 2, only ohmic losses are included, while activation and concentration losses are neglected. Under standard modeling assumptions9,32 (e.g., ideal gases, constant pressure inside the gas channel, choked orifice, uniform temperature distribution and ideal mixing of gas inside the fuel cell stack, negligible heat capacities of the fuel and air, and negligible heat loss to the surroundings), a dynamic model of the following form can be derived for the SOFC stack from material and energy balances:32 Species balances: Ts (qHin - KH2pH2 - 2KrI) p˙H2 ) * τH2T*KH2 2 Ts (qOin - KO2pO2 - KrI) p˙O2 ) * (3) τO2T*KO2 2 Ts (qHin O - KH2OpH2O + 2KrI) p˙H2O ) * τH2OT*KH2O 2 Energy balance: Tin 1 T˙s ) qin Cp,i(T) dT i T ref msCps ˆ ro - VsI - 2KrI∆H

∑ ∫

∑q ∫ out i

Ts

Tref

Cp,i(T) dT (4)

where i ) H2, O2, H2O, pi is the partial pressure of component i, Ts is the stack temperature, qin i is the inlet molar flow rate of component i, ms and Cps are the mass and average specific heat of fuel cell materials excluding gases, Cp,i is the specific heat ˆ ro is the specific heat of reaction, I is of gas component i, ∆H the load current, τi* ) V/KiRT* is a time constant for ith component, Ki is the valve molar constant for component i, and Kr ) N0/4F. The values for the various model parameters can be found in Table 1.32 Throughout the paper, the system of eqs 3-4 will be referred to as the SOFC plant. Remark 1. It should be noted that while the above bare fuel cell model of eqs 3-4 is used throughout the paper to illustrate the design and implementation of the proposed networked control methodology, the results of this paper can also be applied to the complete fuel cell plant which includes (besides the fuel cell stack) additional supporting units such as a reformer and an AC-DC converter (power conditioning unit) to interface the fuel cell with the grid. The focus of this study, however, is not on developing a comprehensive model of the fuel cell plant, but rather on using a simplified (but representative) model that exhibits sufficient complexity to be suitable for illustration purposes. Furthermore, the dynamics of the fuel cell itself are typically slower (and thus are more dominant) than the dynamics of the supporting electronic devices such as the AC-DC converters. Remark 2. As can be seen from eqs 2-4 and Table 1, the main source of temperature nonlinearity in the model is the dependence of the stack voltage Vs on the stack temperature Ts (since the heat capacities, Cp,i, are treated as constants over the range of operation). Since the dependence of Vs on Ts follows an Arrhenius law (similar to the Arrhenius

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Table 1. Process Parameters and Steady-State Values for the SOFC Model ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

N0 KH2 KO2 KH2O R r0 T0 P R F T* Tref Tin ms Cps Cp,H2 Cp,O2 Cp,H2O ˆ ro ∆H τ*H2 τ*O2 τ*H2O xHs 2 xOs 2 xHs 2O T ss qHin,s2 qOin2

384 8.43 × 10-4 2.52 × 10-3 2.81 × 10-4 -2870 0.126 973 12 8.314 96485.34 1273 298 973 83059.2 0.4 31.248 35.84 48.28 -2.418 × 105 26.1 2.91 78.3 0.3973 0.3142 0.2951 1162.83 5 10

kmol/(atm · s) kmol/(atm · s) kmol/(atm · s) Ω K atm KJ/(kmol · K) s · A/mol K K K g J/(g · K) J/(mol · K) J/(mol · K) J/(mol · K) KJ/kmol s s s

K mol/s mol/s

dependence of chemical reaction rates on temperature), the effects of temperature nonlinearity can potentially be significant. As will be shown in the next section, a linear feedback controller can nonetheless be quite adequate to regulate the SOFC plant at the desired set-point. 3.2. Control Problem Formulation for the SOFC Plant. Referring to the SOFC plant of eqs 3-4, the control objective is to regulate the power output of the fuel cell stack at a desired set-point by manipulating the inlet fuel flow rate. The set-point is assumed to be determined by the supervisor based on its knowledge of the load changes in the distributed power network that it manages (this typically requires solving an optimization problem in real-time to coordinate power generation between the DERs and determine the optimal setpoint for each one,33 and is beyond the scope of the current work). Measurements from the SOFC plant are collected and sent to the central controller where the control action is calculated and sent back to the actuator to effect the desired change in power output. To simplify the controller design and implementation, we consider the problem on the basis of the linearization of the fuel cell plant around the desired set-point. Linearizing the plant model around the desired steady state yields x˙ ) Ax + Bu

(5)

where x and u are the state and manipulated input vectors for the plant, respectively, defined by

[ ] xH2 - xHs 2

x)

xO2 - xOs 2

xH2O - xHs 2O

,

u ) qHin2 - qHin,s2

Ts - Tss

where xi is the mole fraction of component i, the superscript s denotes the steady state values of the corresponding states and input, and A, B are constant matrices given by

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[

-0.035 0 A) 0 -8.928

[ ]

0 -0.314 0 -28.673

0 0 -0.012 -3.257

]

0 0 , 0 -0.011

0.0035 0 B) 0 0.635 To regulate the power output of the fuel cell in the absence of communication constraints, a stabilizing feedback controller of the form u ) Kx, where K ) [-18.4 0 0 0.047], is designed to enhance the speed at which the fuel cell meets the desired power demand from the supervisor. Beyond enhancing the response speed, feedback control is also needed to maintain robust operation in the presence of external disturbances. Figure 2 simulates the startup behavior of the fuel cell plant when a load current of I ) 500 A is applied to the stack. The dashed profiles show the closed-loop responses of the stack temperature, the power output, the mole fraction of H2 and the inlet fuel flow rate under the chosen controller gain. For comparison, the open-loop profiles (solid lines) are also included in the plots. It can be seen that, under feedback control, the stack temperature and power output reach their desired set-point faster than in the case of the uncontrolled plant. Remark 3. While not considered explicitly in this work, in practice there are constraints on the inlet fuel flow rate which typically arise from the limited capacity of the control actuators. Such constraints, if tight enough, could influence the stability and performance properties of the control system and must then be accounted for explicitly in the controller design and closed-loop stability analysis. One approach to deal with this problem without redesigning the feedback

controller is to limit the operating region of the plant to an invariant subset of the region where the controller satisfies the constraints. Other approaches that account explicitly for input constraints, such as bounded control and model predictive control, can also be used but lead to substantial complications in the analysis of the networked closed-loop system and preclude the derivation of explicit stability and performance assessment criteria (see Remark 7 for a discussion of the issues that arise when considering both control and communication constraints). In the next two sections, we describe how the feedback control strategy is tailored to take communication constraints in the sensor-controller link explicitly into account. We begin in section 4 with the case when full-state measurements are assumed to be available. Beyond illustrating the main ideas, the results of this case will serve as the basis for tackling the output feedback control problem in section 5. 4. Networked Controller Design and Implementation under Full-State Feedback When considering a networked control system of the type depicted in Figure 1, state measurements from the SOFC plant can be received by the controller only through the network. To reduce unnecessary network usage, we embed a dynamic model of the fuel cell in the supervisor to provide it with an estimate of the evolution of the states of the fuel cell when measurements are not available. The use of a model at the controller/actuator side to recreate the dynamics of the fuel cell allows the on-board sensors of the fuel cell to transmit their data at discrete time instances and not continuously (since the model can provide an approximation of the fuel cell dynamics); thus allowing conservation of network resources. The computational load associated with this step (e.g., solving the model equations and performing the control

Figure 2. Evolution of the stack temperature, the power output, the mole fraction of H2 and inlet fuel flow rate under open-loop (solid) and closedloop (dashed) control in the absence of communication constraints between the fuel cell and the controller.


calculations) is justified and supported by the increasing capabilities of modern software control systems used by the central control authority. Feedback from the fuel cell is then performed by updating the state of the model state using the actual state that is provided by its sensors at discrete time instances. The model-based controller is implemented as follows: t ∈ [tk, tk+1) u(t) ) Kxˆ(t), xˆ(t) ) Aˆxˆ(t) + Bû(t), t ∈ [tk, tk+1) xˆ(tk) ) x(tk), k ) 0, 1, 2, · · ·

(6)

where xˆ is an estimate of x, Aˆ and Bˆ are estimates of A and B, respectively, and tk is the kth transmission time. The model state is used by the controller for as long as no measurements are transmitted over the network, but is updated (or reset) using the true measurement whenever it becomes available from the network. Remark 4. In the networked control architecture considered here, only the sensors of the SOFC plants are assumed to have interruptible access to the central controller over the network, while the controller-actuator communication links are considered to be flawless. In situations where the communication between the control actuators of the plant and the central controller may be interrupted (e.g., due to a malfunction of the central controller or because of network access problems), the proposed networked controller needs to be complemented with local (on-board) control systems that can continue to regulate the SOFC plants at the desired set-points and prevent potential instabilities and performance deterioration resulting from the loss of communication with the central control authority. 4.1. Characterizing the Minimum Allowable Communication Rate over the Network. Our first objective is to determine the largest update period that guarantees plant stability. This corresponds to the minimum rate at which state measurements need to be collected from the SOFC plant and transmitted to the controller over the network. To this end, we define the augmented state vector ξ ) [xT eT]T, where the estimation error is given by e ) x - xˆ, and let tk, k ) 0,1,2, · · · , be the instants when the model states are updated such that the update period tk+1 - tk ) h is constant (generalizations to the problem of time-varying update periods are possible and the subject of other research work). Then it can be shown18 that the closed-loop system of eqs 5-6 can be cast in the following form: ξ˙ (t) ) Λξ(t), x(tk) ξ(tk) ) 0

[ ]

t ∈ [tk, tk+1) (7)

where e(tk) ) 0, for k ) 0,1,2, · · · , since the model states are updated at tk, and Λ)

[

A + BK -BK A˜ + B˜K Aˆ - B˜K

]

(8)

where A˜ ) A - Aˆ and B˜ ) B - Bˆ are the model errors. It can also be shown18 that the system described by eqs 7-8 with initial condition ξ(t0) ) [xT(t0) 0]T ) ξ0 has the following response: ξ(t) ) eΛ(t-tk)(IseΛhIs)kξ0,

for t ∈ [tk, tk+1)

with tk+1 - tk ) h, where Is )

[ ] I O O O

(9)

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and that a necessary and sufficient condition for the stability of this system is to have all the eigenvalues of the test matrix M(h) ) IseΛhIs restricted inside the unit circle. This requirement ensures stability by limiting the possible growth of the closed-loop state during the intervals when communication is suspended and no measurements are transmitted (notice that the term Mk, which appears in the closed-loop response, is a direct consequence of the periodic nature of the sensor-controller transmission). By examining the above expressions, it can be seen that the eigenvalues of M depend on the mismatch between the model and the plant, the controller gain, and the update period. Remark 5. Hybrid (combined discrete-continuous) system tools have been used successfully in the past to formulate and solve important process control problems such as faulttolerant and scheduled control.34-36 The hybrid system formulation of eqs 7-8 differs from previous formulations not only in the problem that it is designed to address but also in the fact that the discontinuous behavior is caused by a jump in the evolution of the augmented state itself (recall that the error between the model state and plant state is reset to zero at transmission times) and not by a switch in the governing dynamics or the control action (i.e., the right-hand side of the differential equation). The hybrid system of eqs 7-8 is therefore more appropriately referred to as a jump hybrid system. To investigate the effect of model uncertainty on the stability of the networked SOFC plant, we consider as an example parametric uncertainty in CpH2 and define δ1 )(CpH2m - CpH2)/ CpH2, where CpH2m is a nominal value used in the model, as a measure of model accuracy (any other set of uncertain parameters can also be considered and analyzed in a similar fashion). Figure 3 panels a and b depict the dependence of the maximum eigenvalue magnitude, λmax, on both δ1 and the update period. In the contour plot a, the area enclosed by the unit contour lines represents the stability region of the linearized plant. In general, the range of stabilizing update periods shrinks as the model uncertainty increases. Equivalently, the range of tolerable parametric uncertainty shrinks as the update period is increased. The predictions of Figure 3a are confirmed by the closed-loop temperature and power profiles in Figure 4 which show that the plant can be stabilized at the desired set-point when the networked control system is operated at a point inside the unit contour zone (δ1 ) 5, h ) 10 s), and becomes unstable when the operating point is chosen outside the unit contour zone (δ1 ) 5, h ) 15.5 s). In the latter case, the sensor-controller communication rate is insufficient to compensate for the significant model inaccuracy, thus leading to instability and fluctuations in the delivered power output. Figure 3b shows the maximum eigenvalue magnitude of the matrix M versus the update period for different values of δ1. As expected, the maximum allowable update period (the one corresponding to λmax ) 1) decreases as the model uncertainty increases. For comparison, included in this plot also is the case when a zero-order hold scheme is used. In this case, the controller holds the last measurement received from the plant until the next time a measurement is transmitted and received through the communication network. This corresponds to using a model with Aˆ ) O and Bˆ ) O. It is clear from the plot that a model-based control scheme with relatively accurate models can yield update periods larger than the zero-order hold scheme. In the case of large plant-model mismatch, however, the zeroorder hold scheme can outperform its model-based counterpart and allow stabilization with smaller sensor-controller communication rates.

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Figure 3. (a) Dependence of the maximum eigenvalue magnitude of the test matrix M, λmax, on plant-model mismatch and the update period, h. (b) Dependence of λmax on h when different compensating models and a zero-order hold scheme are used.

Figure 4. Fuel cell stack temperature and power output profiles under state-feedback networked control with model uncertainty δ1 ) 5 and different update periods.

The dependence of λmax (M) on h depicted in Figure 3b for a given model essentially reflects the trade-off between the effects of communication and model errors on closed-loop stability. For sufficiently small h, the rate of communication is high enough to compensate for model inaccuracies (and therefore the closed-loop system is stable and λmax < 1). This is intuitively expected since the controller is designed to be stabilizing under continuous communication. For sufficiently large h (beyond some critical value which depends on the specific structure of Λ), the destabilizing effects of model errors overpower the limited corrective control action provided by the measurement updates, leading to instability (where λmax > 1). It is interesting to note that within the stabilizing range of update periods λmax starts at 1 and then decreases initially as h increases. This can be understood from the structure of the test matrix M ) IseΛhIs, which suggests that the maximum eigenvalue of M will always start at 1 when h ) 0, and therefore λmax will decrease initially before it eventually increases for large values of h. Note, however, that this behavior does not mean that the plant becomes “more stable” or that the states converge “more quickly” as h increases since the closed-loop system exhibits combined discrete-continuous dynamics and the speed of the hybrid system response is not dictated solely by the eigenvalues of M. In addition to model accuracy, the choice of controller design parameters plays an important role in deciding the maximum allowable update period. To this end, we have also investigated the effect of varying the controller design parameters on the size of the feasible update period for a fixed model uncertainty. Specifically, the parametric uncertainty was fixed at δ1 ) 5, and the eigenvalues of the compensated fuel cell plant were

varied. For simplicity, only the first closed-loop eigenvalue of the plant, which we denote by λ1, was varied while leaving the other three poles fixed. Figure 5a is a contour plot showing the dependence of the maximum eigenvalue magnitude on λ1 and h. The stability region is the one enclosed by the unit contour lines. Note that as the eigenvalue becomes more negative (i.e., a more aggressive controller is used), the range of allowable update periods decreases. The prediction of Figure 5a is further confirmed by the closed-loop temperature profile in Figure 5b, which shows that the plant can be stabilized at the desired setpoint with h ) 15.5 s when λ1 is increased from its original value of -0.05 to -0.04 such that the networked control system is operated at a point inside the unit contour zone. Remark 6. It can be seen from the structure of the matrix Λ in eq 8 that the stability condition requires knowledge of both the true plant dynamics A (which are generally unknown) and the approximate model dynamics Aˆ (which are fixed by the chosen model). While the true plant dynamics may not be directly available, in many cases one can obtain certain bounds on the extent of model uncertainty (e.g., the ranges of possible values that process parameters can take) whether through experiments, preliminary simulations, physical insight, or a combination thereof. These bounds can be used to estimate the plant-model mismatch, Aˆ, which in turn can be used to estimate the true A and apply the stability criterion to determine the range of allowable update periods using the method described in this section. Remark 7. When input constraints are present, the maximum allowable update period between the sensors and the controller will depend not only on the accuracy of the model but also on the size of the control constraints. To see this, recall that the


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Figure 5. (a) Dependence of λmax on the update period and the closed-loop eigenvalues under the model-based scheme with δ1 ) 5. (b) Fuel cell stack temperature profile under state-feedback networked control with update period h ) 15.5 s, model uncertainty δ1 ) 5, and different closed-loop eigenvalues.

notion of a minimum data transmission rate is essentially a measure of the networked control system’s tolerance to the errors resulting from the lack of direct feedback from the sensors over a given period of time. Therefore, when a model is used to compensate for the unavailability of measurements, the minimum transmission rate becomes primarily a function of how well the model captures the true plant dynamics. When control constraints are present, however, the maximum tolerance of the control system to those errors over the same period of time may diminish due to the limited availability of control action. The reason is that actuator constraints impose fundamental limitations on the set of initial conditions starting from where closedloop stability can be achieved (stability region or nullcontrollable region), and therefore if the communication rate based on the unconstrained system is used, the model estimation errors could drive the closed-loop system trajectory during periods of communication suspension to a point outside the stability region from where stabilization is not possible using the limited control action available. Note that even in the absence of sensor-controller communication disruption, input constraints impose limitations on the size of the null-controllable region for the non-networked closed-loop system.37 The size of this region shrinks even further when communication constraints are introduced. In this sense, one can analyze the effect of communication constraints on the “loss” of stability region as compared to the null-controllable region of the non-networked plant. An explicit characterization of the stability properties of the constrained networked closed-loop system requires techniques from nonlinear and hybrid systems38 and is the subject of other research work. Remark 8. In addition to providing a solution to the problem of control over networks, the results presented in this work can be used to analyze other conceptually similar problems involving the on/off loss of communication between the sensor and the controller, such as the problems of measurement sampling and data losses due to sensor faults. The latter problem was recently investigated in ref 24 where the robustness of the control system to (possibly nondeterministic) data losses that arise because of malfunctions in the measurement sensors was analyzed, and a lower bound on the maximum rate of data losses that is sufficient to guarantee closed-loop stability was derived. A key difference that distinguishes the approach pursued in our work is the fact that the sensor-controller communication disruptions considered here are not due to sensor faults but are introduced purposefully (on a deterministic and periodic basis) to reduce the extent of network utilization. Another important difference is that the bound on the data loss rate obtained in ref 24 holds for the

case when the last available measurement is used during periods of data losses. In our work, however, we consider a more general setup in which a dynamic model of the plant is used to compensate for the lack of direct measurements during periods of communication suspension. 4.2. Incorporating Performance Considerations. In addition to stability considerations, the performance of the networked fuel cell plant under external disturbances is of major concern. Our objective in this section is to assess the performance of the networked closed-loop system under disturbances and characterize its dependence on the update period to determine a suitable communication rate that ensures not only stability but also minimal influence of the disturbances on the power output. To this end, we first rewrite the linearized plant in the following form to account explicitly for the presence of external disturbances: x˙ ) Ax + Bu + Ew z ) Fx + Gu

(10)

where w is the external disturbance input and z is the performance output signal of interest. After some manipulations, the networked closed-loop system can be formulated as ) Λξ(t) + Hw(t), t ∈ [tk, tk+1) x(tk) k ) 0, 1, 2, · · · , h ) tk+1 - tk ξ(tk) ) , 0 z(t) ) Nξ(t) (11)

ξ˙ (t)

[ ]

where H ) [ET ET]T and N ) [F +GK -GK]. As the performance index, we will use the extended H2 norm of the performance output.39 This is an H2-like norm that is suitable for analyzing periodic systems and captures the 2-norm of the performance output when an impulse disturbance is introduced in the input at t ) t0 (see ref 39 for the theoretical details and for other types of performance measures that can be used as well). In this case, the response of the closed-loop system to an impulse disturbance w ) δ(t - t0) can be expressed explicitly as z(t) ) NeΛ(t-tk)(IseΛh)kH,

t ∈ [tk, tk+1)

and the extended H2 norm, |G|H2, is given by |G| H2 ) trace(HTXH)1/2

(12)

where X is the solution of the discrete Lyapunov equation:

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Mp(h)TXMp(h) - X + Wo(0, h) ) 0 with Mp(h) ) IseΛh, and Wo(0,h) is the observability Gramian computed as Wo(0, h) )

∫

h ΛTt T

0

c

N NcΛt dt

To test the performance of the networked control system, we initialize the closed-loop SOFC plant at the desired set-point and introduce a unit impulse disturbance in the inlet flow rate of air, qOin2. The fuel cell stack temperature Ts is chosen as the performance output. Figure 6a is a plot of the extended H2 norm of the stack temperature as a function of the update period when a model of the plant with uncertainty δ1 ) 5 is embedded in the controller. It can be seen that the optimal update period that minimizes the size of the extended H2 norm (and hence minimizes the effect of the disturbance on the stack temperature and ensures fastest recovery from the disturbance) occurs near h ) 5 s (as expected this is less than the maximum allowable update period needed for stability which was predicted in Figure 3a). Note that the performance degrades as the update period is increased beyond the optimal value, which is intuitively expected given that the controller has to rely for longer periods of time on the uncertain plant model which is reset only infrequently. Note also that the performance degrades as h falls below the optimal value. This indicates that, for sufficiently small h, the performance of the model-based networked control system is better than that of the non-networked control system (where h ) 0 and measurements are transmitted continuously). This apparent improvement in performance for small h is due to the fact that when h is increased slightly beyond zero, the controller begins to rely on the embedded model of the fuel cell to generate the control action, which leads to a more aggressive control

action for short periods of time. These “bursts” of control action help the performance output recover faster from the disturbance and converge more quickly to the steady state (relative to the case of the non-networked plant). The aggressive control action based on the uncertain model, together with the fact that the model is updated frequently for small h, result in a faster recovery from the disturbance and hence a smaller norm. This trend, however, is expected and observed only when h is within a small range where the destabilizing effects of the model uncertainty are not dominant due to the frequent measurement updates. As h is increased past the optimal value, the length of the time intervals during which feedback is lost becomes large enough such that the destabilizing effects of the model uncertainty begin to dominate and interfere with the ability of the plant to recover from the disturbance resulting in a slower recovery (larger norm) and eventually instability as h reaches its critical limit predicted from the stability analysis. The predictions of Figure 6a are confirmed by the stack temperature, power output, and inlet fuel flow rate profiles shown in Figure 6b-d, which shows that the closed-loop plant with an update period h ) 5 s exhibits an improved response (faster recovery from the disturbance) than the one obtained with an update period h ) 15 s. This is consistent with the fact that the value of the extended H2 norm at h ) 5 s is smaller than it is at h ) 15 s. Remark 9. While in many fuel cell systems the inlet air flow rate can be used as an additional manipulated variable, only the inlet fuel flow rate is used in the simulation example as the manipulated variable since it is sufficient to achieve the desired control objective. The choice to introduce an impulse disturbance in the inlet air flow rate is made to demonstrate how the modelbased networked controller works to minimize the effect of the external disturbance on the closed-loop plant. The proposed

Figure 6. (a) Dependence of |G|H2 on the update period using a model with model uncertainty δ1 ) 5. (b-d) Closed-loop stack temperature, power output, and inlet fuel flow rate profiles under the networked control system for different update periods, when an impulse disturbance is introduced in the inlet air flow rate.


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Figure 7. (a) Dependence of λmax on plant-model mismatch for various update periods under output feedback control. (b) Maximum stabilizing update period using different compensating models and a zero-order hold scheme.

design and analysis methodology, however, is not limited to this choice of disturbances but can be applied to evaluate disturbances from other sources including those from fluctuations in inlet temperatures and electrical loads.

that the closed-loop system with initial condition ξ(t0) ) [xT(t0) jxT(t0) 0]T ) ξ0 has the following response:

5. Networked Controller Design and Implementation under Output Feedback Control

where

In this section, we address the output feedback control problem where the full-state of the fuel cell plant is unavailable for measurement and only the stack temperature can be transmitted from the fuel cell to the controller. To address this problem, a state observer of the following form is designed and embedded within the sensor: x¯˙ ) (Aˆ - LC)xj + Bû + Ly

(13)

where jx is the observer-generated estimate of x, L is the observer gain (chosen as L ) [0 0.0041 - 0.007 0.8008]T in the simulations), y is the measured output of the plant defined as y ) Ts - Tss, and C ) [0 0 0 1]. In this setup, the controller continues to rely on the plant model for as long as communication over the network is suspended. When communication is re-established, however, the observer state (instead of the plant state) is transmitted over the network and used to update the model state in the controller. The model-based networked output feedback controller can then be implemented as follows: u(t) x¯˙(t)

t ∈ [tk, tk+1) ) Kxˆ(t), ) (Aˆ - LC)xj(t) + Bû(t) + Ly(t), t ∈ [tk, tk+1) t ∈ [tk, tk+1) ) Aˆxˆ(t) + Bû(t),

xˆ˙ (t) xˆ(tk) ) jx(tk),

k ) 0, 1, 2, · · · (14)

Substituting the above control law into eq 5, and defining the augmented state variables as ξ(t) ) [xT(t) jxT(t) e(t)]T and the error variable as e(t) ) jx(t) - xˆ(t), it can be shown that the networked closed-loop system under output feedback control can be cast in the form of eq 7, where

[

A BK Λ ) Λo ) LC Aˆ - LC + BˆK LC -LC

-BK -BˆK Aˆ

]

(15)

Notice that the error variable is now redefined as the difference between the model state and the observer state and is reset to zero at each transmission time. Following an analysis similar to that presented in section 4.1 for the state feedback case, it can be shown

ξ(t) ) eΛo(t-tk)(IoeΛohIo)kξ0,

for t ∈ [tk, tk+1)

[ ]

(16)

I O O Io ) O I O O O O

and h ) tk+1 - tk, and that a necessary and sufficient condition for the stability of the output feedback controlled networked closedloop system is to have all the eigenvalues of the following matrix M(h) ) IoeΛohIo restricted inside the unit circle. This stability condition captures the dependence of the minimum allowable communication rate on the plant-model mismatch as well as the controller and observer design parameters. 5.1. Stability-Based Analysis and Controller Design. Figure 7a-b depicts the dependence of the maximum eigenvalue magnitude of M on both the plant-model mismatch and the update period. In the contour plot a, the area enclosed by the unit contour lines represents the stability region of the linearized plant. As expected, a trend similar to that in the state feedback case is observed whereby the range of tolerable parametric uncertainty shrinks as the update period is increased. The predictions of Figure 7a are further confirmed by the closedloop temperature and power profiles in Figure 8 which show that the networked closed-loop plant is stable when operated at a point inside the unit contour zone (δ1 ) 5, h ) 8 s), and unstable when operated outside this region (δ1 ) 5, h ) 12.8 s). Notice also that, for the same plant-model mismatch, the maximum allowable update period is slightly less under output feedback control than it is under full-state feedback control. This reflects the limitations introduced by the observer errors in updating the model state, which require an increased level of sensor-controller communication beyond what is required under full-state feedback. In addition to studying the effect of plant-model mismatch, we have also investigated the effect of varying the observer design parameters on the size of feasible update period for a fixed model uncertainty. To this end, the parametric uncertainty was fixed at δ1 ) 5 and the eigenvalues of the matrix Aˆ - LC were varied. For simplicity, only the third eigenvalue of the matrix, which we denote by λ3, was varied while keeping the other three poles fixed. Figure 9a is a contour plot showing the dependence of the maximum eigenvalue magnitude on λ3 and h. The stability region is the one enclosed by the unit

9600


Figure 8. Fuel cell stack temperature and power output profiles under the networked output feedback control system with model uncertainty δ1 ) 5 and different update periods.

Figure 9. (a) Dependence of λmax on the update period and the observer gain L under the model-based scheme with δ1 ) 5. (b) Fuel cell stack temperature profile under output-feedback networked control with update period h ) 12.8 s, model uncertainty δ1 ) 5, and different observer gains.

Figure 10. (a) Dependence of |G|H2 on the update period using a model with uncertainty δ1 ) 5 under output feedback control. (b-d) Closed-loop stack temperature, power output, and inlet fuel flow rate profiles under the networked control system for different update periods, when impulse disturbance is introduced in the inlet air flow rate.


contour line (unshaded region). Note that as the eigenvalue becomes more negative (i.e., a faster observer is used), the range of allowable update periods increases. The prediction of Figure 9a is further confirmed by the closed-loop temperature profile in Figure 9b, which shows that the plant can be stabilized at the desired set-point with h ) 12.8 s when λ3 is decreased from the original value of -1 to -5 such that the networked control system is operated at a point inside the unit contour zone. 5.2. Performance Assessment under Disturbances. To evaluate the performance of the networked closed-loop system under output feedback control, we consider the linearized plant with measured and performance outputs: x˙ ) Ax + Bu + Ew y ) Cx z ) Fx + Gu

(17)

where w is the disturbance input and z is the performance output signal. The networked closed-loop system can be cast in the form of eq 11 where ξ(tk) ) [xT(tk) jxT(tk) 0], H ) [ET O O]T and N ) [F GK -GK]; and the extended H2 norm, |G|H2, is given by eq 12, where X is the solution of the discrete Lyapunov equation:

9601

cost-effective solution to the deployment and integration of DERs within existing power networks, the effectiveness of this solution requires the development of control strategies that account explicitly for the inherent limitations of the communication medium. In this paper, we explored the application of concepts and techniques from networked control systems to address the problem of managing DERs over a bandwidthconstrained network. As a model system, we considered a SOFC plant and implemented a resource-aware networked control strategy that enforces the desired stability and performance properties with minimal communication between the fuel cell and the controller over the network. The results were illustrated through numerical simulations under both full-state and output feedback control formulations. Extensions of the networked control strategies to account for other communication constraints, such as network-induced delays, signal quantization, and real-time scheduling constraints, are topics under current investigation. Acknowledgment Financial support, in part by NSF CAREER Award, CBET0747954, and in part by the UC Energy Institute, is gratefully acknowledged.

Mp(h)TXMp(h) - X + Wo(0, h) ) 0 Literature Cited with Mp(h) ) IoeΛoh, Wo(0,h) is an observability Gramian computed as Wo(0, h) )

∫

h ΛTt T o

0

e

N NeΛot dt

and Λo is given by eq 15. To characterize the optimal update period for the fuel cell plant, we introduce a unit impulse disturbance in the inlet flow rate of air and choose the fuel cell stack temperature as the performance output; i.e., z(t) ) y(t) with F ) [0 0 0 1] and G ) 0. To characterize the dependence of the performance index of the networked closed-loop system on the update period, we plot |G|H2 in Figure 10a as a function of the update period when a model of the plant with uncertainty δ1 ) 5 is embedded in the controller and an observer of the form of eq 13 is embedded in the sensor. A trend similar to the state feedback case is observed where the performance deteriorates as h increases. This trend is further confirmed by the closed-loop temperature, power output, and inlet fuel flow rate profiles depicted in Figure 10(b-d) where it is seen that the response of the closed-loop system under an update period of h ) 5 s outperforms the response under an update period of h ) 12 s. Note also from Figure 10a that, similar to the state feedback case, there is a range of update periods over which the closed-loop performance exhibits minimal sensitivity to the disturbance (a flat optimum) and that the performance index begins to rise sharply as h approaches the critical stability limit. 6. Concluding Remarks Distributed energy resources (DERs) are a suite of on-site, grid-connected, or stand-alone technology systems that can be integrated into residential, commercial, or institutional buildings and/or industrial facilities. These energy systems offer advantages over conventional grid electricity by offering end users a diversified fuel supply; higher power reliability, quality, and efficiency; lower emissions and greater flexibility to respond to changing energy needs. While control of DERs over communication networks has recently emerged as an appealing

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ReceiVed for reView May 29, 2009 ReVised manuscript receiVed August 14, 2009 Accepted August 26, 2009 IE9008869

Networked Control of Distributed Energy Resources: Application to

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