Dynamics of a Microgrid Supplied by Solid Oxide Fuel Cells 1

Transcription

1 Bulk Power System Dynamics an Control - VII, August 19-4, 007, Charleston, South Carolina, USA 1 Dynamics of a Microgri Supplie by Soli Oxie Fuel Cells 1 Eric M. Fleming Ian A. Hiskens Department of Electrical an Computer Engineering University of Wisconsin - Maison Maison, WI USA Abstract The paper presents a moel for a soli oxie fuel cell (SOFC) stack operating at relatively low pressures. The moel incorporates the electrochemical reaction ynamics an the major voltage losses in SOFCs, but is not concerne with the thermal ynamics. A DC-DC boost converter interfaces the SOFC stack to a DC bus, where an ultra-capacitor provies energy storage. The paper presents a control system for the converter which regulates the DC-bus voltage along with the fuel cell current. A control strategy is propose for the inverter that interfaces the SOFC plant to the AC gri. The controller regulates the gri-sie voltage an the active power elivere to the gri, an takes into account the phase-locke loop (PLL) ynamics. The results of an example in which two SOFC plants provie power to a microgri are presente. The simulation inclues isconnection from the main gri, autonomous operation, an re-synchronization with the main gri. Keywors: Soli oxie fuel cells, microgri ynamics, inverter control. I. INTRODUCTION Distribute generation sources are becoming more prevalent within istribution systems, with a growing role in applications ranging from combine heat an power to reliability enhancement. This tren has motivate the concept of a microgri, which generically consists of multiple generators an loas, interconnecte by a subsection of the istribution system [1], []. Microgris, by efinition, shoul be able to operate when gri-connecte, or islane (autonomous), an shoul transition reliably through the connection/isconnection process. In parallel with the tren towars greater istribute generation, fuel cell technology has been avancing rapily, as a result of significant research an evelopment efforts. It is therefore likely that fuel cells will make an increasingly important contribution to istribute generation. In particular, soli oxie fuel cells (SOFCs) are well suite to stationary applications where there is a requirement for combine heat an power. Microgris that are supplie solely by inverter-base sources, such as fuel cells, may have no rotating inertia. As a result, the frequency of the AC voltages an currents must be establishe by the inverter controls. Generally this is achieve through the use of a phase-locke loop (PLL) which establishes a reference signal for power electronic firing. The ynamic behavior of the PLLs therefore has an important influence on the microgri frequency. Furthermore, with no rotating inertia present, interactions between inverter controls may be potentially estabilizing. 1 Research supporte by the National Science Founation through grant ECS , System Integration of Distribute Generation. The paper investigates the ynamic behavior of microgris that are supplie by SOFCs. Initially, a ynamic moel of an SOFC stack is evelope. The gri connection of an SOFC is then consiere, with particular emphasis on controller esign for the DC-DC converter an the DC-AC inverter. A microgri example is use to explore the behavior of an SOFC-supplie microgri. A. Backgroun II. SOFC STACK MODEL The operation of all fuel cells is base on the reaction of hyrogen with oxygen to prouce water: H + O H O. In orer to extract current, an thus electric power, from a fuel cell, the above reaction must be separate into two halfreactions. In an SOFC, these half-reactions take the form [3], [4], H + O H O + e O + 4e O. The first of these half-reactions takes place at the anoe, an the secon takes place at the cathoe. The electrons release in the anoe half-reaction flow through an external circuit as current an return to the cathoe to react with oxygen ecules. The oxygen ions release at the cathoe flow through an electrolyte to the anoe, where they react with hyrogen [4] [7]. Since one fuel cell prouces only about 1 volt across its terminals, in practice many cells are stacke in series to prouce a useable voltage [7]. This paper presents a moel for an SOFC stack which incorporates the electrochemical reaction ynamics an the major voltage losses. The thermal ynamics are not inclue, as the associate time constants are much longer than those of the electrochemical response [4], [6]. The fuel supplie to the SOFC plant is natural gas, which must unergo reforming in orer to extract the hyrogen use in the fuel cell reaction. The moel presente in this paper assumes that reforming is one in an external fuel processor an pure hyrogen is supplie to the anoe channel. It is also necessary to supply oxygen to the cathoe reaction. The moel assumes that this is achieve via an air compressor which is proviing a constant flow rate of air into the cathoe channel. The electrochemical reaction of a fuel cell is spatially istribute, occurring over the membrane that separates the anoe an cathoe. Therefore, the composition of the gas insie each electroe channel varies along the channel length.

2 For instance, in the anoe, the partial pressure of hyrogen is highest at the inlet orifice an ecreases across the length of the channel as it is consume, an is therefore lowest at the outlet orifice. This iea is use in the SOFC moel presente in [4]. It is common, however, for this istribute reaction to be moele using states that capture average behavior. In particular, average partial pressures are use for the gases within the anoe an cathoe in all moel calculations. The SOFC moels presente in [3], [6] use this iea, as oes the moel presente in this paper. B. Electroe moels An important aspect of the electroe moels is the etermination of the flow rates of gases out of each channel. The outlet flow rate is epenent on the area as well as other properties of the orifice through which the gas flows, which can be represente by a valve constant. If the orifice is fixe, the outlet flow rate is couple with the flow rate of gas into the respective channel as well as the pressure insie the channel. In particular, specifying one of these quantities (inlet flow rate or pressure) ictates the other. The moel in this paper uses a specifie inlet flow rate an a fixe outlet orifice to etermine the channel pressures. This is consistent with SOFC moels presente in [3], [6]. The SOFC moel in [4] uses a ifferent approach in which the inlet flow rate an the total pressure insie each channel are both specifie. This requires the use of a pressure regulator, which acts like an ajustable valve at the outlet orifice an therefore eliminates the nee for an orifice equation. If the pressure ifference across an orifice is relatively small, the flow through the orifice is consiere unchoke, an the volumetric or mass flow rate is proportional to the square root of the pressure ifference [8] [10]. However, if the pressure ifference is large, the flow is consiere choke, an the volumetric or mass flow rate through the orifice can be approximate as being proportional to the upstream pressure. The bounary between these two types of flow is efine by the critical ratio of upstream to ownstream pressure, which is equal to [8]. The SOFC moels in [3], [6] assume the pressure insie the channels is high enough that the outlet flow can be consiere choke, an they make the simplification of efining the outlet flow rate of each species as being proportional to its partial pressure insie the channel. The moel escribe in this paper assumes that the stack is operate at a sufficiently low pressure that the outlet flow is unchoke an therefore uses the square root relationship in calculating the outlet flow. C. Anoe moel The gases insie the anoe are hyrogen an water vapor. With p H an p H O as the partial pressures of hyrogen an water vapor, respectively, in the anoe, the average ar mass of the anoe gas can be foun as follows: M a = p H M H + p H OM H O, (1) p H + p HO where M H an M H O are the ar masses of hyrogen an water vapor, respectively. Assuming the gases obey the ieal gas law, pv = n, () the ensity of the anoe gas can then be foun as ρ a = m a V a = nm a V a = pm a = (p H + p H O) M a, (3) where m a is the mass of gas insie the anoe channel, V a is the volume of the anoe channel, R is the gas constant J (8.31 K ), an T is the operating temperature. Assuming unchoke flow, the volumetric flow rate of gas out of the anoe [8] [10] is given by, Q a = C fa A a ρ a (p H + p H O p atm ) (4) where C fa is the anoe flow coefficient, A a is the anoe valve area, p atm is the atmospheric pressure, an Q a has units of (m 3 /s). Again assuming the ieal gas law hols, the ar outlet flow rates of the two component gases can be etermine as, N o H = Q ap H NH o O = Q ap HO (6) where these quantities have units of (/s). The ar outlet flow rates are use in the ifferential equations governing changes in hyrogen an water vapor partial pressures, which also are base on the ieal gas law [3], [4], [6], t p H = [ NH in V NH o N ] oi (7) a F t p H O = [ ] No I V a F N H o O (8) where NH in is the hyrogen ar inlet flow rate, N o is the number of cells in the stack, I is the stack current, an F is Faraay s constant (96485 Coulombs ). The term N oi F in the first ifferential equation is the ar rate of hyrogen use up in the reaction, an likewise the corresponing term in the secon ifferential equation is the ar rate of water vapor prouce. D. Cathoe moel (5) A similar formulation can be use to etermine the ar outlet flow rates of oxygen an nitrogen from the cathoe. The input air is assume to have a nitrogen-to-oxygen ecular ratio of 78 to 1, an the effects of smaller components of air such as carbon ioxie are ignore. The moel equations are as follows: Average ar mass: Gas ensity: M c = p N M N + p O M O p N + p O (9) ρ c = M c (p N + p O ) (10)

3 3 Volumetric outlet flow rate: Q c = C fc A c ρ c (p N + p O p atm ) (11) Oxygen ar outlet flow rate: NO o = Q cp O Nitrogen ar outlet flow rate: NN 0 = Q cp N Partial pressure variations: [ N in t p O = V c t p N = V c O N ] oi 4F N O o [ ] 78 1 N O in NN o (1) (13) (14) (15) Note that the ar rate of oxygen reacte has a 4 in the enominator rather than a. This is because half as many es of oxygen are use in the reaction as es of hyrogen. Also note that there is no term in the ifferential equation for nitrogen partial pressure corresponing to a reaction rate since nitrogen oesn t participate in the reaction. E. Electrical moel The reversible, or open circuit, potential of an SOFC is given by the Nernst equation [7], [ ( )] E = N o gf o ph p 1/ O + ln (16) F p H O p 1/ atm where gf o is the change in the ar specific Gibbs free energy of formation for the fuel cell reaction, which epens on the operating temperature. For the typical operating temperature range of SOFCs (above 800 o C), the relationship can be approximate by [7], g o f = (T ). (17) There are three main types of losses in a fuel cell, contributing to three ifferent voltage rops which subtract from the open circuit potential. These are known as the activation, ohmic, an concentration losses. The activation loss is cause by the kinetics of the chemical reactions taking place at the fuel cell electroes, which limit how fast the reactants can be consume [7]. The resulting voltage rop is moele in a piecewise fashion, η act = j j o 4F) j o ( F ln j + 4F j j o j (18) j o j > j o where j is the current ensity, which is equal to the stack current ivie by the fuel cell area, j = I/A. Equation (18) is an approximation of the Butler-Volmer equation [5]. At low currents, the relationship between activation rop an current is approximately linear. Above j o, known as the exchange current ensity, the relationship is logarithmic. The ohmic loss is cause by resistance to electron flow through the electroe materials as well as resistance to ion Fig. 1. Constant utilization implementation. flow through the electrolyte [6]. The voltage rop is moele with a resistance, r, whose value epens on the operating temperature of the fuel cell, r = 0. exp [ 870 ( T )]. (19) The concentration loss is ue to the limite rate at which reactant concentrations can change, which limits the rate at which the reactants can be transporte to the electroe surfaces [7]. This voltage rop is moele as [5], [6], η con = 4F ln (1 jjl ) (0) where j l is known as the limiting current ensity. Finally, the terminal voltage of the fuel cell stack is given by [5] [7], V fc = E η act ri η con. (1) F. Moes of operation There are two basic moes in which fuel cells are operate, constant input an constant utilization. In constant input moe, the inlet flow rates of hyrogen to the anoe, NH in, an oxygen to the cathoe, NO in, are set to fixe values. In constant utilization moe, the inlet flow rate of hyrogen is controlle so that the ratio of hyrogen consume in the reaction to hyrogen supplie to the anoe is regulate to a set value. This ratio is known as the fuel utilization an is efine by [4], N 0I F U H = NH in. () The control for constant utilization can be implemente using current feeback to ajust the hyrogen input flow rate [4]. The fuel processor can be moele as a first-orer elay transfer function [4], [11]. A block iagram of the moel implementation of constant utilization is shown in Figure 1. The corresponing equations are, ÑH in = N oi (3) F ŨH ( ) N in 1 t H = in (Ñ T H NH in ). (4) The steay-state plot of current versus voltage for a fuel cell stack is known as the polarization curve. Figure shows polarization curves for the two basic moes of operation. The constant input case uses the parameters in Table I. The constant utilization case uses the same parameters, except that

4 4 TABLE I SOFC MODEL PARAMETERS. 60 Constant input Constant utilization parameter value units T K V a 0. m 3 V c 0. m 3 NH in.0 s s NO in 1.0 N o A 0.1 m A a.005 m A c.005 m C fa.75 - C fc.75 - p atm P a M H kg M H O kg M N kg M O kg A j o 1500 m A j l m T 1 s Voltage (V) Fig. 3. SOFC voltage response to a current step. the constant hyrogen input flow is replace by a hyrogen utilization of Constant input Constant utilization Fig. 4. Boost converter circuit iagram. Voltage (V) Fig Current (A) SOFC polarization curves. The curve for constant input moe has three ientifiable regions [5]. At low currents, the curve is steep. This is known as the activation region, since the activation loss is the main factor contributing to voltage rop. As the current increases, the curve becomes less steep an is nearly linear for a broa range of currents. This is known as the ohmic region, since the preominant voltage rop is ohmic. At high currents, the curve becomes steep again ue to the preominance of the concentration voltage rop, an hence this portion of the curve is calle the concentration region. The constant utilization polarization curve oes not have the steepness at low currents like that for constant input. The constant utilization curve sits below the constant input curve for currents where the utilization in the constant input case is less than that of the constant utilization case, an sits above the constant input curve at higher currents. Figure 3 shows the fuel cell voltage in response to a step change in current from 500 A to 750 A for the SOFC moel operating in the two ifferent moes. The initial jump in voltage in both cases is ue to the change in the three voltage rops cause by the current step. The voltage then settles to its new steay-state value. The constant utilization curve exhibits a secon-orer response ue to the effect of the fuel processor ynamics, while the constant input case oes not. Similar finings are reporte in [4]. III. GRID INTERFACE A. DC-DC converter an controls A c-c converter is use to step up the voltage of the fuel cell stack to that of the c bus, an to provie regulation of the DC bus voltage an of the fuel cell current [1]. The circuit iagram of a boost converter is shown in Figure 4 [13]. The moel in this paper is not concerne with the switching ynamics of the converter. Accoringly, the inuctor ynamics are not moele. It is assume that an ultra-capacitor is use for energy storage on the DC bus. Since a ifferential equation must be inclue for this evice, its value of capacitance can inclue that of the converter capacitor. The topology of the DC bus moel is shown in Figure 5. Aopting an average moel for the boost converter, the

5 5 are as follows: Fig. 5. DC bus topology. Ĩ 1 = P set V fc (9) V err = V c Vc set (30) t (a) = K 6V err (31) Ĩ = a K 5 V err (3) Ĩ = Ĩ1 + Ĩ (33) I err = I fc Ĩ (34) t (b) = K 8I err (35) G = b K 7 I err (36) G = G 0 + G. (37) Fig. 6. DC-DC converter control scheme. corresponing equations are as follows: V c = GV fc (5) I c = I fc G (6) I c = I c + I inv (7) where G is the gain of the converter. Capacitor ynamics are governe by the equation where C is the capacitance. t (V c) = I c C, (8) The control scheme presente in this paper for the DC-DC converter employs two control loops. The block iagram for this system is shown in Figure 6. The inner loop regulates the DC bus voltage to a setpoint value by feeing the voltage error through a PI controller; this strategy is also propose in [1]. The outer loop regulates the fuel cell current to a value corresponing to the power setpoint for the SOFC plant, which is fe from the inverter control system (escribe later). Specifically, the current setpoint is the ratio of the power setpoint to the fuel cell voltage. The output from the voltage controller is then ae to this setpoint current. The current error is fe through a PI controller, the output of which is ae to the nominal gain of the converter to prouce the new value of converter gain. The corresponing moel equations B. Inverter moel an controls 1) Inverter-gri interface moel: The SOFC plant interfaces to the AC system through an inverter. A moel for the inverter-gri interface is given in Figure 7. It consists of an internal bus at which the voltage-source inverter synthesizes an AC voltage waveform, an the terminal bus that is common with the gri. The corresponing voltage phasors are V i δ i an V t δ t, respectively, where the phase angles are specifie with respect to a global reference sinusoi of nominal frequency. These two buses are connecte through a transformer, with impeance jx. All quantities are expresse in per-unit. The inverter seeks to regulate the active power P gen elivere to the gri, an the terminal bus voltage magnitue V t. This is achieve by controlling the moulation inex m of the inverter as well as the inverter firing angle, which is equivalent to the phase δ i of the synthesize voltage waveform. A similar control strategy is propose in [14]. It is important to keep in min that the inverter has no knowlege of the global reference. Accoringly, the absolute phase angle δ i is meaningless. Rather, the phase of the inverter voltage must be establishe relative to a local reference signal. A phase-locke loop (PLL) is often use to provie that local reference. Fig. 7. Inverter-gri interface. ) Phase-locke loop: Synchronizing the inverter an gri AC voltage waveforms can be achieve using a phase-locke loop (PLL). A block iagram isplaying the functional components of a PLL is given in Figure 8. The measure sinusoi is mixe with the cosine generate by the PLL oscillator. This mixing process effectively establishes the phase ifference between the two waveforms. That error signal is filtere an fe back to the voltage-controlle oscillator. The outcome is

6 6 Fig. 8. Generic PLL block iagram. where V set may be constant, or may follow a roop characteristic that is epenent upon the reactive power elivere to the gri. The inverter internal bus voltage is then given by V i = mv c V base (41) where V base is the per-unit base voltage for the DC bus an inverter. From Figure 7, the active power elivere to the gri is given by P gen = V iv t X sin(δ i δ t ). (4) This quantity must also be equal to the power on the DC sie of the inverter after making the per-unit conversion, Fig. 9. Linearize PLL block iagram. a signal that is phase locke to the measurement. Stanar simplifying assumptions [15] allow the PLL to be moelle accoring to the linear block iagram of Figure 9. This moel is commonly use for analysis an esign of PLL-base systems [15]. Note though that the PLL has no knowlege of the global synchronous reference, an hence cannot etermine the absolute angles δ t an δ p. The phase ifference δ t δ p is, however, available locally. Relating back to the inverter-gri connection of Figure 7, the PLL input is given by the terminal bus voltage v t (t), which has phase angle δ t relative to the global reference sinusoi. If the gri frequency experiences a constant offset from nominal, then δ t will by time-varying. To achieve zero offset between δ t an δ p uner such circumstances, the PLL transfer function F (s) shoul take the form F (s) = K s. From Figure 9, this gives δ p = K KK o s (δ t δ p ). Rewriting as a ifferential equation, with K 3 = K KK o, an efining gives δ p = ω p (38) ω p = K 3 (δ t δ p ). (39) An estimate of the eviation of system frequency from nominal is provie by ω p. This can be use in the roop characteristic of the inverter controls. 3) Inverter control: As mentione previously, the control objectives are to regulate the terminal bus voltage magnitue V t, an the active power elivere to the gri P gen. The first objective can be achieve by simple integral control ṁ = K 1 (V set V t ) (40) P gen = V ci inv P base. (43) Assuming V i an V t remain relatively constant, regulation of P gen can be achieve by controlling the angle ifference δ i δ t. The PLL output δ p provies a filtere version of δ t though, so it is preferable to control Integral control gives, θ = δ i δ p. (44) θ = K (P set P gen ). (45) The active power setpoint P set is etermine from the roop characteristic P set = P 0 Rω p, (46) where ω p is the estimate frequency eviation provie by the PLL, see (38), an R is the roop constant. The following analysis shows, however, that interactions between this P gen controller an the PLL ynamics cause sustaine oscillations. Rearranging (44) an substituting into (39) gives ω p = K 3 (δ t δ i + θ). (47) Referring to (4), if P gen, V i an V t are constant, then δ i δ t must also be constant. Uner those conitions, ifferentiating (47) an substituting (45) an (46) gives which implies ω p = K 3 θ = K K 3 (P 0 Rω p P gen ) ω p + K K 3 Rω p = K K 3 (P 0 P gen ). (48) This secon orer system is unampe. Any isturbance will lea to unattenuate oscillations in ω p, even when P gen, V i an V t are all constant. Viable control requires the aition of a amping term to (48). This can be achieve by aing an extra term into (39), ω p = K 3 (δ t δ p ) + K 4 θ. (49) Differentiating, as above, an making similar substitutions results in ω p = K 3 θ + K4 θ = K K 3 (P 0 Rω p P gen ) + K K 4 ( R ω p )

7 7 Fig. 10. PLL an active power regulation block iagram. an hence ω p + K K 4 R ω p + K K 3 Rω p = K K 3 (P 0 P gen ). (50) Fig. 11. Example power system. Further rearranging gives 1 K ω p + K 4 R ω p + K 3 Rω p = K 3 (P 0 P gen ) (51) which reveals that 1/K acts like an inertia constant, an amping is provie by K 4 R. Defining the algebraic relationship allows (49) to be rewritten x = ω p K 4 θ (5) ẋ = K 3 (δ t δ p ). (53) Bringing the complete control strategy together gives, ṁ = K 1 (V set V t ) θ = K (P set P gen ) ẋ = K 3 (δ t δ p ) δ p = ω p 0 = V i mv c V base 0 = P set (P 0 Rω p ) 0 = θ (δ i δ p ) 0 = x (ω p K 4 θ) 0 = P gen V ci inv P base. The PLL an power regulator are escribe by the block iagram of Figure 10. IV. MICROGRID SIMULATIONS The microgri shown in Figure 11 will be use to illustrate the ynamic behaviour of the moel components. SOFC plants are locate at buses an 3, an a constant power loa is connecte to bus 4. Bus 1 forms the interface between the microgri an the rest of the power system, which is moelle as an infinite bus. All of the AC quantities are expresse as per-unit values. A power base of 100 kva is use in connecting the SOFC plant moels to the inverter an gri moels. A voltage base of 40 V was chosen for the DC buses an inverters. Since the DC bus setpoint voltage is 480 V, this causes the nominal value of the moulation inex to be 0.5. The actual voltage of the gri is irrelevant. For this example, the SOFC stacks are running in constant utilization moe with the parameters given in Table I, an a hyrogen utilization setpoint Ũ H = 0.8. The inverters an DC bus components have the parameters given in Table II. Plant 1 has a power setpoint of 0.7 pu (70 kw), an Plant has a power setpoint of 0.6 pu (60 kw). The active an reactive power of the loa, P L an Q L, are 1.7 pu an 0.6 pu, respectively, an the voltage of the infinite bus is set to 1 pu. The circuit breaker (CB) connecting bus 1 to the rest of the gri is initially close. The two fuel cell plants together supply 1.3 pu of the active power emane by the loa. The remaining 0.4 pu active power is rawn from the main gri through bus 1. At 1 s, the CB opens. The constant loa must now be supplie by the fuel cell plants. At 7 s, the CB is signale to close, but closing is prevente until the voltage magnitue appearing across the CB contacts reuces to a given threshol. For this simulation, the threshol is set to 0.05 pu. Consequently, the CB actually closes at s. Figure 1 shows the power generate by each of the SOFC plants over the simulation perio, an Figure 13 shows the frequency eviation given by the inverter PLLs. When the microgri is initially isconnecte from the main gri, the powers generate by the SOFC plants immeiately increase to compensate for the lost gri supply. Microgri frequency rops in accorance with the roop characteristic. Note that Plant 1, which has a higher power setpoint, overshoots when the CB opens, while Plant oes not. The sum of the two power outputs must always equal the active power of the loa while the CB is open. When the CB closes again, the phase relationship between the inverter voltages an the gri is such that active power initially flows from the microgri to the infinite bus. Therefore, both plants see a power spike For numerical reasons, the simulation actually monitors the square of the voltage magnitue.

8 8 TABLE II SOFC-INVEER PLANT PARAMETERS. parameter value K 1 10 K 0 K 3 0 K 4 10 D 0.4 X 0.pu V set 1pu K 5 5 K 6 5 K 7 10 K 8 5 C 0.1F Vc set 480V Frequecy eviation (ra/s) Plant 1 inverter Plant inverter 1.5 x Plant 1 Power Plant Power Fig. 13. PLL frequency eviation of inverters. Power (W) Power (W) 1.5 x SOFC stack power Inverter (plant) power Fig. 1. Power output of SOFC plants. 0.7 immeiately following the reconnection. The inverter controls respon accoringly, with active power outputs quickly returne to their pre-isturbance values, an microgri frequency restore to the nominal value. Figure 14 shows the power output of the SOFC stack for Plant 1 as well as the inverter output. When the power output makes a suen increase, the inverter is able to provie more power than the fuel cell stack alone ue to the presence of the capacitor on the DC bus. As the transients subsie, the stack power overshoots the inverter power at times, allowing the capacitor to recharge. Figure 15 shows the variation of the DC bus voltage of Plant 1 uring the simulation perio. Each time the power output increases, the voltage makes a suen rop ue to the ischarging of the capacitor, but the DC-DC converter controls then restore it to the setpoint value. Figure 16 shows angle behaviour uring the isturbance. (Only Plant 1 inverter quantities are shown.) When the CB opens, the inverter terminal bus voltage unergoes an immeiate phase shift. Over the subsequent perio of autonomous operation, the microgri frequency is below nominal. Accoringly, the microgri phase angle, relative to a global reference at nominal frequency, isplays a steay ecrease. This contin Fig. 14. SOFC stack an plant power for Plant 1. ues until the CB closes. At that instant, the microgri voltages are out of phase with the stronger system. In response, the inverter terminal bus phase angle ajusts very rapily, quickly settling to a value that is exactly π ra behin its initial value. Figure 16 also shows that the PLL angle closely tracks the terminal bus angle. The ifference between these quantities is shown more clearly in Figure 17. It can be seen that the controller is effective in rapily riving this ifference to zero. The angle ifference across the inverter transformer is also shown in Figure 17. Notice that when the CB closes, the resulting phase shift in the inverter terminal bus voltage causes a spike in the angle ifference across the inverter transformer. That spike in angle ifference unerlies the spike in active power P gen observe in Figure 1. V. CONCLUSIONS A moel for an SOFC stack, operating at relatively low pressures, has been evelope. The moel incorporates the

9 PLL tracking error, δ t δ p Angle across transformer, δ i δ t Voltage (V) Fig. 15. DC bus voltage for Plant 1. Angle ifference (ra) Fig Plant 1 inverter angle ifferences. Angle relative to synchronous reference (ra) Terminal bus angle (δ t ) PLL angle (δ p ) Fig. 16. Plant 1 inverter angle behaviour relative to a global reference. electrochemical reaction ynamics an the major voltage losses in SOFCs, but is not concerne with the thermal ynamics. It uses average partial pressure ieas, consistent with [3], [6], but iffers in the moeling of the anoe an cathoe outflows. In particular, unchoke flow is assume. The two basic moes of operation of fuel cells are constant input an constant utilization. In the former, the flow rate of fuel (hyrogen) into the anoe channel is constant, whereas in the latter, it is controlle so as to regulate the ratio of fuel use to fuel supplie. In the latter moe, a step change in SOFC stack current causes a secon-orer response in the stack voltage. This is ue to the ynamics of the fuel processor, which limits the rate at which the hyrogen input flow rate can change. A DC-DC boost converter is use to raise the voltage of the SOFC stack to that of the DC bus, where an ultra-capacitor provies energy storage. The propose control system for the boost converter consists of two loops. The inner loop regulates the DC bus voltage, while the outer loop regulates the fuel cell current to a value consistent with the plant s active power setpoint. An inverter interfaces the SOFC plant to the AC gri. The propose control strategy ajusts the moulation inex of the inverter to regulate the gri-sie voltage magnitue, an ajusts the firing angle of the inverter to regulate the active power elivere to the gri. The latter utilizes a phase-locke loop (PLL) to provie a local phase angle reference for the inverter, since the phase with respect to a global reference is not available to the plant. The PLL introuces aitional ynamics to the system. An example system, in which two SOFC plants provie power to a microgri, has been presente. In the example, the microgri was initially connecte to the main gri, an was subsequently isconnecte. During autonomous operation, the microgri operate at a frequency below nominal ue to roop control. At reconnection, microgri voltages were out of phase with the stronger system. This resulte in a step in the phase angle ifference across the transformers connecting the inverters to the gri, causing a spike in their power outputs. Inverter controls quickly respone, returning the power outputs to their setpoint values, an the microgri frequency to nominal. During power output spikes, the inverters were able to provie more power than the stacks alone because of the energy store in the DC bus capacitors. The DC-DC converter controls allowe the capacitors to subsequently recharge, while riving the DC bus voltages back to their setpoint values. REFERENCES [1] C. Smallwoo, Distribute generation in autonomous an nonautonomous micro gris, in Proceeings of the IEEE Rural Electric Power Conference, Colorao Springs, CO, May 00, pp. D1.1 D1.6. [] F. Katiraei an M. Iravani, Power management strategies for a microgri with multiple istribute generation units, IEEE Transactinos on Power Systems, vol. 1, no. 4, pp , May 006. [3] J. Paulles, G. W. Ault, an J. R. McDonal, An integrate SOFC plant ynamic moel for power systems simulation, Journal of Power Sources, vol. 86, pp , 000. [4] C. Wang an M. H. Nehrir, A physically-base ynamic moel for soli oxie fuel cells, IEEE Transactions on Energy Conversion, to appear.

10 [5] R. O Hayre, S.-W. Cha, W. Colella, an F. B. Prinz, Fuel Cell Funamentals. New York, NY, USA: John Wiley & Sons, Inc., 006. [6] K. Seghisigarchi an A. Feliachi, Dynamic an transient analysis of power istribution systems with fuel cells - Part I: fuel-cell ynamic moel, IEEE Transactions on Energy Conversion, vol. 19, no., pp , June 004. [7] J. Larminie an A. Dicks, Fuel Cell Systems Explaine, n e. Chinchester, Englan, UK: John Wiley an Sons, Lt., 003. [8] J. F. Blackburn, G. Reethof, an J. Shearer, Es., Flui Power Control. Cambrige, MA, USA: The Technology Press of M.I.T. an John Wiley & Sons, Inc., 1960, pp [9] E. funamentals, Orifice flowmeter calculator, Website ( orifice flowmeter.cfm), Apr [10] E. N. Viall an Q. Zhang, Determining the ischarge coefficient of a spool valve, in Proceeings of the American Control Conference, vol. 5, June 000, pp [11] Y. Zhu, Analysis an control of istribute energy resources, Ph.D. issertation, Washington State University, May 00. [1] C. Wang an M. H. Nehrir, Short-time overloaing capability an istribute generation applications of soli oxie fuel cells, IEEE Transactions on Energy Conversion, to appear. [13] P. T. Krein, Elements of Power Electronics. Oxfor University Press, [14] K. Seghisigarchi an A. Feliachi, Dynamic an transient analysis of power istribution systems with fuel cells - Part II: control an stablility enhancement, IEEE Transactions on Energy Conversion, vol. 19, no., pp , June 004. [15] D. Abramovitch, Phase-locke loops: a control centric tutorial, in Proceeings of the American Control Conference, Anchorage, AK, May