Hierarchical Load Tracking Control of a Grid-connected Solid Oxide Fuel Cell for Maximum Electrical Efficiency Operation

: Based on the benchmark solid oxide fuel cell (SOFC) dynamic model for power 21 system studies and the analysis of the SOFC operating conditions, the nonlinear 22 programming (NLP) optimization method was used to determine the maximum electrical 23 efficiency of the grid-connected SOFC subject to the constraints of fuel utilization factor, 24 stack temperature and output active power. The optimal operating conditions of the grid- 25 connected SOFC were obtained by solving the NLP problem considering the power 26 consumed by the air compressor. With the optimal operating conditions of the SOFC for the maximum efficiency operation obtained at different active power output levels, a 28 hierarchical load tracking control scheme for the grid-connected SOFC was proposed to realize the maximum electrical efficiency operation with the stack temperature bounded. The hierarchical control scheme consists of a fast active power control and a slower stack temperature control. The active power control was developed by using a decentralized 32 control method. The efficiency of the proposed hierarchical control scheme was 33 demonstrated by

Abstract: Based on the benchmark solid oxide fuel cell (SOFC) dynamic model for power 21 system studies and the analysis of the SOFC operating conditions, the nonlinear 22 programming (NLP) optimization method was used to determine the maximum electrical 23 efficiency of the grid-connected SOFC subject to the constraints of fuel utilization factor, 24 stack temperature and output active power. The optimal operating conditions of the grid-25 connected SOFC were obtained by solving the NLP problem considering the power 26 consumed by the air compressor. With the optimal operating conditions of the SOFC for 27 the maximum efficiency operation obtained at different active power output levels, a 28 hierarchical load tracking control scheme for the grid-connected SOFC was proposed to 29 realize the maximum electrical efficiency operation with the stack temperature bounded. 30 The hierarchical control scheme consists of a fast active power control and a slower stack 31 temperature control. The active power control was developed by using a decentralized 32 control method. to an external ac power system through a power control unit (PCU), the active power control of the 141 PCU shall be taken into account as well in order to achieve the maximum efficiency load tracking 142 operation and has not been studied. 143 The paper presents a maximum electrical efficiency load-tracking control scheme for the grid-144 connected SOFC in order to improve the operation performance. In Section 2, by using an existing 145 benchmark SOFC dynamic model specifically developed for power system studies, the maximum 146 efficiency of the SOFC can be obtained by solving a non-linear programming problem which is subject 147 to a set of steady-state equality and inequality constraints. Next, the locations of the open-loop poles of 148 the dynamic model lead to the proposed structure of the hierarchical control scheme shown in Section 149 3. In order to achieve the optimal operating state, a decentralized power and temperature control 150 system is proposed and described in Section 4. The performance of the maximum electrical efficiency 151 load tracking control scheme is illustrated through the case studies in Section 5, followed by the 152 conclusions. 153

Nonlinear Programming 155
In order to facilitate the analysis, the following assumptions are made, 156 (1) Hydrogen rich nature gas is converted to hydrogen (H 2 ) through the external-reforming fuel 157 processor. Like in [22], the carbon oxide (CO) shift reaction is ignored in the analysis. Only 158 pure H 2 is fed to the anode; 159 (2) Oxygen in the air is used as the oxidant. The mole ratio of nitrogen (N 2 ) to oxygen (O 2 ) in 160 the air is denoted as k c which is 3.762; 161 (3) Both the fuel and air are preheated to the same temperature before they are transmitted to the 162 cell stack. The detailed thermal management is not studied; 163 (4) The cell stack is well-insulated and the energy losses caused by radiation, convection and 164 conduction are negligible. studies, and the model is shown within the dash lines in Figure 1. In this model, it is assumed the stack 168 temperature T is constant. Considering the cell stack tabular structure, the channels that transport the 169 gases along the electrodes have a fixed volume, but their lengths are small. Hence it is sufficient to 170 define one single pressure value in the cell stack interior. The exhaust of each channel is via a single 171 orifice. The ratio of pressures between the interior and exterior of the channel is large enough and it 172 can be assumed that the orifice is choked and the lumped-parameter model can be derived. Therefore, 173 the mass balance equation, expressed in terms of the partial pressures p i , is given as, 174 where the subscript 'i' denotes either H 2 , O 2 or water (H 2 O), the superscript 'in', 'o' and 'r' denote the 175 input, output and reaction variable, respectively, R is the ideal gas constant, V i is the anode or cathode 176 volume, and q i , K i and  i are the i th gas mole flow rate, valve molar constant and time constant, 177 respectively. Thus,  i can be written as, According to the Faraday's Law of Electrolysis, the reaction flow rates are, 179 where P is the SOFC dc output power, m s C ps is the mass-specific heat product of the stack, i h is the i th 194 gas per mole enthalpy and it can be written as, 195 In (7), std i h , is the i th gas per mole enthalpy at the standard pressure of 0.1MPa and the standard 196 temperature T std of 283K, pi C is the i th gas average constant-pressure specific heat and ΔT is the 197 temperature change. 198 Substituting (3) and (7) into (6), (6) can be rewritten as, 199 In (9), H LHV has the low heat value of 241.83kJ if 1 mole of H 2 is fully combusted to produce 201 gaseous H 2 O at the standard state [1], and T in is the stack inlet gas temperature. As Denote the terminal voltage of the SOFC stack as V dc and the grid voltage as V s . The turns-ratio of 218 the transformer is 1:k T , and the transformer series impedance plus the linking feeder yield the 219 equivalent reactance X. Define . The injected active and reactive power (P+jQ) from 220 the SOFC to the grid system is [27], 221

Operating Variables and Constraints 228
The operating variables of the grid-connected SOFC such as T, V dc , P, in H q 2 , in O q 2 and I r (I FC ) can be 229 calculated based on the energy balance principle, Nernst equation and Figure 1, i.e. through solving the 230 following equations, 231 where E 0 , the ideal standard potential, is a function of T [30], 232 In (16), p 0 is the standard pressure. V act , V r and V con , as shown in (19)-(21), are activation loss, 233 Ohmic loss and concentration loss, respectively [1].The detailed definitions of the parameters and their 234 typical values are given in Nomenclature and Table 1  The cell lifespan and performance are dependent on the operating parameters. Therefore, three 238 operating constraints must be respected for the safe operation of the cell. The most important operating 239 constraint is the fuel utilization factor u, given as, 240 Typically u min = 0.7 and u max = 0.9 [21]. 241 The other two operating constraints are T and P, 242 max min Energies 2014, 7 Typically, T min = 1173K, T max = 1273K, P min = 0.1pu and P max = 1pu of the SOFC rated power [1,23]. 243

Determination of the Optimal Operating Condition 244
The electrical efficiency η of the hydrogen SOFC is defined as the ratio of the net power to the total 245 power obtainable by burning H 2 at the standard state [1], 246 However, if the stack operating pressure p is higher than 0.1MPa, not all the power generated by the 247 SOFC will be delivered to the external circuit. The parasitic losses P loss is dominated by the air 248 compressor in the form [1], 249 where  c is the equivalent efficiency of the air compressor.

250
Under a given pressure p, it can be seen from (26) and (27)  in H q 2 , I r (I FC ) as well as u for a 259 targeted P will be obtained and pre-stored in a look-up table as the reference input signals for the 260 SOFC load tracking control system. 261

Hierarchical Load Tracking Control Scheme for the Grid-connected SOFC 262
With the optimal operating condition of the SOFC determined by the NLPP, the load tracking 263 control scheme for the grid-connected SOFC can be developed to track the power demand and operate 264 at The open-loop poles of the SOFC are analyzed to study the dynamic response of the SOFC and a 266 hierarchical control scheme for the SOFC is proposed based on the dynamic response analysis. 267

Analysis of the Open-loop System Poles
In the load tracking control scheme for the SOFC, the internal dynamics of the PCU can be 269 neglected as the typical response time of the PCU is a few milliseconds. The load tracking speed of the 270 SOFC will be dominated by the dynamic response of devices on the dc side of the power plant where 271 the typical time constants are of the order of 1s or larger. 272 The response characteristics of the SOFC operating at the maximum efficiency can be assessed by 273 examining the locations of the six open-loop poles of the dynamic model shown in Figure 1. The i th 274 pole can be calculated as, 275 The electrochemical reaction and fuel processor contribute to the poles -1/τ e and -1/τ f . They are 276 independent of P and T. However, the location of the poles -1/τ H2 , -1/τ O2 ,-1/τ H2O and -1/ T may change 277 when the SOFC operates at different power levels. As shown in (2) and Figure 3, three gas time 278 constants are the function of T but independent of I FC . Therefore, -1/τ H2 , -1/τ O2 and -1/τ H2O will be 279 away from the origin when T increases. On the other hand, from (10) and (11),  T is seen to be 280 inversely proportional to I FC . Thus, the remaining pole -1/ T is directly proportional to I FC or P. 281 power. In the design of a control system for a multi-input-multi-output plant, it is desirable that the 290 structure of the control system is selected in such a way that possible interactions between the control 291 loops is minimized. therefore results in a single-input-single-output (SISO) stack temperature control scheme, denoted as 303 the T control system in this paper. With the T control in place, the remaining two control variables in fuel q 304 and  can be utilized to perform the load tracking of P while maintaining u at the optimal value. This 305 strategy leads to a two-input-two-output P control system. 306 In summary, as shown in Figure 4, a hierarchical control structure is proposed to achieve both the 307 maximum electrical efficiency operation and stack temperature control of the SOFC when the FC 308 tracks the power demand. The structure is based on the inherent differences in the speeds of response 309 of P, u and T of the SOFC to the demand changes.

Design of the P and T Control Systems 316
The detailed design procedure of the P and T control systems is described in this section. 317

SOFC Dynamic Model for the Design of P Controller 318
According to Section 3.2, T can be assumed constant during the P control process. Therefore, the 319 nonlinear model given in Figure 1 can be linearized around the plant initial operating state. For the 320 convenience of the analysis and controller design, the plant variables are normalized in the following 321 way. The values of the state variables in fuel q max , , u max , P max , δ max , which correspond to the operating 322 condition when the SOFC operates at the maximum P, are selected as the base for the normalization. 323 The normalized output-control model shall be of the form The subscript '0' in (32)-(37) indicates the initial value of the respective variables when the SOFC 329 operates at η max . 330

Selection of P Control Output-input Variables Pairs 331
Although there are many methods of designing a control system for a general two-input-two-output 332 plant, the decentralized control is a widely used approach. The advantages of the decentralized control 333 include hardware simplicity, operation flexibility, and the relative ease in the controller design and 334 tuning. However, the dynamic performance of the resulting two SISO sub-systems may be degraded 335 by any unaccounted interactions between the two control loops. Therefore, in order to design a feasible 336 and robust controller, an important step is to determine the most suitable two output-input variable 337 pairs for the two SISO sub-systems. 338 The With the typical values given in Table 1 and the SOFC operating at η max , the variations of each 345 element of  are shown in Figure 5. It is shown that the values of the off-diagonal elements λ 12 and λ 21 346 are closer to 1 compared to that of the diagonal elements λ 11 and λ 22 , particularly under heavy load 347 conditions. This means the selection of the output-input pair (P-in fuel q , u-) will be more suitable 348 because the interactions of the P- and u-in fuel q loops are smaller and decreases as P increases. 349 Therefore, (P-in fuel q , u-) were selected as the output-input variable pairs when designing the P control 350 system. 351 denotes its reference value. The figure has been configured to reflect the outcome of the pair selection 356 described in the previous sub-section, i.e. the adoption of the (P-in fuel q , u-) output-input variable pairs. 357 The system of Figure 6(a) is then split into two independent SISO systems, with each SISO having the 358 structure shown in Figure 6(b). The so-called multiplicate model factor (MMF) is utilized to account 359 for the loop interactions between the two SISO systems. In Figure 6(b), c i (s) is the respective controller 360 where the subscript "i" denotes either P or u. The design method for c i (s) can be summarized as 361 follows. 362 Step One: Design the c i (s) controllers without considering loop interactions. Suppose the controller 363 c i (s) in Figure 6(b) is the PID type and is tuned using the simple internal mode control (SIMC) method 364 described in [32]. Thus for a second-order system g ii (s) with a dc-gain k i and a time delay c i (s) shall be of the form, 366 It can be seen from (33) and (34) that in g Pf (s) and g u (s),  i =0. Set the desired closed-loop cross-367 over time constant  ci equals to  ii , a practice often used in the process control [32], the PID parameter 368 settings are then given as, 369 With this set of settings, the phase margin of c i (s)g ii (s) is 90º and it meets the typically desirable 370 phase margin of 60º. While it can be seen from (33) that g Pf (s) is independent of P, however, (34) 371 shows the dc-gain of g u (s) will be the maximum when P = P min . Therefore, the c u (s) controller must be 372 designed based on the minimum SOFC output power condition. 373 Step Two: Calculate the MMF by using dynamic Relative Index (dRI) and obtain the equivalent 377 transfer function of each SISO system. In Figure 6(b), it is shown the output of the sub-system i will be 378 superimposed by the output y' i from the neighboring system j. Define the dRI between y' i and the 379 output of the subsystem i as  ij (s).  ij (s) for the P control system can then be derived using the 380 technique described in [32] for a general process system, 381 The MMF of the i th SISO system is then given as, i  and  i are the magnitude and phase angle of the MMF, respectively. Therefore, the rectangular 383 box formed by the delineated lines in Figure 6(b) represents the equivalent transfer function g' ii (s) 384 where 385 Step Three: Redesign each c i (s) based on the equivalent transfer function g' ii (s). In a manner 389 similar to that in designing the SIMC-PID controller in Step One, if the time constant corresponding to 390 the closed-loop cross-over frequency of c i (s)g' ii (s) is selected to be the same as the process maximum 391 time constant  ci , as suggested in [32], the new controller settings for c i (s) are, 392 ));

T Control System Design 393
As explained in Section 3.2, the temperature control involves slower dynamics of the hierarchical 394 control system. Since the SOFC output power P can be maintained at the targeted value through the 395 regulation of the faster P-in fuel q and u- control loops, P can be assumed to have reached a quasi-steady 396 state value, even before the T control loop starts to become active. From (8)-(11) and Figure 1 2 will lead to a decrease in T because 400 T in < T 0 . However, as explained in Section 2, the parameter B will increase when I FC increases. As B 401 also appears in the denominator of (47), the consequence is that the phase margin of the transfer 402 function g To (s) in (47) will be at the minimum when I FC is at the minimum, i.e. when P = P min . 403 Therefore, the parameters of the temperature controller c T (s) can be determined using the same SIMC-404 PID tuning method as that used in the design of the P controller. c T (s) is to be tuned under the most 405 onerous condition when the SOFC is at the minimum load. 406

Overall Load Tracking and Temperature Control Scheme 407
The overall control scheme for the SOFC to achieve  max during the load tracking process is 408 illustrated in the "Control and Optimization System" portion of Figure 2. Based on the above analysis, both P and u will be ahead of T to reach the reference values. It can be seen from (8) and will be 410 illustrated in Section 5 that continuously varying P may lead the transient T to exceed the constraint 411 given in (24). In order to guarantee the cell lifespan, T should be monitored on-line. If the measured T 412 is not within the pre-set band which is close to its operating boundaries, as shown in Figure 2, the "Pre-413 filter" block will convert the error between the targeted power level P t and the SOFC output power P 414 into continuous adjustments P ref .

Case Studies 419
The benchmark SOFC power plant in [21,22,30] was used to carry out case studies to illustrate the 420 efficiency of the proposed hierarchical control scheme. The 100kW power plant is connected to a 421 400V ac system and the associated parameters are given in Table 1. On the 400V and 100kVA base, 422 the SOFC power plant ac terminal voltage is assumed to be constant at 1.05pu. It is also assumed that 423 the link reactance X in Figure 2 is 0.05pu. The simulation tool used is MATLAB/SIMULINK. 424

425
Suppose the SOFC is to operate between 10kW and 100kW. higher than η 1 . The optimal η can be found on the boundaries of T and u when P is at the low level. 429 The highest η max is 43.4% when P=0.3pu. However, the power consumed by the air compressor is over 430 15% of the output power if the cell operating pressure is 0.15MPa. This will cause η max less than 40% 431 under the maximum output power condition. 432 As discussed in Section 4.4, the controller for the T control system is also designed when P = P min . 456 Accordingly, the T controller is c T (s) = -0.64(1+1/292s). 457 Again, the above three controllers designed for the model shown in Figure 1 indicate the SOFC is 458 feasible for slow load tracking application. The tracking speed is firstly limited by the P controllers. As 459 the cross-over time constant of c P (s) is around 0.2s, it will be safe for the SOFC to track the load 460 within this bandwidth. On the other hand, the cross-over time constant of c T (s) is about 0.0035s. The T 461 control system is much slower than the P control system. As shown in (8) Figure 7a, the "Pre-filter" block 471 can generate the new power reference only the measured T is below a pre-set threshold value (say 472 1263K). It will take about 30 minutes to achieve the targeted power due to the variable power ramp 473 rate. If the temperature control is not considered, the targeted power can be reached in about 12 474 minutes. The on-line control strategy proposed in [27] is designed such that the final load level shall be 475 reached within the minimum time. Indeed, the on-line method shown in Figure 7(a) has a higher speed 476 of response, i.e. the 0.9pu power change is reached in about 100s. However, in [27], the ratio of the 477 fuel flow rate to oxygen flow rate is kept constant at 1.145. It is shown in Table 2 that it is impossible  478 to maintain a constant T with the flow rate ratio fixed. Therefore, the constant temperature assumption 479 made in [27] is invalid once the energy balance consideration is included in the dynamic model. 480 An interesting observation is that the direction of the u variation based on the on-line method is 481 opposite to that obtained under the hierarchical control. This is shown in Figure 7  According to the discussion above, it can be concluded that the proposed hierarchical control 499 scheme will be able to track the power demand in a safe manner, and the mutual loop interactions have 500 been included in the control system design. The scheme will also lead to the maximum electrical 501 efficiency operations of the SOFC. 502

Conclusions 503
By considering the power loss caused by the air compressor, the maximum electrical efficiency 504 operating conditions of the grid-connected SOFC can be obtained by solving a nonlinear programming 505 problem which is subject to constraints of stack temperature, fuel utilization factor and output power. 506 In order to accommodate the inherently different dynamical processes within the SOFC, a hierarchical 507 control scheme for the grid-connected SOFC power plant has been proposed. The scheme consists of a 508 P control system and a relatively slower T control system. The case studies verify that the proposed 509 hierarchical control scheme can achieve maximum efficiency load tracking operation for the grid-510 connected SOFC with the stack temperature bounded within the preset constraints. 511 The FC-based DG technology is still far from mature. Continuous improvements on the FC 512 performance, durability and making it economically competitive are needed in order to realize its wide 513 application. For power system analysis, the SOFC model and control strategies should be improved 514 and verified through experiment in the future work. 515

Conflicts of Interest 516
The authors declare no conflict of interest. 517