Based on the previous results, two control strategies emerge as feasible for operating the
SOFC stack. Table 6.2 summarizes these control strategies. While both of these strategies have been considered before at the system level [22,23,57,58], the present work considers these strategies at the stack level. The first strategy involves manipulating the fuel flow to control the power, while manipulating the current density to control the fuel utilization. The second strategy is the reverse of the first—it involves manipulating the current density to control the power, while manipulating the fuel flow rate to control the fuel utilization. Control of the fuel utilization may be achieved by manipulating either the fuel flow rate or the current density because changing either of these variables induces significant changes in the fuel utilization (Figs. 6.2b and 6.4b, respectively). Likewise, control of the power may be achieved by manipulating either the fuel flow rate or the current density because changes in both of these variables induce significant changes in the power (Figs.6.3and6.4a, respectively). In either strategy, the air flow would serve to control SOFC temperature, as the air flow was found to negligibly influence the fuel utilization and power while still having the potential to control the PENtemperature during a transient event.
A major difference between these control strategies is their interdependent quality. As mentioned previously, interdependence may be defined as the inadequacy of a manipulated variable to effectively control a targeted variable, unless tight control of another variable(s) is assumed. Interdependence between pairs of control variables is undesired, as it could lead
to oscillations between control levels in a cascade controller. Importantly, the first strategy (manipulating the fuel flow rate to control the power, while manipulating the current density to control the fuel utilization) gives rise to strong interdependence. When the fuel utilization is maintained at 85%, in particular, Fig. 6.3 demonstrates that manipulating the fuel flow rate influences the power significantly. However, when the fuel utilization is allowed to vary freely, Fig. 6.2b demonstrates that manipulating the fuel flow rate hardly influences the power at all. Hence, controlling the power using the fuel flow rate is sensible only if tight control of the fuel utilization is implemented. In a cascade controller, such as that proposed by Martinez, et al. [23], such a control strategy may result in oscillations between the fuel utilization and power control levels, as these control loops would be highly interdependent.
The second control strategy, on the other hand, appears to minimize interdependence. If the current density controls the power, that is, then the power and fuel utilization operate fairly independently. In particular, it can be seen from Fig. 6.4a that manipulating the current density gives rise to significant changes in the SOFC power, without placing any restrictions on the fuel utilization. Likewise, Fig.6.2b shows that manipulating the fuel flow rate gives rise to significant changes in the fuel utilization, without placing any restrictions on the current density. Because these control loops operate fairly independently, control need not jump back and forth between the power and fuel utilization levels to satisfy control criteria. Of course, restrictions may apply. Changing the current density too rapidly, for instance, may cause the fuel utilization to overshoot or undershoot its bounds (as shown in Fig. 6.4, the fuel utilization responds to the fuel flow in seconds, whereas the power responds nearly instantaneously to the current density). However, rate limitations could be incorporated into the control strategy at the system level [22].
Based on the considerations discussed previously, manipulating the current density is the most effective way to control the SOFC power, while manipulating the inlet fuel flow rate is the most effective way to control the fuel utilization. Relying on one variable (current density) to control power, rather than relying on two variables (current density while holding the fuel utilization fixed), simplifies the control logic. The time required for theSOFCstack to meet a power demand provides further motivation for adopting this strategy. If the current density controls the power, then the power responds instantaneously to a load change (Fig. 6.4a).
If the fuel flow rate controls the power, on the other hand, then the power responds slower to load changes (Fig. 6.3). Buildings experience significant load change over the course of a day [149,160,161], and meeting power demand quickly is important.
6.4 SUMMARY
This chapter investigated the response of key SOFC variables to step changes in the inlet fuel flow rate, current density, and inlet air flow rate. Manipulating the current density significantly changed the SOFC stack’s power without placing any restrictions on the fuel utilization. Manipulating the inlet fuel flow rate, on the other hand, required tight control of the fuel utilization; otherwise, the inlet fuel flow rate exhibited little or no influence on the SOFC stack’s power. Because the former strategy provides greater independence between control loops, it is recommended that this strategy be considered for use in a cascade controller. Consideration has also been given in this study to the time required for theSOFC
to meet a power demand. The SOFC power responded quicker to changes in the current density (near-instantaneous) compared to changes in the inlet fuel flow rate (seconds), thus providing further motivation for adopting the former strategy. The next chapter continues to consider the SOFC stack’s dynamic behavior, looking more closely at electrochemical dynamics.
7.0 CHARGE DOUBLE LAYER
The fuel cell stack model allows for dynamic simulations on the millisecond timescale. Impor- tantly, if the charge double layer extends beyond the millisecond timescale, then it will likely influence the fuel cell stack’s control logic (described in the previous chapter), potentially leading to undesired operation. Although the charge double layer effect has traditionally been characterized as a millisecond phenomenon, longer timescales may be possible under certain operating conditions. The present chapter identifies operating conditions that give rise to unusually long electrochemical settling times inside the SOFC stack. Baseline con- ditions are first defined, followed by consideration of minor and major deviations from the baseline case. The present work also investigates the behavior of the fuel cell stack with a relatively large double layer capacitance value, as well as operation of the SOFC stack under proportional-integral (PI) control. The fuel cell stack model is simulated under step load changes. It is found that high activation and concentration polarizations correspond to unusually long electrochemical settling times, as do large capacitance values. Thus, while neglecting the charge double layer simplifies the fuel cell model, it may also detract from the fuel cell model’s accuracy under certain operating conditions.
7.1 CHARGE DOUBLE LAYER
The charge double layer is a (dual) layer of positive and negative charge that accumulates along the electrode-electrolyte interfaces, giving rise to a capacitor-like effect. Charge may accumulate due to electrochemical reactions or charge diffusion across the interfaces, or possibly another cause [26,43]. An example of such a charge configuration is shown in
Fig. 7.1a, where the negative charges represent oxygen ions being transported from the cathode to the anode through the electrolyte. Clearly, the charge double layer resembles an electric capacitor. Similar to an electric capacitor, the charge double layer may be charged or discharged, depending on the direction of current, or load (Fig. 7.1b). As discussed in Chapter 3, the voltage drop across the charge double layer is treated as an irreversibility in the SOFC, similar to the ohmic polarization. That is, the double layer polarization is subtracted from the Nernst potential when calculating the fuel cell’s operating voltage:
Vop = EN− Vdbl− iRohm (3.25 revisited)
where Vop is the fuel cell’s operating voltage, EN is the Nernst potential, Vdbl is the double layer polarization, i is the electric current, and Rohm is the ohmic resistance. The present study is especially concerned with the time required for the charge double layer to settle following a load change. The time constant of the electrochemical model is given by [64]:
τdbl= (Ract+ Rconc) × Cdbl (7.1)
where τdbl is the electrochemical time constant, Ract is the activation resistance, Rconc is the concentration resistance, and Cdbl is the double layer capacitance. The activation and concentration resistances equal the ratio of the activation and concentration polarizations to the electric current. It is evident from Eqn. (7.1) that increasing Ract, Rconc, or Cdbl slows the fuel cell’s response to load changes. Section 7.2 further explores operating conditions that give rise to high values of Ract and Rconc. A high value of the double layer capacitance is considered during dynamic simulations presented in Section 7.3.
(a)
(b)
Figure 7.1 The charge double layer (adapted from Ref. [43]): (a) Charge double layer,
(b) Equivalent capacitor showing the charging and discharging of the charge double layer.