A linear quadratic regulator control of a stand-alone PEM fuel cell power plant

Abstract

This paper introduces a technique based on linear quadratic regulator (LQR) to control the output voltage at the load point versus load variation from a stand-alone proton exchange membrane (PEM) fuel cell power plant (FCPP) for a group housing use. The controller modifies the optimal gains k_i by minimizing a cost function, and the phase angle of the AC output voltage to control the active and reactive power output from an FCPP to match the terminal load. The control actions are based on feedback signals from the terminal load, output voltage and fuel cell feedback current. The topology chosen for the simulation consists of a 45 kW proton exchange membrane fuel cell (PEMFC), boost type DC/DC converter, a three-phase DC/AC inverter followed by an LC filter. Simulation results show that the proposed control strategy operated at low commutation frequency (2 kHz) offers good performances versus load variations with low total harmonic distortions (THD) , which is very useful for high power applications.

Keywords

modeling of proton exchange membrane fuel cell (PEMFC) / controlling of PEMFC / linear quadratic regulator (LQR) / DC/DC converter / DC/AC inverter

Introduction

As the world seeks to find alternative means to produce clean power, fuel cells (FCs) emerge as a promising source of power generation. Depending on their type, FCs can be utilized for a wide range of applications, varying from a few Watts to mega-Watt applications. They have been recognized as promising candidates for power generating devices in the automotive, distributed power generation and portable electronic applications [1,2]. FCs basically convert chemical energy of hydrocarbon fuels, typically hydrogen directly into DC form of electrical energy. The private sectors and utilities are now concentrating on green power technologies with accrued benefits on account of their cleanliness, modularity, high efficiency and reliability. Among the different green power technologies e.g. wind power, photovoltaic, micro turbine, and FCs, the FC based distributed generation is considered as one of the most promising technology due to its high operating efficiency (40%-60%), reliability and higher potential capability [3,4].

The FC based distributed generation can be placed anywhere in the system to upgrade system integrity, reliability and efficiency. Among the various types of FCs, PEMFC are particularly attractive for residential use due to their relatively low operating temperature (~80ºC) and good dynamic response [5,6]. The open circuit voltage of the single cell is in the range of 0.8-1.2 V. To get higher operating voltage and power; many such cells are stacked and connected in the form of cascaded series and parallel connection. An FC based power system mainly consists of a fuel-processing unit (reformer), FC stack and a power conditioning unit. The power conditioning unit is simply a DC/DC converter used to raise the DC output voltage, which is generally the DC bus voltage, followed by a single-phase or three-phase DC/AC inverter. The FC uses hydrogen as input fuel and produces DC power at the output of the stack [7]. The performance of the stack is expressed by the polarization curve, which gives the relation between stack terminal voltage and load current, as shown in Fig. 1.

It is evident from Fig. 1 that the cell voltage decreases almost linearly as the load current increases. Therefore the output voltage should be regulated at a desired value [2-6]. To enable the FCPP to conform to the load changes, it is essential to control the active and reactive power output of the power plant. Many schemes of control have been developed. In Refs. [8] and [9], a technique has been proposed to control the active and reactive power output from an FCPP. The proposed technique is based on controlling the inverter modulation index (m) to control the voltage level and the reactive power output from the FCPP. The active power flow from the FCPP to the load is controlled by controlling the phase angle of the AC output voltage (δ). In Ref. [10], a neural network (NN)-based controller to control the active and reactive power output of the FCPP has been developed.

In this paper the proposed linear quadratic regulator (LQR) controller modifies the optimal gains k_i by minimizing a cost function, and the phase angle of the AC output voltage to control the active and reactive power output from an FCPP to match the terminal load. The FC output voltage is modeled based on the FC load current. Since numerous FC based power system topology can be chosen to meet the required criterion, in this work, a commonly used 45 kW FC, a boost type DC/DC converter, a three-phase DC/AC inverter followed by an LC filter to perform voltage and real power control as well as power quality performance evaluation are selected with respect to residential load variations. The simulation is performed based on a group of 9 homes, each of which composed of a standards home appliance such as refrigerator, TV, microwave oven, washing machine, air conditioner, etc. as loads. To track the voltage of the system, a linear quadratic controller is proposed for the PEMFC power plant. The most common and hazardous power quality problem is the presence of harmonics. Power electronics-based systems, such as static power converters, are responsible for harmonic pollution of the power distribution system.

PEM FC model

In Ref. [11] Padulles et al. introduced a model for solid oxide fuel cell (SOFC). The model has been modified to simulate a PEMFC [12]. This model is based on simulating the relationship between the output voltage and partial pressures of hydrogen, oxygen, and water. A detailed model of the PEMFC is illustrated in Fig. 2 and the model parameters are given in Notation.

The 45 kW PEM FCPP model parameters are based on a 272 V DC bus voltage with a stack current capacity of 18 A, and a cell voltage of 0.8 V. The PEM FCPP consists of 9 parallel stacks, each stack has 340 cells in series. Using the indicated number of cells and stacks the 45 kW PEM FCPP model parameters are listed in Table 1.

The simulated characteristics of PEM FC stack voltage for the fixed values of input fuel pressures for single cell is shown in Fig. 1. It can be seen that at low current level, the ohmic loss becomes less significant and the increase in output voltage mainly resulted from the slow chemical reactions. So this region is also called active polarization. At very high current density, the voltage falls down significantly because of the reduction of gas exchange efficiency. This is mainly caused by the over flooding of water in catalyst and this region is also called concentration polarization. Intermediate between the active region and concentrations region there is a linear slope which is mainly caused by internal resistance offered by various components of the FC. This region is generally called ohmic region [14, 15].

Power converter topologies

There are numerous power converter topologies for FC applications that can be chosen to meet the overall performance requirement. Since FC operates in the low voltage and the grid voltage is relatively high (either 120 Vrms or 230 Vrms), the voltage must be amplified either in the DC/DC stage or in the AC/AC stage, as depicted in Fig. 3 [16].

The best topology depends on the cost, desired performance, and application of the system. The topology chosen in this study for voltage/real power control and power quality evaluation consists of an 272 V, 45 kW PEMFC, a boost type DC/DC converter, and a three-phase DC/AC inverter followed by an LC filter.

A simple representation of the considered FC based power system consists of the PEMFC, a power conditioning unit, and load, as demonstrated in Fig. 4.

The power control scheme has been developed in Ref. [8] that can be summarized as

(1)P_{\mathrm{a}\mathrm{c}}={\displaystyle \frac{v_{\mathrm{c}}{v}_{\mathrm{L}}}{X}}\sin \delta ,

where P_ac is the AC power; v_c, the capacitor voltage; v_L, the load voltage; X, the line reactance; and δ, the phase angle of the AC voltage.

Assuming a lossless inverter and DC/DC converter is

(2)P_{\mathrm{a}\mathrm{c}}=P_{\mathrm{d}\mathrm{c}}=V_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}{I}_{\mathrm{d}\mathrm{c}},

(3)q_{H_{\mathrm{2}}}={\displaystyle \frac{N_{stack}{N}_{\mathrm{0}}{I}_{dc}}{\mathrm{2}FU}},

where P_dc is the DC power; I_dc, the stack current; q_{H_{\mathrm{2}}}, the input molar flow of hydrogen; N_stack, the number of stacks; N₀, the number of cells per stack; F, the Faraday’s constant; and U, the utilization factor.

(4)\sin \delta ={\displaystyle \frac{\mathrm{2}FUX{V}_{cell}}{v_c{v}_L{N}_{\mathrm{0}}{N}_{stack}}}{q}_{H_{\mathrm{2}}}\mathrm{.}

Assuming a small phase angle is,

(5)\delta ={\displaystyle \frac{\mathrm{2}FUX{V}_{cell}}{v_c{v}_L{N}_{\mathrm{0}}{N}_{stack}}}{q}_{H_{\mathrm{2}}}\mathrm{.}

Equation (5) describes the relationship between output voltage phase angle δ and hydrogen flow q_{H_{\mathrm{2}}} . Equations (1) and (5) show that the active power as a function of the voltage phase angle δ can be controlled using the amount of hydrogen flow. Output voltage can be controlled by the LQR controller. The model of the PEMFC and voltage controllers used in the computer simulation is sketched in Fig. 5.

DC/DC boost converter

The increase in load power decreases the FC output voltage according to the model dynamics. Therefore, a boost type DC/DC converter is used at the FC system bus to maintain the 400 V output voltage. The topology of the boost type DC/DC converter is given in Fig. 6. In the converter, the gate signal of the IGBT is obtained using a band hysteresis controller-based system which determines the duty cycle according to the load side voltage.

The selection of components such as boost inductor value and capacitor value is very important to reduce the ripple generation for a given switching frequency. However, large inductance tends to increase the start-up time slightly while small inductance allows the coil current to ramp up to higher levels before the switch turns off [17].

(6)V_{dc}=V_{cell}{\displaystyle \frac{T}{t_{off}}}={\displaystyle \frac{V_{cell}}{\mathrm{1}-k}},

where t_{of f}= (1-k) T and t_on = kT.

Assuming a lossless circuit, the average input current is,

(7)I_{cell}={\displaystyle \frac{I_l}{\mathrm{1}-k}}.

Then

(8)\mathrm{\Delta }{I}_{cell}={\displaystyle \frac{V_{cell}(V_{dc}-V_{cell})}{f{L}_{dc}{V}_{dc}}}={\displaystyle \frac{V_{cell}k}{f{L}_{dc}}},

where ΔI_cell is the peak-to-peak ripple current of inductor; L_dc, and f indicate the switching frequency.

When the switch S_w is on, the capacitor supplies the load current for t= t_on. The average capacitor current during this time t_on is I_cdc=I_l and the peak-to-peak ripple voltage of the capacitor is

(9)\mathrm{\Delta }{V}_{Cdc}={\displaystyle \frac{I_{l}k}{f{C}_{dc}}}\mathrm{.}

The size of the reactive elements of boost converter can be determined from the rated voltage, current ripple, voltage ripple and switching frequency of the converter based on Eqs. (6), (8) and (9).

Description of the model

The LC filter, the line reactance and the load are considered to be the plant of the system. The inverter is controlled by the unipolar PWM. The power switches are turned on and off at the carrier frequency, where the first simple phase is presented in Fig. 7.

The plant can be modeled by the state space variable v_c and i_L as

(10)\stackrel{•}{\left[\begin{array}{l}{v}_c\\ i_L\end{array}\right]}=\left[\begin{array}{ll}{\displaystyle \frac{-\mathrm{1}}{ZC}}& {\displaystyle \frac{\mathrm{1}}{C}}\\ {\displaystyle \frac{-\mathrm{1}}{L}}& \mathrm{0}\end{array}\right]\left[\begin{array}{l}{v}_c\\ i_L\end{array}\right]+\left[\begin{array}{l}\mathrm{0}\\ {\displaystyle \frac{\mathrm{1}}{L}}\end{array}\right]\mathit{u},\mathit{y}=\left[\begin{array}{ll}\mathrm{1}& \mathrm{0}\end{array}\right]\left[\begin{array}{l}{v}_c\\ i_L\end{array}\right],

or

(11)\dot{\mathit{x}}=A\mathit{x}+B\mathit{u},\mathit{y}=C\mathit{x},

where

Z=Z_{Load}+X\mathrm{.}

Then, a discrete time model of the plant and sample time T_S is given by

(12)\mathit{x}(k+\mathrm{1})=A_d\mathit{x}(k)+B_d\mathit{u}(k),\mathit{y}(k)=C_d\mathit{x}(k)\mathrm{.}

where

\mathit{x}(k)={\left[\begin{array}{ll}{v}_c(k)& i_L(k)\end{array}\right]}^T,

A_d=I+T_{S}A,

B_d=T_{S}B\mathrm{.}

.

Linear quadratic controller

The model of the controllers used in the computer simulation is displayed in Fig. 8. The LQR controller has the objective of tracking the discrete sinusoidal reference r(k) in each sample instant [18]. The system output y(k) is the capacitor voltage in the discrete form of v_c(k). The state variables used in the LQR are the measured output voltage v_c(k), the inductor current i_L(k) integrated tracking error v(k); all with a feedback action and the discrete reference r(k) and its derivative ŕ(k) with a feed forward action. Each state variable has the weighting k_i tuned according to the plant parameters. The control system, as exhibited in Fig. 8, is, therefore, proposed.

Then, in the proposed system, the state vector z(k) is defined as

(13)\mathit{z}(k)={\left[\begin{array}{lllll}{v}_c(k)& \stackrel{}{i_L(k)}& v(k)& r(k)& \dot{r}(k)\end{array}\right]}^T.

And the LQR control signal is given by

(14){\mathit{u}}_{LQR}(k)=-K\mathit{z}(k).

To design the optimal gains k₁, k₂, ..., k₅, the system must be represented in the form of

(15)\mathit{z}(k+\mathrm{1})=\mathit{G}\mathit{z}(k)+\mathit{H}{\mathit{u}}_{LQR}(k),

where each state variable is calculated by a difference equation. The two first variables of vector BoldItalic(k) are obtained by Eq. (12). The signal v(k) is

(16)v(k+\mathrm{1})=e(k+\mathrm{1})+v(k),

where the error is given by

(17)e(k)=r(k)-y(k).

From Eqs. (12), (16) and (17), the difference equation can be obtained:

(18)v(k+\mathrm{1})=v(k)+r(k)+T_S\dot{r}(k)-C_d{A}_{d}x(k)-C_d{B}_d{u}_{LQR}(k).

The continuous time reference variables are

(19)\stackrel{•}{\left[\begin{array}{l}r(t)\\ \dot{r}(t)\end{array}\right]}=\left[\begin{array}{ll}\mathrm{0}& \mathrm{1}\\ -{\omega }^{\mathrm{2}}& \mathrm{0}\end{array}\right]\left[\begin{array}{l}r(t)\\ \dot{r}(t)\end{array}\right],\dot{\mathit{n}}=R\mathit{n}\mathrm{.}

This system generates a sinusoidal reference

(20)r(t)=\mathrm{230}\sqrt{\mathrm{2}}\sin (\omega t+\delta ),

and

\delta ={\displaystyle \frac{\mathrm{2}FUX{V}_{cell}}{V_{ac}{V}_{load}{N}_{\mathrm{0}}{N}_{stack}}}{q}_{H_{\mathrm{2}}}\mathrm{.}

In the discrete form, using a sample period T_S, the subsystem in Eq. (19) is given by

(21)\mathit{n}(k+\mathrm{1})=R_d\mathit{n}(k),

where

(22)\mathit{n}(k)=\left[\begin{array}{ll}r(k)& \dot{r}(k)\end{array}\right],

(23)R_d=I+T_{S}R\mathrm{.}

Then, using the state Eqs. (12), (18) and (21), the closed loop system representation becomes

(24)\mathit{y}(k)=\left[\begin{array}{lll}{C}_d& \mathrm{0}& \mathrm{0}\end{array}\right]{\left[\begin{array}{lll}\mathit{x}(k)& \mathit{v}(k)& \mathit{n}(k)\end{array}\right]}^T\mathrm{.}

The optimal gains of the control law in Eq. (14) are those that minimize the following cost function:

(25)\mathit{J}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \sum _{k=\mathrm{0}}^{\infty }\{{\mathit{z}}^T(k)\mathit{Q}\mathit{z}(k)+{\mathit{u}}^T(k){\mathit{R}}_u\mathit{u}(k)\},}

where BoldItalic and BoldItalic_u are chosen as positive definite matrices that set the weighting of each state and of the control signal.

The BoldItalic gains can be obtained through the evaluating Riccati equations [19]

(26)$$\begin{gathered} \mathit{S}(k)={\mathit{G}}^T\mathit{S}(k+\mathrm{1})\mathit{G}+\mathit{Q}-{[{\mathit{H}}^T\mathit{S}(k+\mathrm{1})\mathit{G}]}^T \\ {[{\mathit{R}}_u+{\mathit{H}}^T\mathit{S}(k+\mathrm{1})\mathit{H}]}^{-\mathrm{1}}[{\mathit{H}}^T\mathit{S}(k+\mathrm{1})\mathit{G}], \end{gathered}$$

(27)\mathit{K}(k)={\mathit{R}}_u^{-\mathrm{1}}{\mathit{H}}^T{({\mathit{G}}^T)}^{-\mathrm{1}}(\mathit{S}(k)-\mathit{Q}).

A good flexibility in the design of the controller is provided by the selection of BoldItalic and BoldItalic_u matrices.

Results and discussions

The model parameters of the PEM FC shown in Fig. 2 are given in Table 1. A linear quadratic controller is used for the FC voltage and active power control, which has the objective of tracking the discrete sinusoidal reference r(k) according to load variations shown in Fig. 9.

The LQR controller parameters and system specifications are presented in Table 2.

The 45 kW PEMFC supplies nine homes evenly distributed in the three phases of the inverter. The data corresponding to each home appliance are tabulated in Table 3. The profile of home appliance varies from linear to nonlinear load with different harmonic levels. While some of the nonlinear home appliances such as television, computer, fluorescent lamp and light dimmer have high current THD content, others such as microwave oven, refrigerator and air conditioner have low current THD content.

The series of step changes in power is converted to stack current to calculate the corresponding FC DC voltage. The transient and steady state response of the system using LQR controller has been found to be excellent, as shown in Figs. 10 and 11. From Figs. 10 and 11, it is observed apparently that the increase in load power increases the FC current, which results in decreased FC output voltage or vice-versa. These results are based on 230 Vrms and 45 kW.

Figure 10 shows the AC output power transferred to the load. From Fig. 10, it is seen evidently that the output active and reactive powers endure time delay in meeting the load demand.

From Figs. 10 to 12, it is noticed clearly that an increase in the load increases feedback current, which in turn decreases output voltage of the FC. The increase in current increases hydrogen flow rate and increases phase angle of output voltage, as shown in Figs. 13 and 14. Figure 15 demonstrates the effect of the DC link voltage for the changes in load current. It is observed that the design of the simple boost converter with the hysteresis controller gives a better performance for changes in load without the use of any storage device. Figure 16 shows that the LQR controller maintains a constant voltage output under load commutations of Fig. 9. Figures 17 to 21 show the corresponding optimal gains.

Following a step increase or decrease in the load current leads to FC voltage drop or increase, below or above its nominal value. Then, the LQR controller stabilizes it with a very short time delay.

Figure 22 shows the instantaneous wave of the first simple output voltage with low total harmonic distortions (THD). Figure 23 shows the THD of output voltage under a series of load commutations. It is evident seen from Fig. 23 that the THD of the output voltage stays below the 3% limit. Figure 24 shows the instantaneous wave of the three simple output voltages.

Conclusion

This paper introduces a technique based on LQR to control the output voltage at the load point versus load variation from a stand-alone proton exchange membrane FC power plant for a group housing use. The paper presents an overall evaluation of a 45 kW PEMFC power plant in terms of stack voltage modeling, voltage control and active power and output voltage quality.

The proposed LQR controller modifies the optimal gains k_i by minimizing a cost function, and the phase angle of the AC output voltage to control the active and reactive power output from an FCPP to match the terminal load.

It is evidently observed from the computer simulation results that satisfactory dynamic responses are obtained from the proposed control scheme. The proposed control strategy offers good performances versus load variations with low THD even at low frequencies (2 kHz) making it very useful for high power applications.

References

[1] Wang L, Husar A, Zhou T, Liu H. A parametric study of PEM fuel cell performances. International Journal of Hydrogen Energy, 2003, 28(11): 1263-1272

[2] Yalcinoz T, Alam M S. Improved dynamic performance of hybrid PEM fuel cells and ultracapacitors for portable applications. International Journal of Hydrogen Energy, 2008, 33(7): 1932-1940

[3] Farooque M, Maru H C. Fuel cells—the clean and efficient power generators. Proceedings of the IEEE, 2001, 89(12): 1819-1829

[4] Ellis M W, von Spakovsky M R, Nelson D J. Fuel cell systems: efficient, flexible energy conversion for the 21st century. Proceedings of the IEEE, 2001, 89(12): 1808-1818

[5] Hashem Nehrir M, Wang C, Shaw S R. Fuel cells: promising devices for distributed generation. IEEE Power & Energy Magazine, 2006, 4(1): 47-53

[6] Larmine J E, Dicks A. Fuel Cell Systems Explained. Chichester, England: John Wiley and Sons, 2000

[7] EG&G Technical Services, Inc.Fuel Cell Hand Book. 7th ed. U.S. Dept. Of Energy, Office of Fossil Fuel, National Energy Technology Laboratory, West Virginia, 2000

[8] El-Sharkh M Y, Rahman A, Alam M S, Byrne P C, Sakla A A, Thomas T. Analysis of active and reactive power control of a stand-alone PEM fuel cell power plant. IEEE Transactions on Power Systems, 2004, 19(4): 2022-2028

[9] Tanrioven M, Alam M S. Modeling, control, and power quality evaluation of PEM fuel cell-based power supply system for residential use. IEEE Transactions on Industry Applications, 2006, 42(6): 1582-1589

[10] El-Sharkh M Y, Rahman A, Alam M S. Neural networks-based control of active and reactive power of a stand-alone PEM fuel cell power plant. Journal of Power Sources, 2004, 135(1-2): 88-94

[11] Padulles J, Ault G W, McDonald J R. An integrated SOFC plant dynamic model for power systems simulation. Journal of Power Sources, 2000, 86(1-2): 495-500

[12] El-Sharkh M Y, Rahman A, Alam M S, Sakla A A, Byrne P C, Thomas T. Analysis of active and reactive power control of a stand-alone PEM fuel cell power plant. IEEE Transactions on Power Systems, 2004, 19(4): 2022-2028

[13] Hamelin J, Agbossou K, Laperriere A, Laurencelle F, Bose T K. Dynamic behavior of a PEM fuel cell stack for stationary application. International Journal of Hydrogen Energy, 2001, 26(6): 625-629

[14] Arsov G L. Improved parametric PSpice model of a PEM fuel cell. In: 11th International Conference on optimization of Electrical and Electronics Equipment, Brasov, Romania, 2008, 203-208

[15] Cheng K W E, Sutanto D, Ho Y L, Law K K. Exploring the power conditioning system for fuel cell. In: 32nd IEEE Annual Power Electronics Specialists Conference. Vancouver, Canada, 2001, 2197-2202

[16] Andersen G K, Klumpner C, Kjaer S B, Blaabjerg F. A new green power inverter for fuel cells. In: IEEE 33rd Annual Power Electronics Specialists Conference. Queensland, Australia, 2002, 727-733

[17] Mohan N, Undeland T M, Robbins W P. Power Electronics Converters, Applications and Design. 3rd ed. Jon Wiley & Sons, 2001

[18] Boualaga R, Amar B, Ammar M, Loron L. Parameters and states estimation with linear quadratic regulator applied to uninterruptible power supplies (UPS). In: IEEE 32 th Annual IEEE Conference on Industrial Electronics. Paris, France, 2006, 2055-2060

[19] Ogata K. Discrete-Time Control Systems. Prentice-Hall, 1987

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg