Closed-Loop Design for Standalone Photovoltaic-Battery Hybrid Power System
Xiong Xiaoling
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (North China Electric Power University), Beijing, China
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To cite this article:
Xiong Xiaoling. Closed-Loop Design for Standalone Photovoltaic-Battery Hybrid Power System. Journal of Electrical and Electronic Engineering. Vol. 4, No. 5, 2016, pp. 131-138. doi: 10.11648/j.jeee.20160405.17
Received: November 1, 2016; Accepted: November 22, 2016; Published: November 24, 2016
Abstract: The photovoltaic-battery hybrid power system is constructed with photovoltaic panels and a battery, which are connected to the load via a boost converter and a bidirectional buck/boost converter, respectively. Depending on the amount of available solar power, the photovoltaic panels may operate in a maximum power point tracking (MPPT) mode or off-MPPT mode to regulate the output voltage, and at the same time, the battery may provide power to the load or store energy from the solar power. The whole system is thus designed to operate with multiple operating modes. Compared to the system of single operating mode, the stability analysis and closed-loop design is much more difficult, as the control loops are usually coupled with each other and the design for stable operation of such system requires consideration of the stability conditions for all possible operating modes. To design the closed-loop, the commonly used small-signal analysis method based on averaged state-space is helpless here, and then the nonlinear analysis method with discrete-time mapping model is performed to evaluate the stability boundaries of the system. The parameters of the closed-loop are chosen in the stable region of the stability boundary diagrams. Moreover, a prototype is built and the experimental results are shown to verify the nonlinear analysis method in the design of closed-loop.
Keywords: Renewable Power Generation System, Photovoltaic Panels, Multiple Operating Modes, Nonlinear Analysis, Closed-Loop, Stability
1. Introduction
In order to avoid the possibility of energy crisis and solve global environment problems, development renewable power generation systems with solar energy, wind energy and so on has gained more and more attention in recent years [1]. However, as climate, solar radiation, wind speed, etc. vary from time to time, the amount of available energy from renewable energy sources is fluctuating and even intermittent. Thus, a power system that depends entirely upon one kind renewable energy source can be unreliable. In order to provide continuous electrical power to the load, multiple renewable energy sources and energy storages, such as batteries, super capacitors, etc, are usually adopted to construct hybrid renewable power generation systems [2,3]. As the power distribution of the multiple sources is controlled reasonably, the stability and flexibility of the system can be improved.
The renewable power generation systems can be grid-connected systems or standalone system [4]. For localized applications in some remote non-electrified regions, constructing stand-alone power generation systems is often more cost-effective than the conventional deployment of the grid-connected power generation systems. The renewable energy sources and energy storages can be connected to a common dc bus or a common ac bus via a number of power converters. In the field of power supply, transportation, communication and so on, the renewable power generation system based on dc bus is more commonly used [5, 6]. Fig. 1 shows one of standalone hybrid renewable power generation system based on a common dc bus, in which photovoltaic (PV) panels are connected the dc bus via a dc/dc power converter, and the energy storage is connected to the dc bus with a bidirectional dc/dc power converter, respectively.
There are two sources in Figure 1, and then the power from each source needs to be managed, such as the solar power should be used as much as possible to save up the energy from the back up source, i.e., the battery. At the same time, the energy storage needs to be protected safely from damaging. Then, the PV panels may operate in a maximum power point tracking (MPPT) mode or off-MPPT mode to regulate the output voltage, and at the same time, the battery may provide power to the load or store energy from the PV panels. The whole system is thus designed to operate with multiple operating modes [7-10]. Compared to the system of single operating mode, the stability analysis and closed-loop design is much more difficult, as the control loops are usually coupled with each other and the design for stable operation of such system requires consideration of the stability conditions for all possible operating modes.
To design the parameters of the closed-loop, the commonly used small-signal analysis method based on averaged state-space [11-13] is helpless here. The two reasons are: (1) there are several control loops to regulate the several duty cycles in each operating mode, and the control loops are usually coupled with each other, which make the closed-loop design become difficult. The literature [10] provided an ideal to design the regulators first in the operating mode without control loop coupling, and then transmitted the designed parameters into other operating modes and checked the stability of the system. If they are not satisfied, the parameters of the regulators need to be adjusted until they meet the stability requirements. However, this method cannot be applied when the control loops are coupled in each operating mode. The literature [14] introduced a decoupling matrix to design the regulatorsof the coupled control loops, but this method is useless in the multi-operating-mode system, because the decoupling matrixes are different in each operating mode and introduce new couple for another operating mode. (2) The small signal model based on the state space averaging method can capture and analyze the low-frequency oscillation of the system, but cannot predict the instability at and near the switching frequency [15,16]. Therefore, this paper will analyze the stability of the photovoltaic battery hybrid power system (PBHPS) with nonlinear theory base on a discrete-time model. Through this method, the stability boundaries can be found and used to design the parameters of the closed-loop.
In this paper, we will first describe the PBHPS with multiple operating modes and the energy management strategy in Section 2. The nonlinear stability analysis with a discrete-time model is presented in Section 3. The stability boundaries under different choices of compensation parameters are graphically derived in Section 4. The analysis results can thus be used to design the parameters of regulators in the closed loops. In Section 5, we will give the experiment results to verify the nonlinear analysis method. At last, a conclusion is given in Section 6.
2. PBHPS Description
2.1. The Operating Modes of PBHPS
Figure 2 shows the photovoltaic-battery hybrid power system, which provides power to the dc load based on a solar power system and a battery connected to an output dc bus via a boost converter and a bidirectional buck/boost converter, respectively. The solar power is regulated by boost converter, while the bidirectional converter controls the power of the battery. When the solar power is insufficient than the load power, MPPT control strategy for the PVpanels is employed to collect power from the PV panels, and the bidirectional buck/boost converter operates as a boost converter to power the load. When the solar power becomes larger than the load power, the excessive power is used to charge the battery. At this time, if the charge current or charge voltage of the battery achieves the maximum value recommended by the manufacturers, the charge process needs to be controlled with the bidirectional buck/boost converter operating as a buck converter, and meanwhile, the PV panels needs to quit MPPT control and change to regulate the output voltage. Thus, the PBHPS is designed to operate with multiple operating modes.
According to the maximum power supplied by the PV panels and the state of the battery, the PBHPS has three normal operating modes and one shunt-down mode, as shown in Table 1, where P_{o} is the output power, P_{m} is the maximum solar power, V_{bmax} and I_{bmax} are the recommended maximum charge voltage and charge current, and V_{bmin} is the permitted minimum battery voltage. v_{bat} and i_{bat} are the voltage and current of the battery, as shown in Figure 2.
P_{m}<P_{o} | P_{m}≥Po i_{bat} | P_{m}≥P_{o} i_{bat}≥I_{bmax} | |
v_{bat}≤V_{bmin} | OFF | M_{1} | M_{2} |
V_{bmin} | M_{1} | M_{1} | M_{2} |
v_{bat}≥V_{bmax} | M_{1} | M_{3} | M_{3} |
If P_{m}<P_{o} and the battery is deeply discharged, the whole system is shut down.
Operating Mode M_{1} — If P_{m}<P_{o} and the battery is able to provide power, the PV panels power the load with the MPPT algorithm enabled. In this mode, the input current for the boost converter, i.e. i_{pv} is controlled. The battery provides complementary power through the bidirectional buck/boost converter operating in boost mode to regulate the output voltage v_{o}.
If P_{m}>P_{o} by a small margin, the excessive solar power is used to charge the battery, and the charge current i_{bat} is smaller than I_{bmax}. In this case, the control system is the same as operating mode M_{1}. The difference is that the inductor current i_{Lb}_{2} in Figure 2 reverses.
Operating Mode M_{2} — If P_{m}>P_{o} and i_{bat} reaches the limit I_{bmax}, it is necessary to control the charge current to protect the battery, so the bidirectional buck/boost converter operates in buck mode to control the charge current. Meanwhile, the PV panels need to quit MPPT control and regulate the output voltage.
Operating Mode M_{3} — If P_{m}>P_{o} and v_{bat} achieves the limit V_{bmax}, the bidirectional buck/boost converter operates in buck mode to control the charge voltage. Meanwhile, the boost converter still regulates the output voltage.
2.2. The Control System of PBHPS
In this paper, the Perturb & Observe algorithm [17] is chosen for MPPT implementation, as shown in Figure 3. The output current of the PV panels, denoted as i_{pv}, is compared with a reference current I_{pvref} which is calculated from the MPPT algorithm. Thus, the loop controls the PV power p_{pv} in the event of any change in duty cycle of boost converter and forces i_{pv} to follow the reference current I_{pvref}, then the PV panels output the maximum power P_{m}.
In this paper, the Perturb & Observe algorithm [14] is chosen for MPPT implementation, as shown in Figure 3. The output current of the PV panels, denoted as i_{pv}, is compared with a reference current I_{pvref} which is calculated from the MPPT algorithm. Thus, the loop controls the PV power p_{pv} in the event of any change in duty cycle of boost converter and forces i_{pv} to follow the reference current I_{pvref}, then the PV panels output the maximum power P_{m}.
Figure 4 shows the control system of PBHPS, where V_{of} = v_{o}H_{of} and v_{batf}= v_{bat}H_{bf}. Here, H_{of} and H_{bf} are the sampling coefficients for the output voltage and the battery voltage, respectively. They are proportionality constants implemented by resistor dividers. Also, K_{s}is the sampling coefficient for inductor current i_{Lb}_{1} and is also a proportionality constant. The compensation ramp V_{p} shown in Figure 4(b), is applied to increase the stability range of peak current controlled converters. With an additional diode D_{c} and a resistor R connected to ground, the control loop forthe boost converter can switch from the peak current loop controllingi_{Lb}_{1} to a dual control loop controlling v_{o}. At the same time, Q_{2} is the mainly controlled switch, regardless of the bidirectional buck/boost converter operating as a boost converter or a buck converter, and the duty cycle of Q_{3} is always complementary to Q_{2}, so a negative logic is inserted in the charge current andcharge voltage control loops to share the PWM comparator.
When the system operates in M_{1}, diodes D_{c}, D_{2} and D_{3} are off, D_{1} is on, the input current i_{pv} or i_{Lb}_{1} is controlled with the MPPT algorithm under a peak current control loop, and output voltage v_{o} is regulated through a voltageloop by the bidirectional converter which operates in boostmode. If P_{m}>P_{o} and i_{bat} reaches the limit I_{bmax}, then the charge current control loop is enabled, i.e., D_{2} is on, D_{1} is off. This will lead to an increase in v_{o} as excessive power is delivered to the load. Thus, the error signal of regulator1, v_{e}_{1}, will drop. When it falls below zero, diode D_{c} is turned on, and v_{o} is regulated through a dualloopconsisting of an inner peak current loop and an outervoltage loop. In this case, I_{ref} =I_{pvref} + v_{e}_{1}, then the MPPT algorithm disabled. At this point, the system is switched to M_{2} smoothly. If P_{m}>P_{o} and v_{bat} achieves the limit V_{bmax}, it will enable the charge voltagecontrol loop, causing D_{3} to turn on and D_{2} to turn off, and the system switches to M_{3}.
(a) The control system
(b) Typical waveforms for peak current control loop
3. The Discrete-Time Model
As an illustration, the operating mode M_{1} is analyzed with the discrete-time model in detail, and the same analysis procedure can be applied to other operating modes. In our study, the switching frequency of the two converters is the same and trailing edge modulation is used for "PWM Comparator2". Continuous Current Mode (CCM) operation is assumed, so diode D_{pv} and switch Q_{3} are always in complementary states to switches Q_{1} and Q_{2}. Hence, in one switching period, the system toggles among three switch states and the sequence takes the following order: (i) Q_{1} and Q_{2} are on; (ii) Q_{1} is on and Q_{2} is off (if d_{1}³d_{2}) or Q_{1} is off and Q_{2} is on (if d_{1}£d_{2}); (iii) Q_{1} and Q_{2} are off. Here, we assume that d_{1}³d_{2}. Then, the state equations for the main circuit in Figure 2 are
(1)
Where q_{1} and q_{2} are the switching functions of Q_{1} and Q_{2}, given as
(2)
Proportional-Integral (PI) regulators are employed, and the transfer function is
(3)
Here k_{p} and τ_{i} are the proportional gain and the integral time constant. In addition, k_{pj} and τ_{ij} (j = 1, 2, 3) are the parameters for "Regulator1", "Regulator2" and "Regulator3" in Figure 4(a), respectively.
In M_{1}, the boost converter operates with MPPT control, I_{ref}=I_{pvref}, whereas the bidirectional buck/boost converter operates in boost mode to control v_{o}. Thus, according to the control strategy in Figure 4(a), the modulation signal v_{c} is equal to the error signals v_{e}_{1}, and v_{c} satisfy
(4)
Therefore, the state variable vector of operating mode M_{1} is. The corresponding state equations for the three switch states are generally given by
(5)
Where n= 1, 2, 3 correspond to switch state (i) (ii) and (iii), respectively. E_{1} is the voltage vector, and is given as, so A’s and B’s are given by
(6)
(7)
We denote the sampled state value at starting and endinginstants of the nth period as x_{n}= x(nT_{s}), x_{n}_{+1}= x((n+1)T_{s}), respectively. For each state equation in (5), we can find the solution, i.e.,
(8)
(9)
(10)
Where I is unit matrix, F_{j}(x) is the transition matrix and is given by
(11)
To complete the discrete-time map, the relationship between the duty cycles and the state variables should be derived. During the period of Q_{1} is on, inductor current i_{Lb}_{1} rises, and when it reaches the reference level I_{ref}, switch Q1 is turnedoff. Thus, the switching function for Q_{1} can be derived as
(12)
Where C_{1} = [0 −K_{s} 0 0 0]，and m_{c} is the compensation slope given by m_{c} = V_{p}/T_{s}, as shown in Figure 4(b).
In the output voltage control loop, the "PWM comparator2" compares v_{c} with the ramp signal:
(13)
Where V_{L} and m_{ramp} are the lower threshold and rising slope of the ramp signal, respectively. Basically, Q_{2} is turned on if v_{c}>v_{ramp}, and is off otherwise. Then, we can also get a switching function for Q_{2}, as
(14)
Where C_{2} = ].
Combining (10), (12) and (14), an exact discrete-time model is derived. Suppose the equilibrium point is given by x_{n} = x_{e}. By setting x_{n}_{+1} = x_{n} = x_{e}, s_{1} = 0 and s_{2} = 0, the equilibrium point x_{e} and steady-state duty cycles D_{1} and D_{2} can be obtained. Then, the Jacobian of the discrete-time model evaluated at the equilibrium point is
(15)
All the eigenvalues of the Jacobian can be obtained with the characteristic equation, as
(16)
If all the eigenvalues are inside the unit circle, the equilibrium state is stable. Any eigenvalue crossing the unit circle to outside indicate the system becomes unstable.
4. The Stability Boundaries of PBHPS
In our study, the system specifications and the parameters of the converters are chosen, as shown in Table 2.
parameters | values | parameters | values |
V_{pv} | 20~34/V | I_{pvref} | 0~3/A |
V_{ocbat} | 34~40/V | r_{bat} | 8/mΩ |
V_{bmax} | 40/V | I_{bmax} | 3/A |
V_{o} | 48/V | I_{o} | 0~5/A |
L_{b1} | 48/μH | L_{b2} | 48/μH |
C_{f} | 1.88/mF | C_{bat} | 400/μF |
K_{s} | 0.1 | f_{s} | 100/kHz |
In this section, we will generate stability boundaries information. In general, the PI control parameters and slope compensation V_{p} are important deciding parameters, as k_{p} need to be set sufficiently largeto reduce the steady-state error while τ_{i} should be small toimprove the dynamic response. However, if k_{p} is too large or τ_{i} is too small, the system will lose stability. We will first find the worst operating condition for M_{1} with V_{pv}, I_{pvref}, I_{o} and V_{ocbat} serving as bifurcation parameters, and then the stable parameter space of the regulators are derived.
In order to find the worst operating condition for output voltage loop, we keep k_{p}_{2 }= 1, τ_{i}_{2} = 50μs and V_{p} = 0.1, which keeps the charge current control loop and peak current loop stable. Corresponding to the choice of component values given in Tables 2, Figure 5 shows the specific stability boundaries obtained through analysis at full, where B_{12} is the mode boundary between M_{1} and M_{2}.
From Figure 5, the following observations are made:
• The variation of PI parameters will cause the system loses its stability. As k_{p}_{1} becomes larger or τ_{i}_{1} becomes smaller, the output voltage control loop loses stability more readily.
• Parameters k_{p}_{1} and τ_{i}_{1} affect the system’s stability in all operating modes (M_{3} is the same to M_{2}), and they have narrower operating ranges in M_{1}. Hence, they should be carefully designed in M_{1}.
• When the system operates in M_{1}, the worst operating condition is V_{pv} and I_{pvref} are the smallest.
With the same analysis, we can get V_{ocbat} is the smallest at full load makes the output voltage control loop lose stability more easily. Moreover, V_{pv} decreases, the stable area of the peak current control loop shrinks
Now, the stability boundaries for "Regulator1" and slope compensation V_{p} at the worst operating condition can be derived, as shown in Figure 6. From Figure 6(a), it can be found that if V_{p}> 0.08V, the peak current control loop is always kept stable. According to Figure 6(b), we can see that if k_{p}_{1} = 1, τ_{i}_{1} needs to be larger than 1.1mS to keep the output voltage control loop stable, in this paper, τ_{i}_{1} is set to be 1.2mS.
For brevity, we omit the derivation of the stability boundaries for "Regulator2" and "Regulator3", and give the last results here: k_{p}_{2} = 1, τ_{i}_{2} = 10μs, k_{p}_{3} = 10, τ_{i}_{3} = 22μs.
(a) Stability boundaries in M_{1}
(b) Stability boundaries in M_{2}
(a) The parameter space of V_{p}
(b) The parameter space of "Regulator1"
5. Experiment Verification
A 240 W prototype, as shown in Figure 7, was built to verify the nonlinear analysis method based on discrete-time model.
Figure 8(a)-(c) shows the steady-state experiment waveforms at full load in three operating modes when V_{pv} = 31V. In Figure 8(a) and (b), the waveforms from the top to the bottom are the output voltage V_{o}; the drive signals of Q_{1} and Q_{2}, V_{GS}_{1} and V_{GS}_{2}; inductor current i_{Lb}_{1} and i_{Lb}_{2}, respectively. In Figure 8(d), the waveforms are V_{o}, battery voltage V_{bat}, i_{Lb}_{1} and i_{Lb}_{2}, respectively. From the figure, we can see that the system is stable in three operating modes. When I_{pvref}= 0.7A, the DC component is I_{Lb}_{1}= 5.3A, so P_{pv}<P_{o}, and the battery provides part of the power to the load, at this time, I_{Lb}_{2}> 0. When I_{pvref}= 3A, suppose the peak value of i_{Lb}_{1} can reach 30A, but the actual peak value is measured to be 14.2A. This is caused by the charge current is controlled and I_{Lb}_{2}= −3A, then the PV panels quit MPPT control and operates in constant voltage mode. Under the same condition, if the battery is fully charged, the charge voltage control loop is enabled, so the charge current decreases, as shown in Figure 8(c).
(a) M_{1}: I_{pvref} = 0.7A，V_{ocbat} = 39V
(b) M_{2}: I_{pvref} = 3A，V_{ocbat} = 39V
(c) M_{3}: I_{pvref} = 3A, V_{ocbat} = 40V
A step load change is applied to the prototype when V_{pv} = 30V. Figure 9 shows the experiment waveforms of i_{o}, v_{o}, i_{Lb}_{1}, i_{Lb}_{2} and v_{bat} under the load stepping up or stepping down between 10% load and full load, where v_{o} and v_{bat} are given as AC component. From Figure 9, we can see that the system can properly respond to the step change and switches its operating mode correctly and keeps stable after the transient process. Take Figure 9(a) as an example, when the load suddenly steps up to full load and then steps down to 10% load, the system switches from M_{1} to M_{2} and then switches to M_{1} again, still keeps stable.
The experiment results show that the control parameters designed with nonlinear analysis method can make the system operate correctly and stably.
(a) I_{pvref} = 1.1A (P_{m} = 280W), V_{ocbat} = 39V
(b) I_{pvref} = 1.1A (P_{m} = 280W), V_{ocbat} = 40V
6. Conclusions
Due to the intermittent nature of the availability of solar power and the bidirectional flow of the battery power, a commonly used standalone photovoltaic-battery hybrid power system is designed to operate with three operating modes. This system is constructed with photovoltaic panels and a battery is connected to the load via a boost converter and a bidirectional buck/boost converter, respectively. As the control loops are coupled with each other in every operating mode and make the closed-loop design become difficult and impossible with small-signal analysis method based on averaged state-space. This paper fixes this problem using the nonlinear analysis method based on discrete-time mapping model, which is performed to find the worst operating condition for each control loop and then evaluate the stability boundaries of the parameters of the closed-loops at the worst operating condition. At last, the parameters of the closed-loop are chosen in the stable region.. Moreover, a prototype is built and the experimental results are shown to verify the nonlinear analysis method in the design of closed-loop.
Acknowledgements
Project is supported by "the Fundamental Research Funds for the Central Universities" (2016BJ0092).
References