Current-Mode Controlled Battery Emulator

Abstract

This paper proposes a battery emulator based on a bidirectional non-inverting buck-boost power electronics converter. With the capability of bidirectional operation, it can emulate both charging and discharging processes. The proposed emulator is controlled with the advanced I² dual current-mode control (I²DCMC) algorithm, combined with a feedforward control, which ensures fast and accurate tracking of the voltage and current characteristics of the batteries. The emulator is universal in terms of the various mathematical models of the batteries, which can be implemented in real time. It has no limitations regarding different battery types. Detailed analysis and the design procedure of the proposed battery emulator are presented. The performances of the emulator are validated with simulation and experimental results for three battery types: polymer Li-ion, conventional Li-ion, and lead–acid battery. Both steady and transient states are analyzed, especially transitions between charging and discharging phases. The possibility of simple time scaling of charging/discharging processes is successfully achieved and demonstrated, which is very important in making tests faster, with preserved battery characteristics. Considering its low-cost and user-friendly operation, the proposed emulator can be a good alternative to the real batteries in experimental tests of different power electronics systems. The prototype, which is developed for the experimental verification of the emulator, is designed for and limited to the research of lower power ratings systems of up to 100 W. It is suitable in education to easily demonstrate the behavior of the batteries in multiple scenarios in controlled laboratory conditions.

1. Introduction

Batteries are energy storage devices which have a very important role in various power electronics applications today, especially in electric vehicles. Among different battery types, lithium-ion (Li-ion) batteries are one of the most promising and increasingly used storage devices in electric vehicles, grid energy storage systems, and other applications because of their numerous advantages, such as light weight, high energy density, no memory effect, and long lifetime [1,2,3]. However, major drawbacks of the Li-ion batteries are their high cost and safety, which limit their usage. Therefore, working with real batteries during laboratory tests of different battery-based systems requires high investment and care [4].

Except for price, there are other inevitable limitations and problems with using real batteries in experimental tests of power electronics systems: impossibility to obtain repeatable results, time-consuming charging/discharging processes, dependence on the ambient factors, necessity for battery maintenance and management, etc. To overcome these problems, in recent years, researchers have proposed different devices which can mimic the behavior of the batteries. These devices are called battery emulators. The output voltage and current of the battery emulator behave the same as the voltage and current of the real battery. Therefore, the battery emulator can entirely replace the real battery or battery pack in laboratories during the experimental prototyping and verification of various battery energy storage systems, which brings numerous benefits. By using the battery emulators, the test conditions can be controlled and repeated, which makes the entire testing process easier. Unlike real batteries, which take a long time to charge and discharge, the battery emulators can be configured with much shorter charging/discharging times. This allows a significant reduction in the research duration. Also, the battery emulators exclude the use of the battery management systems (BMSs), which are especially needed when working with the real Li-ion batteries, primarily because of their safety issues. On the other hand, if the task is to test the BMS itself, it can be easily tested on the battery emulators, because various scenarios can be quickly obtained and demonstrated without the need for time-consuming work with the real batteries [5,6,7,8,9]. Considering the high cost of the batteries, possibility of their damage, need for their replacement, and safety problems, the use of the battery emulators significantly reduces the research costs and improves research safety.

Usually, the commercially available battery emulators are very expensive, hard to purchase and do not meet all the users’ needs, especially in research and development. For these reasons, researchers have proposed many different types of battery emulators in the literature. Some of them are based on commercial programmable power supplies, which are controlled by various control and data acquisition platforms [8,9,10,11,12,13,14,15,16,17,18,19,20,21]. These power supplies often have high prices. They can be programmed to follow the characteristics of the battery models executed in real time, but in terms of their internal hardware and control modularity, they have limited usage. The battery emulators, consisting of different electronic modules, are proposed in [5,6,7,22,23,24,25,26,27]. They are mostly used for BMS validation, so it is not critical for them to have the power ratings of real batteries. Nevertheless, these modules have power amplifiers or regulators at their outputs to adjust the output voltage and current to the necessary real levels. However, the applied amplifiers have limited output voltage and current, which can be insufficient if the emulators are tested as energy sources instead of real batteries.

Among the existing battery emulators, the most suitable are the emulators based on power electronics converters, because of their modularity, scalability, possibility to obtain desirable voltage, and current levels appropriate for the real batteries and battery packs, wide control possibilities, etc. Various converter topologies have been applied: synchronous buck converter [2,28,29]; two-phase [30], three-phase [31,32,33,34,35], four-phase [36,37], six-phase [38], and eight-phase [39] interleaved synchronous buck converter; boost converter [40]; synchronous boost converter [4,41,42]; three-phase interleaved synchronous boost converter [43]; flyback converter [44,45]; cascaded dual-active bridge converter [46]; and other complex combinations of converters [3,47,48,49,50,51]. When using a buck or boost converter as a main power stage of the battery emulator, there is a limitation in its output voltage range, depending on the input voltage. This can be an issue if the emulator needs to have the possibility of implementation of different battery types and packs with various voltage levels. Interleaving, which brings numerous benefits, comes with other issues, such as increased complexity of topology and control structure of the emulator, and smaller efficiency due to the increased number of switching elements. A flyback converter can achieve a larger output voltage range, but it has its own limitations, especially regarding its maximum output power, which is a problem in the application of emulators in higher power systems. Other proposed converters and their combinations increase the complexity of the emulators, specifically their control.

It can be concluded that there is a gap in finding an optimal battery emulator which satisfies all criteria, from the price, complexity, modularity, and application to the working performances. Most of the research from the literature is focused on developing battery emulators of specific battery types, with specific and narrow applications. A more general approach is needed to make the emulator replicate the behavior of different battery types and technologies, in a user-friendly manner, which will extend and adapt its application to wider research areas. Therefore, this is still a very open scientific topic with many challenges to solve. In order to make a cost-effective, simple, and modular battery emulator, with enhanced control features, this paper proposes the battery emulator with a bidirectional non-inverting buck-boost converter as its main power stage. This converter is chosen rather than some other mentioned converters in the literature for several reasons. The first one is its ability to easily achieve a wide output voltage range. By adjusting the duty cycle of its switching signals, it can work in both buck or boost modes, i.e., its output voltage can be smaller or larger than its input voltage. This makes the design of the emulator easier in terms of the modularity criteria, i.e., if the emulator needs to have an option to select different battery types with various voltage levels. The most important reason is the bidirectional operation of the non-inverting buck-boost converter. Unlike synchronous buck or boost converters, which can also work in both directions, the bidirectional non-inverting buck-boost converter has a symmetrical structure. It looks and behaves the same from its input to output, regardless of the energy flow direction. A very important criterion in design of the proposed emulator is a simple converter topology. In comparison to the other implemented, more complex converters in the literature, especially interleaved converters, a simple topology of the bidirectional non-inverting buck-boost converter does not require a complicated switching control, which makes it more suitable than others.

Special attention must be devoted to the control structure of the battery emulator. It must provide precise and fast tracking of the nonlinear voltage characteristics, which are continuously computed in real time from the battery model. The output voltage of the selected bidirectional non-inverting buck-boost converter must accurately follow the calculated referent battery voltage, which is time-varying according to the implemented battery model. Also, the computed referent battery voltage can change much faster if the charging/discharging times of the emulator are scaled, i.e., decreased on purpose, to speed up the testing process. In addition, considering that the emulators are often tested under different profiles of the battery current, especially pulsed currents, the converter’s control must be robust against these load disturbances. In this paper, the advanced I² dual current-mode control (I²DCMC) method from [52] is applied for the control of the proposed battery emulator. The I²DCMC is selected because it satisfies all the requirements, in terms of necessary stability, accuracy, dynamical response, and robustness.

The main advantages of the proposed battery emulator are as follows:

• Capability of bidirectional operation, which ensures emulation of both charging and discharging phases;
• Generality in terms of the various battery types and models which can be emulated and implemented;
• Excellent accuracy and dynamical properties, thanks to the application of the I²DCMC;
• Modularity in terms of the emulator’s topology, battery model and control structure;
• Capability of charging/discharging time scaling;
• Possibility of rapid control prototyping with user-friendly interface suitable for researchers and especially students, which also provides its application for educational purposes.

This paper is organized as follows. The topology of the proposed battery emulator, with a description of all key blocks, is presented in Section 2. The simulation results of the proposed emulator are presented and discussed in Section 3. Section 4 describes the experimental verification of the emulator, with the corresponding experimental results and discussion. The conclusion is given in Section 5.

2. Topology of the Proposed Battery Emulator

The block-diagram of the proposed battery emulator is shown in Figure 1.

The basic parts from Figure 1 are briefly described in the following subsections.

2.1. Bidirectional Non-Inverting Buck-Boost Converter

The main power stage of the proposed battery emulator is the bidirectional non-inverting buck-boost converter, shown in Figure 2.

This converter is symmetrical with respect to the input and output sides. The pairs of power switches (T₁, T₂) and (T₃, T₄) are simultaneously controlled with the same switching signals with frequency f_s and their inverted values, respectively. When both T₁ and T₂ are switched on, T₃ and T₄ are both switched off, and vice versa. Depending on the duty cycle value D of the switching signals, this converter can work in the buck mode, when the output voltage v_bat is smaller than the input voltage v_g, or in the boost mode, when v_bat is larger than v_g:

$$v_{bat}={\displaystyle \frac{D{v}_g}{1-D}}.$$

(1)

This feature is very useful for the emulation of different battery types and packs with different voltage levels. The output voltage v_bat and current i_bat represent the emulated battery voltage and current, respectively. In other words, the output of the proposed battery emulator, i.e., the output of the non-inverting buck-boost converter from Figure 2 should behave as the output of the real battery. For that purpose, a proper control structure should be implemented to make the output voltage v_bat follow the referent battery voltage v_{bat_ref}, calculated from the battery model. In this paper, the I²DCMC method from [52] is used. The first criterion is to apply current-mode control, because it has numerous advantages over conventional voltage-mode control. The I²DCMC, being advanced current-mode control, is proven to be a good choice for the various types of converters, thanks to its following features: constant switching frequency, stability for all duty cycle values (no subharmonic oscillations in the inductor current i_L), zero error between the average value of the inductor current and referent current on each switching period, fast dynamical response, robustness to different disturbances and variations in the converter parameters, simple design procedure, etc. [52].

A key characteristic of the converter from Figure 2 is its possibility of bidirectional operation, which is ensured with the used power switches. When emulating the discharging mode, the current, i.e., energy, flows from the left (input voltage v_g) to the right (output voltage v_bat). During charging mode, the energy flows in the opposite direction.

2.2. I²DCMC of Non-Inverting Buck-Boost Converter

The structure of the I²DCMC, applied on the proposed battery emulator, is shown in Figure 3.

The I²DCMC method is explained in [52], with the detailed control design procedure. The role of the outer voltage loop compensator G_co(s) is to regulate the output voltage v_bat, in accordance with the referent battery voltage v_{bat_ref}. A simple proportional–integral (PI) compensator is applied:

$$G_{co}(s)=k_p+{\displaystyle \frac{k_i}{s}}.$$

(2)

For better dynamical properties, especially for the improvement in the response during battery load changes, i.e., during the changes in the output current i_bat, a feedforward (FF) block, proposed in [53], is added to the output of the compensator G_co(s) (Figure 3):

$$i_{FF}=i_{bat}\left(1+{\displaystyle \frac{v_{bat}}{v_g}}\right).$$

(3)

The output of the modified voltage loop represents the referent current i_ref, which is used as the reference for the inner current loop. The inner current loop compensator G_ci(s), together with a set of logical circuits (Figure 3), provides a very accurate and fast regulation of the inductor current i_L, which ensures excellent dynamical properties and robustness of the proposed battery emulator. A simple integral (I) compensator is implemented:

$$G_{ci}(s)={\displaystyle \frac{K_i}{s}}.$$

(4)

Design guidelines for both compensators G_co(s) and G_ci(s) are presented in [52]. As proven in [52], the gain value K_i of the compensator G_ci(s) is equal to the selected crossover frequency of the inner current loop gain. This means that the transient response time of the inner current loop is inversely proportional to the gain K_i. Therefore, the gain K_i should be large to obtain fast transient response. On the other hand, the outer voltage loop is usually designed to be slower than fast inner current loop. The gain values (k_p, k_i, K_i) are given in

Table 1. The parameters used in simulations and experiments.

	Name	Value
DC source/load	vDC	6 V
	R1	20 Ω
	C1	1000 µF
Non-inverting buck-boost converter	L	220 µH
	C	1000 µF
Programmable DC load/source	L1	1 mH
	C2	470 µF
	R2	20 Ω
	vDC1	10 V
Polymer Li-ion battery model	Rself-discharge	+∞ (open circuit)
	Ccapacity	3060 F
Conventional Li-ion battery model	E0	3.366 V
	R	0.01 Ω
	K	0.0076 Ω or V/(Ah)
	Q	2.3 Ah
	A	0.26422 V
	B	26.5487 1/(Ah)
Lead–acid battery model	E0	12.4659 V
	R	0.04 Ω
	K	0.047 Ω or V/(Ah)
	Q	7.2 Ah
	A	0.83 V
	B	125 1/(Ah)
I2DCMC	fs	25 kHz
	Ib	0.4 A
	Ib1	0.1 A
	kp	0.499
	ki	59.835
	Ki	5000
	Ki1	500

. In accordance with these values, the settling time of the outer voltage loop is set to approximately 25 ms, while for the inner current loop, it is about 1 ms.

The parameter I_b (Figure 3) represents a constant current offset added to the control signal i_c, so the inductor current i_L is bounded between the i_c + I_b and i_c − I_b limits. The bandwidth 2I_b must be larger than the maximum peak-to-peak ripple of the inductor current i_L, to ensure stable operation of the converter, without subharmonic oscillations.

2.3. DC Source or Load

The non-inverting buck-boost converter must have its power supply, which is capable of both sourcing (discharging mode) and sinking (charging mode) current, as shown in Figure 4.

A circuit from Figure 4 is proposed in [4]. During emulation of battery discharging mode, the power switch T₅ is switched off, the diode D₁ conducts, and the current flows from the DC source v_DC to the right, supplying the non-inverting buck-boost converter. Therefore, in discharging mode, the circuit from Figure 4 behaves as the DC source of the non-inverting buck-boost converter. During emulation of charging mode, the power switch T₅ is switched on and the current flows from the non-inverting buck-boost converter to the left, through the switch T₅ and resistor R₁. The diode D₁ does not conduct, protecting the DC source v_DC from receiving the current. In this way, the circuit from Figure 4 behaves as a load of the non-inverting buck-boost converter during the charging phase.

2.4. Programmable DC Load or Source

As will be explained later, the mathematical model of the battery requires the measured battery current i_bat (Figure 1). This battery current, which is the output current of the non-inverting buck-boost converter (Figure 2), is set and controlled by using a programmable DC load shown in Figure 5.

The main part of the circuit from Figure 5 is the bidirectional synchronous buck/boost converter (inductor L₁, power switches T₆ and T₇). In discharging mode, the power switch T₈ is switched on, the power switch T₉ is switched off, the diode D₂ does not conduct, and the circuit from Figure 5 behaves as the active load of the non-inverting buck-boost converter. Since the current i_bat flows from the left to the right during discharging mode, the converter from Figure 5 works as the synchronous boost converter, with the load resistance R₂ and output capacitance C₂. In charging mode, the power switch T₈ is switched off, the power switch T₉ is switched on, the diode D₂ conducts, and the converter from Figure 5 behaves as the synchronous buck converter, with the input DC source v_DC1‚ input capacitance C₂, and the output voltage v_bat. During charging mode, the current i_bat flows from the right to the left side, and the circuit from Figure 5 behaves as the DC source of the non-inverting buck-boost converter.

To provide different test cases and profiles of the battery current i_bat, the circuit from Figure 5 requires proper control. The battery current i_bat is the current through the inductor L₁, which enables the application of the I²DCMC, as shown in Figure 6.

2.5. I²DCMC of Programmable DC Load/Source

The structure of the I²DCMC, which is used for regulation of the battery current i_bat, is given in Figure 6.

The control structure from Figure 6 is same as the one from Figure 3, except that the outer voltage loop is not used in this case, because the referent battery current i_{bat_ref} is directly set by the user. For discharging mode, the referent battery current i_{bat_ref} is positive (the battery current i_bat flows from the left to the right), while for charging mode, it is negative (the direction of the battery current i_bat is opposite). The proposed I²DCMC ensures very fast and accurate tracking of the given referent current i_{bat_ref}, which means that different profiles, i.e., waveforms of the battery current i_bat can be achieved. As for the I²DCMC of the non-inverting buck-boost converter (Figure 3), a simple I compensator is also implemented here:

$$G_{ci1}(s)={\displaystyle \frac{K_{i1}}{s}}.$$

(5)

The designed settling time of the current loop is about 10 ms. The current offset I_b1 from Figure 6 has the same meaning as the current offset I_b from Figure 3.

2.6. Battery Model

The quality and characteristics of the battery emulator very much depend on the mathematical model of the real battery. The battery model needs to be accurate and comprehensive, to approximately match the behavior of the real battery. The modeling of batteries is a very popular scientific topic, with many different proposed approaches. The battery emulator proposed in this paper is universal in terms of the possibility of implementation of different battery models in real time. Therefore, the modeling procedures are not the focus of this paper, nor the comparison with the response of the real batteries. Without loss of generality, three different models of batteries are implemented in this paper: the polymer Li-ion battery model proposed in [54], the conventional Li-ion, and the lead–acid battery model proposed in [55].

2.6.1. Polymer Li-Ion Battery Model

The equivalent electrical circuit of the polymer Li-ion battery model is shown in Figure 7.

The state of charge (SOC) of the proposed battery model from Figure 7 can be determined from the following equation:

$${\displaystyle \frac{d\mathrm{SOC}}{dt}}=-{\displaystyle \frac{1}{C_{capacity}}}\left(i_{bat}+{\displaystyle \frac{\mathrm{SOC}}{R_{self\text{-}discharge}}}\right).$$

(6)

The values of all model parameters from Figure 7 are taken from [54]. The values of C_capacity and R_{self-discharge} are given in Table 1. Other model parameters are expressed as follows:

$$\begin{array}{l}{V}_{oc}(\mathrm{SOC})=-1.031e^{-35·\mathrm{SOC}}+3.685+0.2156·\mathrm{SOC}-0.1178·{\mathrm{SOC}}^2\\ +0.3201·{\mathrm{SOC}}^3.\end{array}$$

(7)

$$R_{series}(\mathrm{SOC})=0.1562e^{-24.37·\mathrm{SOC}}+0.07446.$$

(8)

$$R_{transient\_S}(\mathrm{SOC})=0.3208e^{-29.14·\mathrm{SOC}}+0.04669.$$

(9)

$$C_{transient\_S}(\mathrm{SOC})=-752.9e^{-13.51·\mathrm{SOC}}+703.6.$$

(10)

$$R_{transient\_L}(\mathrm{SOC})=6.603e^{-155.2·\mathrm{SOC}}+0.04984.$$

(11)

$$C_{transient\_L}(\mathrm{SOC})=-6056e^{-27.12·\mathrm{SOC}}+4475.$$

(12)

The output voltage v_{bat_ref} of the proposed battery model from Figure 7 is equal to

$$v_{bat\_ref}=V_{oc}(\mathrm{SOC})-R_{series}{i}_{bat}-v_{transient\_S}-v_{transient\_L}.$$

(13)

The voltages v_{transient_S} and v_{transient_L} are the voltages of the capacitors C_{transient_S} and C_{transient_L}, respectively:

$${\displaystyle \frac{d{v}_{transient\_S}}{dt}}={\displaystyle \frac{1}{C_{transient\_S}}}\left(i_{bat}-{\displaystyle \frac{v_{transient\_S}}{R_{transient\_S}}}\right).$$

(14)

$${\displaystyle \frac{d{v}_{transient\_L}}{dt}}={\displaystyle \frac{1}{C_{transient\_L}}}\left(i_{bat}-{\displaystyle \frac{v_{transient\_L}}{R_{transient\_L}}}\right).$$

(15)

It is obvious from (6)–(15) that the model of the polymer Li-ion battery has single input—the battery current i_bat. The output of this model is the referent battery voltage v_{bat_ref}, which is used as the reference for the outer voltage loop compensator in Figure 3.

A very important feature of the proposed battery emulator is the possibility to speed up the charging/discharging process, by a simple multiplication of the battery current i_bat in (6) with a scaling factor k (k > 1):

$${\displaystyle \frac{d\mathrm{SOC}}{dt}}=-{\displaystyle \frac{1}{C_{capacity}}}\left(k{i}_{bat}+{\displaystyle \frac{\mathrm{SOC}}{R_{self\text{-}discharge}}}\right).$$

(16)

$$d\mathrm{SOC}=-{\displaystyle \frac{1}{C_{capacity}}}\left(k{i}_{bat}+{\displaystyle \frac{\mathrm{SOC}}{R_{self\text{-}discharge}}}\right)dt.$$

(17)

If self-discharge in (17) is neglected, i.e., R_{self-discharge} = +∞ (open circuit), (17) becomes

$$d\mathrm{SOC}=-{\displaystyle \frac{1}{C_{capacity}}}k{i}_{bat}dt.$$

(18)

When the battery current i_bat is constant, the SOC can be obtained from (18) as:

$$\mathrm{SOC}(t)=\mathrm{SOC}(0)-{\displaystyle \frac{1}{C_{capacity}}}k{i}_{bat}t.$$

(19)

It is obvious from (19) that the slope of the SOC is k-times increased. This means that the SOC will change k-times faster, which results in a faster charging/discharging process. This enables much faster tests of the battery emulator, unlike time-consuming work with real batteries.

2.6.2. Conventional Li-Ion Battery Model

The conventional Li-ion battery model, proposed in [55], is based on a simple electrical circuit, consisting of controlled voltage source and resistor with resistance R, connected in series. The current i_bat, which flows through this circuit, represents the battery current, and it controls the value of the voltage source. The output voltage of the circuit v_{bat_ref} can be represented with the following equations [55]:

$$v_{bat\_ref}=E_0-R{i}_{bat}-{\displaystyle \frac{KQ}{Q-{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}}}\left({\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}+{i^{\ast }}_{bat}\right)+A{e}^{-B{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}}$$

(20)

for discharging, and

$$v_{bat\_ref}=E_0-R{i}_{bat}-{\displaystyle \frac{KQ}{{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}-0.1Q}}{i^{\ast }}_{bat}-{\displaystyle \frac{KQ}{Q-{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}}}{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}+A{e}^{-B{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}},$$

(21)

for charging. The parameters E₀, R, K, Q, A, and B represent battery constant voltage, internal resistance, polarization constant or resistance, battery capacity, amplitude of the exponential zone, and inverted time constant of the exponential zone, respectively. Their values are taken from [55] and given in Table 1. The variable i*_bat is the filtered, i.e., average battery current i_bat.

The SOC of the conventional Li-ion battery model can be calculated in the following way:

$$\mathrm{SOC}={\displaystyle \frac{1}{Q}}\left(Q-{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}\right)=1-{\displaystyle \frac{1}{Q}}{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt},$$

(22)

for discharging, and

$$\mathrm{SOC}={\displaystyle \frac{1}{Q}}\left(Q-Q-{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}\right)=-{\displaystyle \frac{1}{Q}}{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt},$$

(23)

for charging. This model also has the possibility to scale duration of the charging/discharging process, by using the scaling factor k which multiplies the battery current i_bat in (20)–(23), except in part Ri_bat.

2.6.3. Lead–Acid Battery Model

The model of the lead–acid battery, proposed in [55], is also derived from the same simple electrical circuit described in Section 2.6.2. The only difference in equations for v_{bat_ref} is in the last exponential term with parameters A and B:

$$v_{bat\_ref}=E_0-R{i}_{bat}-{\displaystyle \frac{KQ}{Q-{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}}}\left({\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}+{i^{\ast }}_{bat}\right)+Exp,$$

(24)

for discharging, and

$$v_{bat\_ref}=E_0-R{i}_{bat}-{\displaystyle \frac{KQ}{{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}-0.1Q}}{i^{\ast }}_{bat}-{\displaystyle \frac{KQ}{Q-{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}}}{\displaystyle \underset{0}{\overset{t}{\int }}{i}_{bat}dt}+Exp,$$

(25)

for charging. The term Exp from (24) and (25) represents a hysteresis phenomenon between the charging and discharging of the lead–acid battery, and can be expressed as:

$${\displaystyle \frac{dExp}{dt}}=B\left|i_{bat}\right|\left(-Exp+Au\right)$$

(26)

where u = 0 for discharging and u = 1 for charging.

The SOC is calculated with same Equations (22) and (23). The parameters E₀, R, K, Q, A, and B have the same meaning as in the previous model, and they are listed in Table 1.

The time scaling of discharging/charging process with the scaling factor k can be achieved in identical ways as described in the previous models.

3. Simulation Results and Discussion

The proposed battery emulator from Figure 1 is modeled and simulated in Matlab/Simulink, to validate its performances. All three described battery models are implemented. The values of all used parameters are given in Table 1.

The implemented simulation model is very complex and requires a lot of computational burden. It would be impossible to perform simulation tests for typical real charging/discharging times (in hours), because the simulations would last enormously long, producing unbearable burden for the computer. For that reason, the charging/discharging time is scaled by using the factor k, which multiplies the battery current in (6), for the polymer Li-ion battery model, in (20)–(23) for the conventional Li-ion battery model, and in (24)–(26) for the lead–acid battery model. For the following simulations tests, k is set to 7500, 20,000, and 40,000 for the polymer Li-ion, conventional Li-ion, and lead–acid battery model, respectively.

3.1. Simulation Results for the Polymer Li-Ion Battery Model

In Figure 8, the battery voltage and current, i.e., the output voltage and current of the proposed battery emulator, are shown for both charging and discharging processes. For a better view of the transient response during charging/discharging, Figure 9 shows only the output voltage for the same test case.

The referent battery current i_{bat_ref} is set to 1 A for discharging, and −1 A for charging. It is obvious from Figure 8 that the output current i_bat follows the referent current i_{bat_ref} well, thanks to the applied I²DCMC. The current i_bat enters a steady state in approximately 10 ms, which is equal to the designed settling time. It has overshoots during transient response of about 13% at the beginning of charging and 98% at the beginning of discharging. This large current overshoot during discharging can be attributed to the simulation start-up, when Simulink components could exhibit unrealistic behavior in a very short period after simulation starts, especially because of algebraic loops and snubber elements.

Since the current i_bat flows through the inductor L₁ (Figure 5), it contains a small peak-to-peak ripple, which can be neglected. It is worth noting that the average value of the battery current i_bat, which is equal to the referent battery current i_{bat_ref}, must be larger than the half of its maximum peak-to-peak ripple in order to maintain continuous conduction mode of the programmable DC load/source. In this case, for the parameters from Table 1, this minimum value is about 0.1 A.

It can be seen, especially from Figure 9, that the output voltage v_bat looks as expected, as the typical voltage of the Li-ion batteries, during both charging and discharging. The voltage v_bat follows the referent battery voltage v_{bat_ref} well, which is calculated from the model. There is a tracking delay of approximately 25 ms at the beginning of the charging and end of the discharging process, which is expected and equal to the transient response time of the designed outer voltage loop compensator G_co(s), as discussed in Section 2.2. This compensator can be easily made faster to minimize these delays. However, that is unnecessary because it is fast enough for more realistic, much larger charging/discharging times, so these delays during transient responses will be negligibly small.

If only the battery model is simulated separately, without all other components from Figure 1, the input battery current is manually set as a constant (i_bat = ±1 A) and the output battery voltage of the model is computed (green dashed line on Figure 9, denoted as “Model”). In the complete model of the entire battery emulator, the battery model receives the battery current i_bat, which is the output current of the non-inverting buck-boost converter. As shown in Figure 8, the current i_bat has overshoots during transient responses and needs about 10 ms to reach the steady states (±1 A). However, it is obvious from Figure 9 that these transient responses have a very small influence on the accuracy and dynamics of the proposed emulator, since v_{bat_ref}, which is calculated using such current i_bat, is precisely matched with the output battery voltage obtained from the battery model with constant input. Only at the beginning of discharging, there is about 10% over/undershoot in v_{bat_ref} because of the already mentioned battery current overshoot.

To verify the behavior of the proposed battery emulator during step changes in the battery current, the following simulation test is conducted. The referent battery current i_{bat_ref} is changed from −1 A (charging) to 1 A (discharging), and again to −1 A (charging). The duration of the current pulse, which is equal to 0.3 s, is chosen on purpose to avoid full charging and discharging. The simulation results are given in Figure 10.

It can be concluded from Figure 10 that the proposed battery emulator has a very good transient response in terms of both output voltage and current, which can be attributed to the implemented I²DCMC, especially the added FF block (Figure 3). The battery voltage v_bat has an expected settling time of around 25 ms. The battery current i_bat also has an expected settling time of around 10 ms, while its overshoots are 32% (charging-to-discharging transient) and 15% (discharging-to-charging transient).

3.2. Simulation Results for the Conventional Li-Ion Battery Model

The tests presented in Section 3.1 are also performed for the battery emulator with the conventional Li-ion battery model. The simulation results are shown in Figure 11 and Figure 12.

Similar conclusions can be derived from the given results, as in the case of the battery emulator with the polymer Li-ion battery model. This confirms the modularity of the proposed battery emulator in terms of the implemented battery type and model.

3.3. Simulation Results for the Lead–Acid Battery Model

The simulation results for the emulator with the lead–acid battery model are shown in Figure 13 and Figure 14, under the same test scenarios from Section 3.1 and Section 3.2.

It is important to note that in the case of the lead–acid battery model, the value of the DC source voltage v_DC1 of the programmable DC load/source is set to 20 V instead of 10 V used for the previous two models. The reason is, in steady state, when the lead–acid battery is fully charged, or at the beginning of discharging (Figure 13), the battery voltage v_bat is approximately 13.34 V. A topology of the programmable DC load/source (Figure 5) requires that the voltage v_bat is always smaller than v_DC1.

It can be concluded that the proposed battery emulator successfully replicates the output voltage of the lead–acid battery model, in both test cases. The applied I²DCMC ensures a zero steady-state error in both battery voltage and current, with expected transient responses in accordance with the selected control parameters.

The results from Figure 13 confirm the benefits of using bidirectional non-inverting buck-boost converter. Its input voltage v_DC is equal to 6 V during discharging, and the emulated battery voltage v_bat goes up to 13.34 V, which is more than three times larger than the maximum emulated battery voltage for two previous models. Therefore, this converter is suitable for implementation of various battery models with different output voltage ranges.

4. Experimental Results and Discussion

After simulation analysis, the proposed battery emulator is tested experimentally on the developed prototype, shown in Figure 15.

The switching part of the bidirectional non-inverting buck-boost converter (power switches T₁–T₄ from Figure 2) consists of four power MOSFETs IRF540N (International Rectifier, El Segundo, CA, USA) (100 V, 33 A), which satisfy defined voltage and current levels, and nominal power of the converter of about 100 W.

The bidirectional synchronous buck/boost converter from Figure 5 is made from commercially available SPM-HB (Taraz, Rawalpindi, Pakistan) half-bridge power module, with power MOSFETs ST21N90K5 (STMicroelectronics, Geneva, Switzerland) (900 V, 18.5 A).

The power switches T₅ (Figure 4), T₈ and T₉ (Figure 5) are implemented with SIMOPAC modules BSM 191F 4012 (Siemens, Munich, Germany). A diode module DSEI2X31-06C (IXYS, Milpitas, CA, USA) (600 V, 30 A) is used for diodes D₁ (Figure 4) and D₂ (Figure 5).

One of the main requirements in the design of the experimental setup is galvanic isolation between the converter and control electronics. For that purpose, a two-channel galvanically isolated gate driver is developed, which can independently control the output channels, but also synchronously control the power switches in half-bridge, with a programmable “dead time”. The drivers have approximately 0.6 µs turn on/off delay.

All the voltage and current measurements are achieved with galvanic isolation between the converter and control part of the emulator. The input and output voltage of the non-inverting buck-boost converter are measured using optocouplers IL300 (Vishay, Malvern, PA, USA). The inductor and output current are measured with HX 10-NP (LEM, Meyrin, Switzerland) current transducers, having open loop Hall effect technology.

All three mathematical models of the batteries, presented in Section 2.6, are implemented in real time in Matlab/Simulink environment, by using Real Time Windows Target (MathWorks, Matlab R2023b, Natick, MA, USA) library and built-in MF624 (Humusoft, Prague, Czech Republic) data acquisition card. The input and output voltage, and output current of the non-inverting buck-boost converter v_g, v_bat, and i_bat, respectively, are measured and fed to the 14-bit A/D converter inputs of the MF624 card. The A/D converters have approximately 2 µs conversion time and ±10 V input voltage range. The MF624 card enables real-time operation of the implemented battery models in Simulink. The Simulink model calculates in real time the referent battery voltage v_{bat_ref}, which is used for the outer voltage loop of I²DCMC (Figure 3). The outer voltage compensator, combined with the feedforward control, is implemented in this same Simulink model. The generated referent current i_ref, together with the current offset I_b, are exported from the Simulink model via 14-bit D/A converter outputs (±10 V output voltage range, 10 V/µs slew rate) of the MF624 card and fed to the electronic module for implementation of the inner current loop of I²DCMC. The referent battery current i_{bat_ref} (Figure 6) is manually set by the user within the same Simulink model and exported from the MF624 card, together with the current offset I_b1. They are used by another channel of the electronic module for implementation of I²DCMC of programmable DC load/source. The sample time of the Simulink model is set to 50 µs.

Experimental waveforms in the following subsections are recorded with Tektronix MSO 2014 (100 MHz, 1 GS/s) oscilloscope (Beaverton, OR, USA).

4.1. Experimental Results for the Polymer Li-Ion Battery Model

In the first test case, the scaling factor k is set to 500. The experimental results for charging at i_bat = −1 A and discharging at i_bat = 1 A, are shown in Figure 16.

It is obvious from Figure 16a that the output current i_bat follows accurately and fast the step changes in the referent battery current from −1 A to 1 A, and vice versa. The interval of these changes is enough for both charging and discharging processes to be finished. The output voltage v_bat for the entire discharging and charging process is shown in Figure 16b and Figure 16c, respectively. The output voltage v_bat has expected waveforms typical to the Li-ion batteries during charging/discharging. It can be noticed in Figure 16b,c that the output voltage v_bat has constant portions, which correspond to fully charged and discharged thresholds.

In the second test case, the scaling factor k is set to 7500, as in the simulations. The experimental results are shown in Figure 17, for the step changes in the referent battery current from −1 A (charging) to 1 A (discharging), and vice versa. The interval of the current pulse is the same as in the simulations, and equal to 0.3 s. The goal is to test the dynamical behavior of the proposed battery emulator during switching between charging and discharging, and vice versa, when there is not enough time to perform full charging/discharging processes.

The experimental waveform of the battery current i_bat (Figure 17a) is very similar to the simulation waveform from Figure 10. The current overshoots are about 40% and 20% for charging to discharging transient and vice versa, respectively. In the experimental waveform of the battery voltage v_bat (Figure 17b), there are larger undershoots and overshoots in the moments of step changes in the battery current, in comparison to the simulation waveform from Figure 10. This difference can be attributed to the nonidealities (real parameters) in the components of the prototype of the battery emulator, which are not considered in the simulation model, such as switching delays, delays and errors introduced with measurement and filtering of the voltages and currents, printed circuit board trace parasitic elements, losses in semiconductor switches, inductors, capacitors, etc.

4.2. Experimental Results for the Conventional Li-Ion Battery Model

The same experimental tests are conducted for the conventional Li-ion battery model. First, the scaling factor k is set to 20,000/15 = 1.33 × 10³. The experimental results for charging at i_bat = −1 A and discharging at i_bat = 1 A, are shown in Figure 18.

Similar conclusions can be made for the waveforms from Figure 18, as in the case of the polymer Li-ion battery.

In the second test case, the scaling factor k is set to 20,000, like in the simulation tests. The experimental waveforms of the output voltage v_bat are shown in Figure 19. They are similar to the corresponding simulation waveforms from Figure 12, but they have larger under/overshoots during the transients.

It can be concluded, as in the simulation verification, that the modularity of the proposed battery emulator in terms of the implemented battery type and model, is also experimentally validated.

4.3. Experimental Results for the Lead–Acid Battery Model

Following exact testing procedure for the previous two battery models, the battery emulator based on the lead–acid battery model is experimentally verified. In the first test case, the scaling factor k is set to 4000. The corresponding experimental waveforms for charging (i_bat = −1 A) and discharging (i_bat = 1 A), are shown in Figure 20.

In the second test case, the scaling factor k is set to 40,000, as in the simulations. The experimental results are shown in Figure 21.

The experimental waveforms of the output voltage v_bat have expected characteristics, similar to the simulation waveforms from Figure 13 and Figure 14.

4.4. Comparison Between Experimental and Simulation Results

It is mandatory that both simulation and experimental verification of the proposed battery emulator produce well-matched results, i.e., waveforms of the battery voltage and current, under same test conditions. A high quality of the simulation model, which matches reality well, is crucial in the proper design of the emulator’s control structure and for the reliable validation. The following graphs compare the corresponding experimental and simulation responses for all three battery types considered. The given experimental waveforms (blue lines) from Figure 22, Figure 23 and Figure 24 represent the real-time measured values of the output voltage v_bat, obtained from the A/D converters of the MF624 card. It is evident that the experimental and simulation waveforms are well matched for all three implemented battery models, with negligible static errors. This also confirms the benefits of using I²DCMC, which guarantees precise and robust regulation of the battery voltage and current.

5. Conclusions

In this paper, the battery emulator, which is based on the current-mode controlled bidirectional non-inverting buck-boost converter, is proposed, with a detailed description of all its parts. The performances of the proposed battery emulator are successfully validated with both simulation and experimental tests, for three different battery models.

The main contributions of this paper refer to the achieved excellent characteristics and advantages of the proposed battery emulator:

• Excellent tracking accuracy and dynamics—the implemented I²DCMC ensures precise and fast tracking of the referent battery voltage and current;
• Possibility of charging/discharging time scaling—using simple scaling factor k that speeds up the change in SOC, and thus increasing the charging/discharging velocity, which enables much easier and faster laboratory tests of the battery-based systems;
• Modularity in terms of the implemented real-time battery model—various mathematical models of the batteries can be easily configured and applied in real time, in a user-friendly Matlab/Simulink environment;
• Modularity in terms of the implemented battery types—in this paper, the polymer Li-ion, conventional Li-ion, and lead–acid batteries are considered as examples, although other types of batteries with different voltage and current levels can be implemented, thanks to the applied non-inverting buck-boost converter, which can work with wide range of input/output voltages and currents;
• Modularity in terms of the hardware topology—all hardware parts of the emulator can be easily configured and modified, including the controller, i.e., instead of a used MF624 control board, other control platforms can be implemented.

One of the main goals of this paper is to develop a new, user-friendly, and cost-effective platform for battery emulation, with reliable advanced control. The proposed battery emulator successfully provides the desired voltage and current values and waveforms, in accordance with the battery models implemented in real time. The next task for future works will be to develop the model and identify its parameters for the specific real battery, to implement that model in the proposed battery emulator, and to compare the experimental response from the real battery and emulator.