A Generalized State Space Average Model for Parallel DC-to-DC Converters

The high potentiality of integrating renewable energies, such as photovoltaic, into a modern electrical microgrid system, using DC-to-DC converters, raises some issues associated with controller loop design and system stability. The generalized state space average model (GSSAM) concept was consequently introduced to design a DC-to-DC converter controller in order to evaluate DC-toDC converter performance and to conduct stability studies. This paper presents a GSSAM for parallel DC-to-DC converters, namely: buck, boost, and buck-boost converters. The rationale of this study is that modern electrical systems, such as DC networks, hybrid microgrids, and electric ships, are formed by parallel DC-toDC converters with separate DC input sources. Therefore, this paper proposes a GSSAM for any number of parallel DC-to-DC converters. The proposed GSSAM is validated and investigated in a time-domain simulation environment, namely a MATLAB/SIMULINK. The study compares the steady-state, transient, and oscillatory performance of the state-space average model with a fully detailed switching model.


Introduction
Due to the revolutionary development of modern electrical systems such as electric vehicles, ships, and microgrids, DC-to-DC converters are widely used in industry [1][2][3][4][5][6]. Moreover, DC-to-DC converters have become the cornerstone for active power systems, especially in DC and hybrid microgrids [7][8][9][10][11]. The longterm study of a fully detailed DC-to-DC converter requires not just a great deal of time but also an expensive computer unit [12]. Therefore, the conventional concept of a state-space average model of a DC-to-DC converter is used to minimize the simulation time as well as to design the converter controllers.
Most mathematical calculations become easier if an exponential function form is applied (e int ) instead of using trigonometric functions. For simplicity, the general exponential representation of a Fourier series can be written in terms of the exponential function in (2): where the F n coefficient is a constant real number when n = 0, and is a complex number for all values of n > 0, which appear in a conjugate pair. A conventional average state-space variable can be achieved by considering only one cycle of the switching signal [16], which is a DC component term that is given by (3): where T s is the switching period. Fig. 2 presents the waveform of the switching signal u(t). The complex Fourier coefficients are given by (4) for the DC component and by (5) for the AC harmonics. The switching frequency is donated via ω = 2π/T s . Rearranging and simplifying (5) allows the fundamental switching signal harmonic in terms of the sine function to be expressed as follows: Therefore, the averaging AC harmonics of index-n can be determined by differentiation f(t) with respect to time and computing the product of the switching signal u(t) with the circuit state variable index-n, using the following formulas, respectively [16,18]: dhf i n ðtÞ dt ¼ df dt n ðtÞ À jnxhf i n ðtÞ (7) According to (8), the average product of index-n for signals can be calculated using a discrete convolution relationship, which leads to the following expressions for the DC component and the fundamental component, respectively [16]: Eq. (9) gives the DC component of the state variable, which is referred to index-0. The complex state variables based on the fundamental component can be determined from (10). In (10), the conjugate component can be calculated by changing the index from a positive to a negative sign [16].

Buck Converter Modelling
The following generalized differential equations describe the voltage and current of n-parallel buck converters: In this paper, a parallel buck switching model converter, which is shown in Fig. 3, is used for validation by comparing it with a state space average model. The differential equations of a large signal model can be found via applying both Kirchhoff's voltage and current laws. These laws should be applied during the twoswitch state. The state variable equations of the parallel buck converter are given in (13)-(15) while the switching function u(t) is defined in (16).
720 CSSE, 2022, vol.41, no.2 Since circuit topology changes according to the switch mode of operation, the switching function determines the circuit topology over each switching cycle. The GSSAM of a buck converter mainly depends on the Fourier coefficients of state variables which, in this case, are iL 1 , iL 2 , and v c . By including more terms of the Fourier series harmonics, it is possible to reduce any approximation error [15], but the use of higher order approximations increases the complexity of the model. Considering only the first-order approximation for a state-space model is beneficial for the sake of simplicity. Therefore, the first-order Fourier coefficient approximation of (13)- (15) can be written for the state variable as follows [18]: where 〈i L1 〉 0 , 〈i L1 〉 0 , and 〈v c 〉 0 are the DC components of the Fourier coefficient while 〈i L1 〉 1 , 〈i L1 〉 1 and 〈v c 〉 1 are the complex first-order harmonic. The negative sign of the complex first-order harmonic indicates the complex conjugate quantity. For mathematical simplicity of driving the generalized state-space average model, the quantities in (17)- (19) can be expanded and written as follows:  Figure 3: Parallel buck converters The differential equations that form the state-space average model can be found by applying (7) and (8) and substituting (4), (5), (20)-(22) into (13)- (15). Therefore, the differential equations of the state-space model of a parallel buck converter are: The circuit state variables of a parallel buck converter are calculated from (23)-(31) by applying the following equations:

Boost Converter Modelling
The following generalized differential equations describe the voltage and current of n-parallel boost converters: An n-parallel boost switching model converter, which is shown in Fig. 4, is used for validation by comparing it with a state space average model. The large signal model of the two parallel boost converters can be described as: In order to avoid repetition, the same procedure that is explained in Section 3.1 can be applied. Hence, the differential equations of a parallel boost converter, as described in (40)-(48), are as follows: The circuit state variables of a parallel boost converter are calculated according to (20)-(22).

Buck-Boost Converter Modelling
The following generalized differential equations describe the voltage and current of n-parallel buckboost converters: An n-parallel buck-boost switching model converter, which is shown in Fig. 5, is used for validation by comparing it with a state space average model. The differential equations of a large parallel buck-boost converter signal model, based on the state variables, are given as: Ln Lm S1 Sn Sm Vn Figure 5: Parallel buck-boost converters The state variables of buck-boost converters are similar to boost converters. Therefore, the differential equations that form state space representation, based on the multi-frequency averaging model, are found as follows: Based on the previous equation, the circuit state variables of a parallel buck-boost converter are determined using (20)- (22).

Generalized State-Space Average Model of DC-to-DC Converters
This section presents the GSSAM of a DC-to-DC converter based on matrix representation, which is a beneficial representation for controller design. The GSSAMs of the n-parallel buck, boost, and buck-boost converters are illustrated in (63)-(65), respectively. The GSSAM of two parallel buck converters is illustrated in (63): where n in the subscript indicates the number of parallel converters, and A buck sys is the system matrix, which is given in Appendix (A.2). Hence, the expanded states and input matrixes are shown as follows: The GSSAM of two parallel boost converters is illustrated in (66): where the expanded states and input matrixes of the boost converter are given below, and the system matrix A boost sys is given in Appendix (A.3).
The GSSAM of two parallel buck-boost converters is illustrated in (69): were the expanded states and input matrixes of the buck-boost converter are identical to (67) and (68), while A buckÀboost sys is given in Appendix (A.4).
The previous Eqs. (63)-(65) can be expanded for n-number of parallel DC-to-DC converters, namely by replacing only three-dots (⋮) in the matrices with the current state differential equations in the system state matrix as well as in the input matrix. Furthermore, considering more Fourier harmonics will increase the system and the input matrix order. Hence, the accuracy of the GSSAM will be improved.

Simulation Results and Discussion
The GSSAM of parallel DC-to-DC converters is conducted in a MATLAB/SIMULINK environment, which is compared with a detailed switching model. Fig. 6 presents the electrical quantities of the two parallel DC-to-DC buck converters, namely the inductor currents and capacitor voltage. The inductor currents of the parallel buck converters are depicted in Figs. 6a and 6b, respectively while the capacitor voltage is shown in Fig. 6c. The total load current is the summation of both inductor currents, which is presented in Fig. 6d.

Buck Converter Behavior
The zero-order model is mainly based on the DC component of the Fourier series, which corresponds to (4), as depicted in Fig. 6 in the gray color. The moving average complex states, including the DC component of the currents and the voltage, are referred to (6) and (8). The results show that there is no significant difference between the state-space average models and the fully switched detailed model. In fact, the dynamic and oscillatory behavior of the average state-space models can be captured. Therefore, including more harmonic coefficients of the Fourier series leads to an increase in the model order, which improves the model accuracy and increases the model complexity.
In this case, the DC load voltage is assumed to be fixed due to a constant duty cycle. Initially, at t = 0 s, the converters start to supply the DC load according to its resistance which, in this case, is equal to 10 Ω. At t = 1 ms, the DC load changes to be equal to 5 Ω, therefore the load becomes double and both converters equally share the load current. The transient response of the proposed model is obviously seen to be identical to the switched model. At t = 2 ms, the DC load is reduced to be equal to 10 Ω. During the load step change, the DC voltage remains at the same level except during the load transition, which has regular transient response. It is noted that the proposed GSSAM of the parallel buck converter presents the transient, steady state, and switching dynamic of the converters, based on the operation conditions.

Boost Converter Behavior
The electrical quantities of the parallel DC-to-DC boost converter are presented in Fig. 7. These quantities are the inductor current and capacitor voltage. Figs. 7a and 7b, respectively, present the inductor currents of the two parallel boost converters. In this converter topology, it can be seen that there is a coupling term in the state variable equations, which raises the converter model complexity [18]. In other words, the state variables are multiplied by the duty cycle signal, therefore the GSSAM is just approximated by considering only the zero-and first-order of Fourier series harmonics [12]. The GSSAM based on zero-and first-order harmonics still captures the converter' current and voltage behaviors. Hence, the capacitor voltage of the parallel DC-to-DC boost converter is shown in Fig. 7c while the total load current is presented in Fig. 6d. Since the output DC voltage of the boost converter is based on the input voltage divided by (1 − d), this relationship increases the complexity of the boost converter.
Initially, at t = 0 s, both the GSSAM and switching models start to feed the connected DC load according to the applied duty cycle which is, in this case, equal to 0.25. It can be clearly observed that both boost converters have an identical response. A double increase of the DC load step change takes place at t = 1 ms. Therefore, both converter currents are increased accordingly and the DC voltage for both the  Fig. 7c. The DC load is reduced by half at t = 2 ms, so the transient response of both converters matches the switching converter model. Fig. 8 presents the model difference between the GSSAM and the fully detailed switching model when the duty cycle is equal to 0.7. It can be clearly seen that, for both parallel boost converters, the inductor current shows some error difference compared to the switching model, as illustrated in Figs. 8a and 8b, respectively. Hence, the inductor currents will affect the total load current, as shown in Fig. 8d. The output DC voltage is less affected than the inductor currents due to an increase in duty cycle, as depicted in Fig. 8c. The GSSAM of the boost converter shows some differences compared to the switching model once the duty cycle is increased. The reason behind this issue is that the switching frequency should be selected to be higher than a converter's RLC natural frequency in order to eliminate the high order harmonics [12]. Therefore, this converter topology requires proper choice of switching frequency or inclusion of more Fourier harmonics in order to achieve accurate performance of the average model when the duty cycle becomes more than 0.5 due to the existence of even harmonics [13]. Fig. 9 shows the parallel DC-to-DC buck-boost converter inductor current and capacitor voltage. In Figs. 9a and 9b, respectively, the currents of the parallel buck-boost converters are depicted while the capacitor voltage is shown in Fig. 9c. The summation of both inductor currents gives the total load current, as presented in Fig. 9d.   This case evaluates parallel buck-boost converters during a step change of input voltages. During the interval t = 0 to t = 1 ms, both converters are fed via a DC voltage equal to 20 V and duty cycles equal to 0.5. The GSSAM captures the dynamic and oscillatory behavior of the switching model. At t = 1 ms, the input DC voltages are increased to 30 V, hence the results show that the GSSAM is still capable of mimicking the converters' behavior even when the input DC source is reduced to be 20 V at t = 2 ms.

Buck-Boost Converter Behavior
Buck-boost converters have a less effective coupling term compared to boost converter behavior. Therefore, the difference between the GSSAM of the buck-boost and the detailed model is very low even when the duty cycle is increased to be equal to 0.75 compared to the boost converter, as shown in Fig. 10. At t = 0 ms, the buck-boost converter begins to feed the DC load, and the inductor current and capacitor voltage increases gradually, due to the increasing duty cycle. Therefore, the transient response of the proposed model is identical to the switched model. At t = 2 ms, the requested power from the DC load reduces by half, and the proposed model is still capable to capture the transient response of the switched model. It can be clearly noted that the proposed GSSAM of the parallel buck converter presents the transient, steady state, and switching dynamic of the converters.

Conclusion
This paper proposes a generalized state-space averaging model of parallel DC-to-DC converters. The study draws a comprehensive comparison of a generalized state-space averaging model of parallel DC-to-DC converters with a fully detailed switching model. Several advantages of the proposed generalized state-space averaging model can be achieved in order to design the controller loops and to conduct stability studies. This paper exposes some limitations of the generalized state-space averaging model of the DC-to-DC converter, especially for boost converter topology due to its complexity. The study in this paper compares the DC-to-DC converter's dynamic, oscillatory, and steady-state performance with a fully Funding Statement: The author received no specific funding for this study.

Conflicts of Interest:
The author declares that he has no conflicts of interest to report regarding the present study.