Modeling and Performance Assessment of the Split-Pi Used as a Storage Converter in All the Possible DC Microgrid Scenarios. Part I: Theoretical Analysis

Abstract

The integration of an electrical storage system (ESS) into a DC microgrid using a bidirectional DC/DC converter provides substantial benefits but requires careful design. Among such converter topologies, the Split-pi converter presents several merits at the cost of non-isolated operation. However, the few works in the literature on the Split-pi presented only closed-loop control with a single control loop; furthermore, they neglected the reactive components’ parasitic resistances and did not perform any experimental validation. This work aimed at investigating the use of the Split-pi converter as a power interface between an ESS and a DC microgrid. Five typical microgrid scenarios are presented, where each of which requires a specific state-space model and a suitable control scheme for the converter to obtain high performance. In this study, two different state-space models of the converter that consider the parasitic elements are presented, the control schemes are discussed, and criteria for designing the controllers are also given. Several simulations, as well as experimental tests on a prototype realized in the lab, were performed to validate the study. Both the simulation and experimental results will be presented in part II of this work. The proposed approach has general validity and can also be followed when other bidirectional DC/DC converter topologies are employed to interface an ESS with a DC microgrid.

1. Introduction

Over the last few years, DC distribution in terrestrial and marine power systems has attracted a growing interest in view of the implementation of the smart microgrid paradigm due to its advantages in terms of simpler and more efficient electrical architectures. Consequently, power electronic converters that interface distributed generation units, loads, and above all, electrical storage systems (ESSs) with a common DC bus are the subject of renewed interest [1,2,3]. ESSs have manifold beneficial impacts on DC microgrids: they allow for improving stability and resiliency, compensate for the intermittency of renewable generation, provide ramping support to generators, and act as backup power sources. Furthermore, ESSs ensure a power buffer that can be leveraged to apply suitable energy management strategies to microgrids. In particular, energy management systems (EMSs) can be used to compute the optimal values of power flows among the microgrid devices, which allow for pursuing chosen objectives, such as a minimum electricity bill, maximum efficiency, minimum fuel consumption, or minimum greenhouse gas emissions [1,4,5,6].

Depending on the designer’s choice, the microgrid voltage can be controlled either stiffly using a single voltage generator or in a droop scheme using one or more voltage generators with a predefined power-sharing ratio, usually in proportion to their power rating (grid-forming generators). When an EMS is used, the other active devices of the microgrid must be controlled as current generators based on the optimal power flows that are computed by the EMS [1,5]. Therefore, depending on the specific configuration, the converter interfacing the ESS with the microgrid must be operated as a stiff voltage generator, a non-stiff voltage generator, or a current generator.

The scientific literature provides several contributions on bidirectional DC/DC converters (BDCs). Reference [7] presented an overview of BDCs, where, besides the review of both non-isolated and isolated configurations, the most relevant control schemes and switching strategies were analyzed. In some applications, galvanic isolation between the input and output side of the converter is required; in such cases, the most frequent choice is the dual active bridge converter (DAB) due to its many advantages [7,8,9]. However, isolation is mandatory only when very high voltage gain is needed. Non-isolated converters are thus more attractive when the goal is to improve the efficiency, size, weight, and cost of the system [7,10]. A review of non-isolated BDCs topologies was presented in [10]. The advantages and disadvantages of the considered converters were properly highlighted. For example, some converters provide an output voltage with opposite polarity than the input, some draw discontinuous current from the battery, while others require a tapped inductor or exhibit weak regulation capability or a high switch count. Overall, Tytelmaier et al. identified the half-bridge converter (HBC) and the related interleaving variants with coupled inductors as the most promising solutions from the efficiency and robustness standpoints [10]. Furthermore, Odo compared three non-isolated BDC topologies with the classical HBC in terms of their suitability for energy storage in a DC microgrid and identified the cascaded buck-boost HBC topology as the best compromise [11].

An interesting alternative is offered by the Split-pi converter, which is a non-isolated BDC that is based on two cascaded HBCs with a common bulk capacitor instead of a common inductor. As such, it is the dual topology of the cascaded buck-boost HBC that was analyzed in [7,10,11]. The Split-pi was initially developed for electric vehicles and patented in 2004 [12], and it is receiving increasing attention due to its distinct advantages [13,14,15,16,17,18,19,20]. It exhibits high efficiency, like the DAB, but with a reduced switch count (eight vs. four switches, of which, only two are actively commutated). Furthermore, LC filters at both ports of the Split-pi allow for small reactive components and reduce the switching noise. On the other hand, the additional phase delay introduced by such filters can slightly complicate this converter’s control with respect to simpler converters [17,18]. Nonetheless, the control of the Split-pi is less complicated than that of the DAB because it requires conventional duty cycle control of the pulse width modulator (PWM) instead of phase shift control. An additional advantage of the Split-pi is its suitability for multiphase systems, where a significant reduction in component size and cost can be attained. These features make it attractive when high power density is required, such as in hybrid electric vehicles, renewable energy systems, and aerospace/marine/military applications [16,17].

Only three papers in the technical literature proposed applications in which the Split-pi converter was not controlled using an open-loop [16,17,19]. The aim of [17] was to study a system in which the Split-pi interfaced a flywheel with a 12 V, 120 W DC bus connecting a photovoltaic generator with a passive load. However, in the related control scheme, only one control loop was active at a time (either for the output voltage or the input current) using a shared controller, and the choice was made using a rule-based approach. Singhai et al. did not focus on a specific application and presented a state-space model of the Split-pi converter and a transfer function to design a closed-loop controller for its output voltage [16]. However, they considered the converter connected to a passive load and neglected the parasitic resistances. Finally, Monteiro et al. considered a Split-pi converter with a multilevel structure and controlled current or voltage in any of the two ports in an open-loop, whereas the common DC-link voltage was controlled in a closed-loop [19]. Only a few details about the control systems were given because the study mainly focused on comparing the multilevel Split-pi topology and the interleaved topology of [20].

Apart from [17], all these works only presented simulation results without experimental validation. Furthermore, they all studied closed-loop control with a single control loop, neglecting the reactive components’ parasitic resistances. However, when the converter is used to interface an ESS with an active load, such as a DC microgrid, it is essential to control both the ESS current and the output voltage/current of the converter, as required by the microgrid designer. Furthermore, the parasitic elements cannot be neglected because they affect the losses and the maximum gain attainable using the converter.

To cover these aspects, in this work, the use of the Split-pi converter in such an application was investigated with particular attention to its model, the design of its control system, and the assessment of the expected performance in all the possible microgrid configurations. In this first part of the work, it was shown that the Split-pi must be modeled in two different ways depending on the microgrid scenario and that control schemes involving a different number of control loops are needed. In the case of the output voltage control, a feed-forward action was also required to obtain high performance. Furthermore, it was shown that conventional PI regulators alone were not sufficient to obtain the desired performance for output current control in stiff microgrids. The two state-space models considering the parasitic elements and the transfer functions of interest were given in the study, together with criteria to design the controllers. The chosen case study was a DC microgrid that was representative of both terrestrial and marine applications. In Part II of the work, a comprehensive performance assessment is presented based on simulations and experimental tests that validate the study. Finally, the proposed approach has general validity and can also be followed when other BDC topologies are used to interface a storage system with a DC microgrid.

2. Topology, Operation Modes, and Sizing of the Split-Pi Converter

The schematic of a symmetrical Split-pi converter is sketched in Figure 1, including its reactive components’ parasitic resistances. Such a converter can be viewed as the cascaded connection of a first HBC at port 1, a bulk capacitor, and another HBC at port 2. Since the two HBCs are bidirectional, the whole Split-pi converter is also a BDC. If the two HBCs have equal reactive components, the Split-pi is said to be symmetrical.

The four switches (S1–S4) were sketched in Figure 1 as ideal switches. In the original formulation of [12], they were implemented using MOSFETs to exploit the advantage of synchronous rectification. According to [12], the Split-pi has four operation modes, depending on the relationship between V₁ and V₂ and on the power direction, as summarized in the first four rows of

Table 1. Summary of the operating modes of the Split-pi converter.

Mode	V1 ≤ V2	Power Direction	Split-Pi Operation	Duty Cycle of S1	Duty Cycle of S2	Duty Cycle of S3	Duty Cycle of S4	Gain of the Split-Pi in Direction 1 → 2
1	Yes	1 → 2	Boost for 1 → 2	d	1 − d	0	1	$\frac{1}{1-d}$
2	Yes	2 → 1	Buck for 2 → 1	d	1 − d	0	1	$\frac{1}{1-d}$
3	No	1 → 2	Buck for 1 → 2	0	1	1 − d	d	$d$
4	No	2 → 1	Boost for 2 → 1	0	1	1 − d	d	$d$

, which is an extension of the table reported in [18]. In this work, the Split-pi converter was used to interface a storage system (connected to port 1) with a DC microgrid (connected to port 2). It is worth noting that, whenever possible, the storage system is chosen so that V₁ ≤ V₂ for both technical and safety reasons. Hence, the present work considered the Split-pi converter operating in modes 1 and 2.

The Split-pi components can be sized according to the classical formulas used for buck and boost converters [17,21]. Specifically, the minimum inductance value is expressed by (1) for both boost and buck operations; on the other hand, the minimum capacitance of the input and output capacitors C_e can be computed using (2), which is valid for a buck converter, whereas the minimum value of the bulk capacitance C is expressed by (3), which is valid for a boost converter:

$$L_{min}=\frac{100 V_2\left(1-d\right)d}{2F_{sw}{r}_{i\%}{I}_1},$$

(1)

$$C_{e, min}=\frac{100\left(1-d\right)}{8F_{sw}^2{r}_{ve\%}L},$$

(2)

$$C_{min}=\frac{100 I_{2}d}{F_{sw}{r}_{v\%}{V}_2}.$$

(3)

In (1), (2), and (3), F_sw is the switching frequency; d is the duty cycle; V₂ and I₂ are the output voltage and current, respectively; I₁ is the input current; r_i% is the desired inductor current ripple; and r_v% and r_ve% are the desired voltage ripple values on the bulk and external capacitors, respectively.

3. Possible DC Microgrid Scenarios, Load Models, and Required Control Schemes for the ESS Converter

In the present study, it was supposed that the Split-pi converter was used to interface an ESS with a DC microgrid. Under this hypothesis, several scenarios could be considered depending on the control mode chosen for both the storage-side converter and the grid-side converters interfacing other microgrid generators, if any. Depending on the combinations of the control modes of such converters, five scenarios were identified and, for each of them, a different type of load model and control scheme must be used. The different DC microgrid scenarios and the required load models and control schemes are discussed in the following subsections.

3.1. Possible DC Microgrid Scenarios

In general terms, one of the following situations can occur as a control mode for the microgrid converters:

• All the generator converters are controlled in droop mode to regulate the microgrid voltage with a predefined power-sharing ratio. Thus, each generator behaves as an ideal voltage generator with a series-connected resistance. The droop scheme leads to a simple but very reliable microgrid and does not require a communication infrastructure. If a storage system is present, its droop characteristic is usually chosen so that no current is supplied at half the rated load of the microgrid, whereas charging/discharging occurs for load power below/above such a value. If there is only one active device in the microgrid (either a storage system or a generator), it could also be controlled with a null droop resistance (stiff microgrid) so that the microgrid voltage does not vary with load power
• The microgrid follows a master–slave architecture: some converters for generators or storage systems (i.e., the masters) are controlled in droop mode (a null droop resistance is possible only with a single master); the others (i.e., the slaves) behave as current generators and are managed by an EMS to pursue one or more predefined goals.

Therefore, three control modes are possible for both the storage-side and grid-side converters: stiff droop control, non-stiff droop control, and current control. The five scenarios resulting from the combinations of the control modes of such converters are described in the first three columns of

Table 2. Possible combinations of microgrid scenarios, converter and load models, and control schemes for the Split-pi.

Microgrid Scenario	Storage Converter	Other Generators	State-Space Model of the Split-Pi	Load for the State-Space Model of the Split-Pi	Control Scheme
#1 (SS-GN)	Droop mode with Rd = 0 (stiff microgrid)	No other generator present (passive load) or all current controlled by the EMS	Model A	R and Ieq, as in Figure 2b	2 loops + FF, as in Figure 3 with Rds = 0
#2 (SD-GN)	Droop mode with Rd ≠ 0	No other generator present (passive load) or all current controlled by the EMS	Model A	R and Ieq, as in Figure 2b	3 loops + FF, as in Figure 3
#3 (SD-GD)	Droop mode with Rd ≠ 0	At least one is droop controlled and none has Rd = 0	Model A	R (low) and Ieq, as in Figure 2b	3 loops + FF, as in Figure 3
#4 (SC-GD)	Current mode	At least one is droop controlled and none has Rd = 0	Model A	R (low) and Ieq, as in Figure 2b	2 loops, as in Figure 4
#5 (SC-GS)	Current mode	One is droop controlled and has Rd = 0 (stiff microgrid); the others, if present, are current controlled by the EMS	Model B	Ed, as in Figure 2c	2 loops, as in Figure 4

. For the sake of clarity, such scenarios are referred to also by means of an abbreviation in the form Sx-Gy, where x and y specify the control mode for the storage-side and the grid-side converters, respectively. The range of options for x is:

• C for current control.
• D for non-stiff droop control.
• S for stiff droop control.

As for the options for y:

• N if no grid-side generator is present or if none of the grid-side generators are operated with droop control.
• D if at least one generator is droop controlled but not in a stiff way.
• S if there is one generator operated with stiff droop control; in such a case, the other grid-side generators, if present, must be current controlled by the EMS.

3.2. Possible Load Models for the ESS Converter

The dynamic behavior of the storage converter is affected by the load that it must supply, which depends not only on the passive loads of the DC microgrid but also on the possible presence of grid-side generators. In the most general case, the load for the ESS converter can be modeled following these steps: (1) aggregating all the droop-controlled generators of the microgrid using Thevenin’s theorem, obtaining parameters E_d and R_d; (2) combining all the current-controlled generators managed by the EMS to compute the overall current I; (3) including an aggregated passive load R_load. The resulting circuit model is shown in Figure 2a. However, it is convenient to reduce such a model to a suitable equivalent form comprising only one generator. Toward this aim, it is necessary to distinguish between the five scenarios. In scenarios #3 (SD-GD) and #4 (SC-GD), using Norton’s theorem, the couple E_d, R_d can be substituted with an equivalent current generator E_d/R_d and a parallel-connected resistance R_d. Then, the load can be reduced to that of Figure 2b with the following assumptions: I_eq = I + E_d/R_d and R = R_load//R_d, where // denotes the parallel connection of circuit elements. In scenarios #1 (SS-GN) and #2 (SD-GN), the same scheme of Figure 2b can be used assuming I_eq = I and R = R_load. Furthermore, in scenario #5 (SC-GS), the converter’s output voltage is set equal to E_d because R_d = 0. Since both R_load and I are now parallel connected to an ideal voltage generator, they do not influence the converter’s dynamics. Thus, the load can be modeled as shown in Figure 2c.

In the following, the state-space model of the storage converter connected to the load of Figure 2b is denoted as model A, whereas model B refers to the converter connected to the load of Figure 2c. The derivation of such models is given in Section 4. The relationship between the possible DC microgrid scenarios and the storage converter and load models is described in the first five columns of Table 2. It is worth noting that the value of the equivalent load resistance R considered in scenarios #3 (SD-GD) and #4 (SC-GD) is much lower than that of scenarios #1 (SS-GN) and #2 (SD-GN) because it results from the parallel connection of R_load and R_d. Finally, the last column of Table 2 reports the required control scheme for the ESS converter in each microgrid scenario according to the considerations given in the following section.

3.3. Required Control Schemes for the ESS Converter

In general, the closed-loop control scheme for the storage converter depends on the microgrid scenario. However, regardless of the scenario, an important goal is to control the current I_L1 of the leftmost inductor to ensure that it is compatible with the storage system’s current state. Thus, in the related control loop, its reference value should be dynamically saturated to avoid overcharging or overdischarging the storage system.

In the first three scenarios, i.e., SS-GN, SD-GN, and SD-GD, besides the current loop for I_L1, a voltage loop is needed to regulate the output voltage V₂ at the nominal voltage V_2n. A suitable controller is required in each control loop. Typically, PI or PID regulators with an anti-windup action are used, which must be designed to obtain a stable system with the desired dynamics. In scenarios #2 (SD-GN) and #3 (SD-GD), a third loop is also required to implement the droop characteristic of the storage converter by computing the voltage reference V_2ref based on the output current I₂ according to the equation V_2,ref = E_ds − R_ds·I₂, where E_ds and R_ds are the parameters of the droop characteristic of such a converter. The control scheme used in scenarios #1 (SS-GN), #2 (SD-GN), and #3 (SD-GD) is depicted in Figure 3. In particular, the external loop is opened in scenario #1 (SS-GN) because R_d = 0. The presence of the current generator I_eq in parallel to the load resistance is considered a disturbance that affects the system’s output, which will be suitably compensated for by the control system, regardless of the related transfer function. As will be shown in Section 6, when the output voltage V₂ is controlled, a feed-forward (FF) action is also required in addition to the voltage loop to suitably reduce the overshoot. Since I_eq cannot be measured, the output current I₂ is chosen as the FF action input.

On the other hand, the converter is current-controlled in scenarios #4 (SC-GD) and #5 (SC-GS). In these cases, besides the inner loop for I_L1, another current loop is needed to regulate the output current I₂ based on a reference I_2ref that is computed by the EMS. The related control scheme is shown in Figure 4. Again, the external voltage or current generator is considered a disturbance, and the related transfer function is irrelevant.

The type of control scheme to be used in each microgrid scenario and the required number of control loops are reported in the last column of Table 2. It is worth noting that suitable saturators are required in the controllers of each loop (G_ci1, G_cv2, G_ci2). Specifically, the output of the controller G_ci1 (i.e., the duty cycle d) is bounded by the interval [0; 0.9] to avoid overcurrents due to prolonged transients with d = 1. On the other hand, the reference value for I_L1 (i.e., the output of G_cv plus the FF term or the output of G_ci2) is bounded by the interval [−I_cx; I_dx], where I_cx and I_dx are the maximum charging/discharging currents of the storage system. Finally, the upper or lower bound of such an interval is dynamically replaced with zero if the battery SOC reaches 100% or goes below the minimum allowed SOC, respectively.

4. State-Space Models of the Split-Pi Converter

The two state-space models of a Split-pi converter that interfaces a storage system with a non-stiff (scenarios #1~#4) or stiff (scenario #5) microgrid and operates with V₁ ≤ V₂ are presented in the following. They consider the parasitic elements and were determined according to the state-space averaging technique [22].

4.1. State-Space Model A: Split-Pi Converter Connected to a Non-Stiff Microgrid

The state-space model of a Split-pi converter connected to a non-stiff microgrid and operating with V₁ ≤ V₂ can be expressed in matrix form as follows:

$$\{\begin{array}{c}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

(4)

$$x={\left[I_{L1}, I_{L2}, V_c, V_e\right]}^{\prime }$$

(5)

$$u={\left[V_1,I_{eq}\right]}^{\prime }$$

(6)

$$y={\left[I_{L1},V_2,I_2\right]}^{\prime }$$

(7)

$$\{\begin{array}{c}A=d{A}_{on}+\left(1-d\right)A_{off}\\ B=d{B}_{on}+\left(1-d\right)B_{off}\\ C=d{A}_{on}+\left(1-d\right)C_{off}\\ D=d{D}_{on}+\left(1-d\right)D_{off}\end{array}$$

(8)

$$A_{on}=\left[\begin{array}{cccc}-\frac{R_L}{L}& 0& 0& 0\\ 0& -\frac{R_{tot}}{L}& \frac{1}{L}& -\frac{R}{L{R}_{sum}}\\ 0& -\frac{1}{C}& 0& 0\\ 0& \frac{R}{R_{sum}{C}_e}& 0& -\frac{1}{R_{sum}{C}_e}\end{array}\right]$$

(9)

$$A_{off}=\left[\begin{array}{cccc}-\frac{R_L+R_c}{L}& \frac{R_c}{L}& -\frac{1}{L}& 0\\ \frac{R_c}{L}& -\frac{R_{tot}}{L}& \frac{1}{L}& -\frac{R}{L{R}_{sum}}\\ \frac{1}{C}& -\frac{1}{C}& 0& 0\\ 0& \frac{R}{R_{sum}{C}_e}& 0& -\frac{1}{R_{sum}{C}_e}\end{array}\right]$$

(10)

$$B_{on}=B_{off}=\left[\begin{array}{cc}\frac{1}{L}& 0\\ 0& -\frac{R_p}{L}\\ 0& 0\\ 0& \frac{R}{R_{sum}{C}_e}\end{array}\right]$$

(11)

$$C_{on}=C_{off}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& R_p& 0& \frac{R}{R_{sum}}\\ 0& \frac{R_e}{R_{sum}}& 0& \frac{1}{R_{sum}}\end{array}\right]$$

(12)

$$D_{on}=D_{off}=\left[\begin{array}{cc}0& 0\\ 0& R_p\\ 0& -\frac{R}{R_{sum}}\end{array}\right]$$

(13)

where R_p = R//R_e, R_sum = R + R_e, and R_tot = R_p + R_L + R_c.

Since $B_{on}=B_{off}$ and $D_{on}=D_{off}$, the small-signal behavior of the converter does not depend on the input values. According to the method described in [21], the model can be linearized around a chosen operating point corresponding to the duty cycle $\overline{d}$ and the state $\overline{x}={\left[I_{L10}, I_{L20}, V_{c0}, V_{e0}\right]}^{\prime }$, obtaining the following transfer functions:

$$G_{p1}\left(s\right)=\frac{{\tilde{I}}_{L1}}{\tilde{d}}=\frac{n_3{s}^3+n_2{s}^2+n_{1}s+n_0}{d_4{s}^4+d_3{s}^3+d_2{s}^2+d_{1}s+d_0}$$

(14)

$$G_{p2}\left(s\right)=\frac{{\tilde{I}}_2}{{\tilde{I}}_{L1}}=\frac{R_k{I}_{L10}}{L^{2}C{C}_e{R}_{sum}}·\frac{\left(1+s{R}_{c}C\right)\left(1+s{R}_e{C}_e\right)\left(1-s\frac{L}{R_k}\right)}{n_3{s}^3+n_2{s}^2+n_{1}s+n_0}$$

(15)

whose coefficients n_i, d_i, and R_k are given in Appendix A. G_p1(s) and G_p2(s) are small-signal transfer functions that express the effect of perturbations of a variable on another one (both denoted with a tilde). According to Figure 3, G_p1(s) expresses the relationship between d and I_L1, whereas G_p2(s) describes the dependence of I₂ on I_L1. Clearly, the relationship between I_L1 and V₂ is expressed by G_p2(s)∙R.

4.2. State-Space Model B: Split-Pi Converter Connected to a Stiff Microgrid

In the case of a Split-pi converter operating with V₁ ≤ V₂ and connected to a stiff microgrid, (4) and (8) are still valid, but (5), (6), (7),(9), (10) and (11), (12), (13) are replaced by (16), (17), (18), (19), (20) and (21), (22), (23), respectively.

$$x={\left[I_{L1}, I_{L2}, V_c, V_e\right]}^{\prime }$$

(16)

$$u={\left[V_1,E_d\right]}^{\prime }$$

(17)

$$y={\left[I_{L1},I_2\right]}^{\prime }$$

(18)

$$A_{on}=\left[\begin{array}{cccc}-\frac{R_L}{L}& 0& 0& 0\\ 0& -\frac{R_c+R_L}{L}& \frac{1}{L}& 0\\ 0& -\frac{1}{C}& 0& 0\\ 0& 0& 0& -\frac{1}{R_e{C}_e}\end{array}\right]$$

(19)

$$A_{off}=\left[\begin{array}{cccc}-\frac{R_c+R_L}{L}& \frac{R_c}{L}& -\frac{1}{L}& 0\\ \frac{R_c}{L}& -\frac{R_c+R_L}{L}& \frac{1}{L}& 0\\ \frac{1}{C}& -\frac{1}{C}& 0& 0\\ 0& 0& 0& -\frac{1}{R_e{C}_e}\end{array}\right]$$

(20)

$$B_{on}=B_{off}=\left[\begin{array}{cc}\frac{1}{L}& 0\\ 0& -\frac{1}{L}\\ 0& 0\\ 0& \frac{1}{R_e{C}_e}\end{array}\right]$$

(21)

$$C_{on}=C_{off}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& \frac{1}{R_e}\end{array}\right]$$

(22)

$$D_{on}=D_{off}=\left[\begin{array}{cc}0& 0\\ 0& -\frac{1}{R_e}\end{array}\right]$$

(23)

Again, the small-signal behavior does not depend on input values. The model can be linearized around a chosen operating point corresponding to $\overline{d}$ and $\overline{x}$, obtaining the transfer functions (24) and (25), whose coefficients n_i, d_i, and R_k are given in Appendix B. According to Figure 4, G_p1(s) expresses the relationship between d and I_L1, whereas G_p2(s) describes the dependence of I₂ on I_L1. It is worth noting that a zero-pole cancellation occurs due to the stiff voltage imposed on the R_eC_e branch at port 2; thus, the resulting system’s order is three instead of four and the voltage and current on C_e cannot be controlled.

$$G_{p1}\left(s\right)=\frac{{\tilde{I}}_{L1}}{\tilde{d}}=\frac{n_2{s}^2+n_{1}s+n_0}{d_3{s}^3+d_2{s}^2+d_{1}s+d_0}$$

(24)

$$G_{p2}\left(s\right)=\frac{{\tilde{I}}_2}{{\tilde{I}}_{L1}}=\frac{R_k{I}_{L10}}{L^{2}C}·\frac{\left(1+s{R}_{c}C\right)\left(1-s\frac{L}{R_k}\right)}{n_2{s}^2+n_{1}s+n_0}$$

(25)

5. Case Study and Converter Sizing

In this section, the chosen case study is described and the droop characteristics of both the storage converter and the voltage generator of the microgrid are discussed. Then, the Split-pi converter’s reactive components are sized based on the chosen case study’s parameters.

With no loss of generality, the proposed investigation was performed by referring to a 48 V, 750 W storage system that was interfaced with a 180 V DC microgrid using a Split-pi converter. The chosen case study can represent the onboard grid of an unmanned marine vehicle or a scaled prototype of a residential DC microgrid with a 120 V, 60 Hz, single-phase, grid-connected inverter, whose DC link voltage must be higher than 170 V DC. The rated values of the system under study are shown in

Table 3. Rated values of the system.

Parameter	Symbol	Value
Switching frequency	Fsw	20 kHz
Nominal output voltage	V2n	180 V
Nominal input voltage	V1n	50 V
Nominal power	Pn	750 W
Nominal load resistance	Rn	43.2 Ω
Nominal input current	I1n	15 A
Max. charge/discharge current	Icx, Idx	18 A
Nominal output current	I2n	4.167 A
Nominal duty cycle	$\overline{d}$	0.722

.

As for the chosen droop characteristic of the storage converter, in scenario #1 (SS-GN) it was defined by E_ds = 180 V and R_ds = 0, whereas in scenarios #2 (SD-GN) and #3 (SD-GD), it was expressed by E_ds = 180 V and R_ds = 2.2 Ω, i.e., imposing a 5% voltage reduction at the nominal current. For the equivalent droop-controlled microgrid generator, the same parameters as those of the storage converter were chosen in scenario #4 (SC-GD). On the other hand, in scenario #3 (SD-GD), the following parameters were used for the droop-controlled microgrid generator: E_d = 198 V and R_d = 9 Ω; with this choice, the storage system did not supply any power for half the rated load of the microgrid, as desired. Finally, a constant voltage generator E_d = 180 V was considered in scenario #5 (SC-GS) to model the microgrid’s stiff voltage generator.

The Split-pi’s reactive components were sized using (1), (2), (3), and the following parameters were set: r_i% = ±6.0%, r_ve% = ±0.2%, r_v% = ±0.2%. Consequently, the following minimum inductor and capacitor ratings were obtained: L_min = 803 µH, C_e,min = 113 µF, C_min = 419 µF. Aiming to build a converter prototype and due to component availability, slightly higher values were chosen for the reactive components; they are reported in

Table 4. Reactive components of the Split-pi.

Parameter	Symbol	Value
Inductance value of L	L	1000 µH
Parasitic resistance of L	RL	65 mΩ
Capacitance value of Ce	Ce	200 µF
Parasitic resistance of Ce	Re	260 mΩ
Capacitance value of C	C	540 µF
Parasitic resistance of C	Rc	125 mΩ

, together with their parasitic resistances.

6. Control System Design

The control system of the Split-pi converter must be designed by considering the specific microgrid scenario in which it will be used. However, in any case, it must be assumed that the converter supplies its rated power without any contribution from the external current or voltage generators, which are seen as disturbances. If the imposed stability margins are sufficiently wide, the designed controllers will be effective also at lighter loads, i.e., with higher values of R. Thus, models A and B must be linearized around the operating point corresponding to the rated values of the duty cycle and state variables: $\overline{d}=0.722$ and $\overline{x}={\left[I_{L10},I_{L20},V_{c0},V_{e0}\right]}^{\prime }={\left[15, 4.167, 180, 180\right]}^{\prime }$. No other condition is needed for model B in scenario #5 (SC-GS). As for model A, instead, it is R = R_n in scenarios #1 (SS-GN) and #2 (SD-GN) and R = R_n//R_d in scenarios #3 (SD-GD) and #4 (SC-GD).

All the above values must be substituted into (A1)–(A6), given in Appendix A, to obtain the coefficients of the transfer functions of interest. Then, designing of the controllers G_ci1, G_cv2, and G_ci2 can be performed with classic techniques that involve imposing suitable values of the crossover frequency ${\mathsf{\omega }}_{\mathrm{c}}$ and phase margin ${\mathrm{m}}_{\mathsf{\phi }}$ and ensuring a suitable gain margin m_g [23]. For each control loop and scenario, the imposed values and the obtained PI coefficients and gain margins are summarized in

Table 5. Coefficients of PI regulators in the case of model A.

Loop	Scenario	Value of R	Controller	Imposed ωc and mφ	PI Coefficients	Obtained mg
Current IL1	Baseline #1 (SS-GN) #2 (SD-GN)	Rn	Gci1	${\mathsf{\omega }}_{\mathrm{c}}{=3000\mathrm{rad}/\mathrm{s}\mathrm{and}\mathrm{m}}_{\mathsf{\phi }}=85\text{°}$	Kpi = 0.0160 Kii = 5.3703	∞
Current IL1	#3 (SD-GD) #4 (SC-GD)	Rn//Rd	Gci1	${\mathsf{\omega }}_{\mathrm{c}}{=3000\mathrm{rad}/\mathrm{s}\mathrm{and}\mathrm{m}}_{\mathsf{\phi }}=85\text{°}$	Kpi = 0.0161 Kii = 5.2481	∞
Voltage V2 without FF	Baseline	Rn	Gcv2	${\mathsf{\omega }}_{\mathrm{c}}{=100\mathrm{rad}/\mathrm{s}\mathrm{and}\mathrm{m}}_{\mathsf{\phi }}=85\text{°}$	Kpv = 0.2659 Kiv = 18.6209	12.7 dB
Voltage V2 with FF	#1 (SS-GN) #2 (SD-GN)	Rn	Gcv2	${\mathsf{\omega }}_{\mathrm{c}}{=100\mathrm{rad}/\mathrm{s}\mathrm{and}\mathrm{m}}_{\mathsf{\phi }}=85\text{°}$	Kpv = 0.2712 Kiv = 10.4112	13.1 dB
Voltage V2 with FF	#3 (SD-GD)	Rn//Rd	Gcv2	${\mathsf{\omega }}_{\mathrm{c}}{=100\mathrm{rad}/\mathrm{s}\mathrm{and}\mathrm{m}}_{\mathsf{\phi }}=85\text{°}$	Kpv = 0.3284 Kiv = 14.6218	24.2 dB
Current I2	#4 (SC-GD)	Rn//Rd	Gci2	${\mathsf{\omega }}_{\mathrm{c}}{=100\mathrm{rad}/\mathrm{s}\mathrm{and}\mathrm{m}}_{\mathsf{\phi }}=85\text{°}$	Kpi2 = 0.3746 Kii2 = 389.045	23.7 dB

and

Table 6. Coefficients of PI regulators in the case of model B.

Loop	Scenario	Controller	Imposed ωc and mφ	PI Coefficients	Obtained mg
Current IL1	#5 (SC-GS)	Gci1	${\mathsf{\omega }}_{\mathrm{c}}=3000\mathrm{rad}/\mathrm{s}\mathrm{and}{\mathrm{m}}_{\mathsf{\phi }}=85\text{°}$	Kpi = 0.0161 Kii = 5.0699	∞
Current I2	#5 (SC-GS)	Gci2	${\mathsf{\omega }}_{\mathrm{c}}=100\mathrm{rad}/\mathrm{s}\mathrm{and}{\mathrm{m}}_{\mathsf{\phi }}=85\text{°}$	Kpi2 = 1.1953 Kii2 = 362.22	13.6 dB with Gadd(s)

. Furthermore, a baseline scenario employing the control scheme of Figure 3 with R_d = 0 and without the FF action was also considered for comparison purposes to show the usefulness of such an action.

It is worth noting that the design of the controllers must be very conservative. By design, the currents I_L1 and I_L2 have a significant switching ripple compared to the input/output currents and voltages. Thus, the desired crossover frequency for the I_L1 loop must be suitably lower than F_sw to avoid the switching ripple being processed by the controller. The crossover frequency of the loop for V₂ or I₂ must be even lower for proper decoupling with respect to the inner loop. When the converter supplies a passive load (R_load), some ringing can be tolerated, and the usually adopted phase margin (50–60°) is satisfactory. Instead, in the case of an active microgrid, the combined variations of I and R_load could determine a significant excursion from the nominal operating point and pronounced under/overshoots; thus, a higher phase margin (i.e., ${\mathrm{m}}_{\mathsf{\phi }}$ > 80°) is required to ensure stability under all the operating conditions. As for the gain margin, the usually adopted criterion (i.e., m_g > 12 dB) is enough to ensure robustness against parameter variations.

As for the chosen case study, some noteworthy remarks can be made:

• Regardless of the value of R, models A and B exhibited nearly the same dynamics for I_L1; therefore, the corresponding PI regulators G_ci1 had similar coefficients. On the other hand, the dynamics of I₂ were quite dissimilar and required different G_ci2 controllers.
• In the case of model B, it could be possible to achieve the desired ${\mathsf{\omega }}_{\mathrm{c}}$ and ${\mathrm{m}}_{\mathsf{\phi }}$ with a PI or I regulator alone, but the gain margin would be around 6.6 dB due to a resonance peak; thus, the second-order transfer function (26) must be included after the PI regulator to attenuate such a peak and achieve a gain margin of 13.6 dB.
• The dynamics of I₂ were very sensitive to the value of R in model A; thus, an unstable system was obtained in scenario #4 (SC-GD) if the controllers were designed assuming R = R_n instead of R = R_n//R_d.
• Without the FF action, the dynamics of V₂ were sensitive to the value of R: a significantly slower behavior was obtained if the controllers were designed assuming R = R_n in scenario #3 (SD-GD) instead of R = R_n//R_d; on the other hand, a significant ringing was obtained if the controllers were designed considering R = R_n//R_d in scenarios #1 (SS-GN) and #2 (SD-GN) instead of R = R_n.
• Using the FF action, the dynamics of V₂ were pretty insensitive to the value of R; thus, almost no variation was obtained in scenarios #1 (SS-GN), #2 (SD-GN), and #3 (SD-GD) if the controllers were designed when considering either R = R_n//R_d or R = R_n.

Several simulations were performed to assess the performance of the controlled system in all the scenarios. Then, a prototype of the Split-pi converter was built, and experimental tests were performed in several conditions that covered the baseline scenario and all the other five scenarios, obtaining successful results. The simulation and experimental results validating the study are presented in part II of this work.

$$G_{add}\left(s\right)=\frac{1}{\left(1+\raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$533$}\right.\right)\left(1+\raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$606$}\right.\right)}$$

(26)

7. Conclusions

The Split-pi converter is a suitable choice to interface electrical storage systems with DC microgrids. It offers distinct advantages, such as high efficiency, reduced switch count and switching noise, and suitability for multiphase systems at the cost of non-isolated operation. However, to obtain high performance, its control system must be suitably designed according to the specific microgrid scenario in which it will be used.

In this study, five typical microgrid scenarios were identified and analyzed, where each of which required a specific state-space model and a suitable control scheme for the converter. Two different state-space models were presented for the Split-pi converter operating with the storage-side voltage being lower than the grid-side voltage. Both models considered the parasitic elements of the reactive components. As for the control scheme, the number of required control loops depended on the scenario. It was shown that feed-forward action is needed to obtain a high performance in the case of voltage control and that, sometimes, conventional PI regulators alone were not sufficient for stable current control. The most relevant transfer functions of the Split-pi converter were given, together with criteria to design the controllers. Several simulations, as well as experimental tests on a prototype realized in the lab, were performed to validate the study, whose results will be presented in part II of this work.

The approach followed in this study has general validity and can also be followed to devise the state-space model of a Split-pi operating with a storage-side voltage that is higher than the grid-side voltage or when other bidirectional DC/DC converter topologies are employed to interface an ESS with a DC microgrid. Furthermore, the presented study builds the premises for designing unconventional control systems for the Split-pi that are suitable for operating in more than one microgrid scenario.