*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 *IL*<sup>1</sup> 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 *IL*1, a voltage loop is needed to regulate the output voltage *V*<sup>2</sup> at the nominal voltage *V*2*n*. 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*2*ref* based on the output current *I*<sup>2</sup> according to the equation *V*2,*ref* = *Eds* − *Rds*·*I*2, where *Eds* and *Rds* 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 *Rd* = 0. The presence of the current generator *Ieq* 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 *V*<sup>2</sup> is controlled, a feed-forward (FF) action is also required in addition to the voltage loop to suitably reduce the overshoot. Since *Ieq* cannot be measured, the output current *I*<sup>2</sup> 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 *IL*1, another current loop is needed to regulate the output current *I*<sup>2</sup> based on a reference *I*2*ref* 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 (*Gci*1, *Gcv*2, *Gci*2). Specifically, the output of the controller *Gci*<sup>1</sup> (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 *IL*<sup>1</sup> (i.e., the output of *Gcv* plus the FF term or the output of *Gci*2) is bounded by the interval [−*Icx*; *Idx*], where *Icx* and *Idx* 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 *V*<sup>1</sup> ≤ *V*<sup>2</sup> 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 *V*<sup>1</sup> ≤ *V*<sup>2</sup> can be expressed in matrix form as follows:

$$\begin{cases} \dot{\mathbf{x}} = A\mathbf{x} + Bu\\ \mathbf{y} = \mathbf{C}\mathbf{x} + Du \end{cases} \tag{4}$$

$$\mathbf{x} = [I\_{L1\prime} \ I\_{L2\prime} \ V\_{\mathbf{c}\prime} \ V\_{\mathbf{c}}]^{\prime} \tag{5}$$

$$
\mu = \begin{bmatrix} V\_1 \ I\_{\text{eq}} \end{bmatrix}' \tag{6}
$$

$$y = [I\_{L1}, V\_{2}, I\_{2}]'\tag{7}$$

$$\begin{cases} A = dA\_{\text{on}} + (1 - d)A\_{off} \\ B = dB\_{\text{on}} + (1 - d)B\_{off} \\ C = dA\_{\text{on}} + (1 - d)C\_{off} \\ D = dD\_{\text{on}} + (1 - d)D\_{off} \end{cases} \tag{8}$$

$$A\_{on} = \begin{bmatrix} -\frac{R\_L}{L} & 0 & 0 & 0\\ 0 & -\frac{R\_{tot}}{L} & \frac{1}{L} & -\frac{R}{LR\_{sum}}\\ 0 & -\frac{1}{L} & 0 & 0\\ 0 & \frac{R}{R\_{sum}C\_c} & 0 & -\frac{1}{R\_{sum}C\_c} \end{bmatrix} \tag{9}$$

$$A\_{off} = \begin{bmatrix} -\frac{\overline{R\_c + R\_c}}{L} & \frac{\overline{R\_c}}{L} & -\frac{1}{L} & 0\\ \frac{\overline{R\_c}}{L} & -\frac{\overline{R\_{tot}}}{L} & \frac{1}{L} & -\frac{R}{L\overline{R\_{inv}}}\\ \frac{1}{\overline{C}} & -\frac{1}{\overline{C}} & 0 & 0\\ 0 & \frac{\overline{R\_c}}{\overline{R\_{conv}}\overline{C\_c}} & 0 & -\frac{1}{\overline{R\_{conv}}\overline{C\_c}} \end{bmatrix} \tag{10}$$

$$B\_{on} = B\_{off} = \begin{bmatrix} \frac{1}{L} & 0\\ 0 & -\frac{R\_p}{L} \\ 0 & 0 \\ 0 & \frac{R}{R\_{sum}C\_\varepsilon} \end{bmatrix} \tag{11}$$

$$\mathbf{C}\_{on} = \mathbf{C}\_{off} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & R\_p & 0 & \frac{R}{R\_{sum}} \\ 0 & \frac{R\_r}{R\_{sum}} & 0 & \frac{1}{R\_{sum}} \end{bmatrix} \tag{12}$$

$$D\_{on} = D\_{off} = \begin{bmatrix} 0 & 0 \\ 0 & R\_p \\ 0 & -\frac{R}{R\_{sun}} \end{bmatrix} \tag{13}$$

where *Rp* = *R//Re*, *Rsum* = *R* + *Re*, and *Rtot* = *Rp* + *RL* + *Rc*.

Since *Bon* = *Boff* and *Don* = *Doff* , 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 *d* and the state *x* = [*IL*10, *IL*20, *Vc*0, *Ve*0] , obtaining the following transfer functions:

$$G\_{p1}(s) = \frac{\widetilde{I}\_{L1}}{\widetilde{d}} = \frac{n\_3s^3 + n\_2s^2 + n\_1s + n\_0}{d\_4s^4 + d\_3s^3 + d\_2s^2 + d\_1s + d\_0} \tag{14}$$

$$G\_{p2}(s) = \frac{\overline{I\_2}}{\overline{I\_{L1}}} = \frac{R\_k I\_{L10}}{L^2 \mathcal{C} \mathcal{C}\_\varepsilon R\_{sum}} \cdot \frac{(1 + sR\_\varepsilon \mathcal{C})(1 + sR\_\varepsilon \mathcal{C}\_\varepsilon)\left(1 - s\frac{L}{R\_k}\right)}{n\_3 s^3 + n\_2 s^2 + n\_1 s + n\_0} \tag{15}$$

whose coefficients *ni*, *di*, and *Rk* are given in Appendix A. *Gp*1(*s*) and *Gp*2(*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, *Gp*1(*s*) expresses the relationship between *d* and *IL*1, whereas *Gp*2(*s*) describes the dependence of *I*<sup>2</sup> on *IL*1. Clearly, the relationship between *IL*<sup>1</sup> and *V*<sup>2</sup> is expressed by *Gp*2(*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 *V*<sup>1</sup> ≤ *V*<sup>2</sup> 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.

$$\propto = \begin{bmatrix} I\_{L1\prime} & I\_{L2\prime} & V\_{c\prime} & V\_{c} \end{bmatrix}^{\prime} \tag{16}$$

$$
\mu = [V\_1, E\_d]'\tag{17}
$$

$$y = [I\_{L1}, I\_2]'\tag{18}$$

$$A\_{on} = \begin{bmatrix} -\frac{R\_L}{L} & 0 & 0 & 0\\ 0 & -\frac{R\_c + R\_L}{L} & \frac{1}{L} & 0\\ 0 & -\frac{1}{C} & 0 & 0\\ 0 & 0 & 0 & -\frac{1}{R\_c C\_c} \end{bmatrix} \tag{19}$$

$$A\_{off} = \begin{bmatrix} -\frac{R\_c + R\_L}{L} & \frac{R\_c}{L} & -\frac{1}{L} & 0\\ \frac{R\_c}{L} & -\frac{R\_c + R\_L}{L} & \frac{1}{L} & 0\\ \frac{1}{C} & -\frac{1}{C} & 0 & 0\\ 0 & 0 & 0 & -\frac{1}{R\_c C\_c} \end{bmatrix} \tag{20}$$

$$B\_{on} = B\_{off} = \begin{bmatrix} \frac{1}{L} & 0\\ 0 & -\frac{1}{L} \\ 0 & 0 \\ 0 & \frac{1}{R\_c C\_r} \end{bmatrix} \tag{21}$$

$$\mathbb{C}\_{on} = \mathbb{C}\_{off} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \frac{1}{\mathbb{R}\_r} \end{bmatrix} \tag{22}$$

$$D\_{on} = D\_{off} = \begin{bmatrix} 0 & 0 \\ 0 & -\frac{1}{\mathbb{R}\_{\epsilon}} \end{bmatrix} \tag{23}$$

Again, the small-signal behavior does not depend on input values. The model can be linearized around a chosen operating point corresponding to *d* and *x*, obtaining the transfer functions (24) and (25), whose coefficients *ni*, *di*, and *Rk* are given in Appendix B. According to Figure 4, *Gp*1(*s*) expresses the relationship between *d* and *IL*1, whereas *Gp*2(*s*) describes the dependence of *I*<sup>2</sup> on *IL*1. It is worth noting that a zero-pole cancellation occurs due to the stiff voltage imposed on the *ReCe* branch at port 2; thus, the resulting system's order is three instead of four and the voltage and current on *Ce* cannot be controlled.

$$G\_{p1}(s) = \frac{\tilde{I}\_{L1}}{\tilde{d}} = \frac{n\_2s^2 + n\_1s + n\_0}{d\_3s^3 + d\_2s^2 + d\_1s + d\_0} \tag{24}$$

$$G\_{p2}(s) = \frac{\overline{I}\_2}{\overline{I}\_{L1}} = \frac{R\_k I\_{L10}}{L^2 \mathbb{C}} \cdot \frac{(1 + sR\_c \mathbb{C})\left(1 - s\frac{L}{\mathbb{R}\_k}\right)}{n\_2 s^2 + n\_1 s + n\_0} \tag{25}$$

#### **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.

As for the chosen droop characteristic of the storage converter, in scenario #1 (SS-GN) it was defined by *Eds* = 180 V and *Rds* = 0, whereas in scenarios #2 (SD-GN) and #3 (SD-GD), it was expressed by *Eds* = 180 V and *Rds* = 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 droopcontrolled microgrid generator: *Ed* = 198 V and *Rd* = 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 *Ed* = 180 V was considered in scenario #5 (SC-GS) to model the microgrid's stiff voltage generator.

**Table 3.** Rated values of the system.


The Split-pi's reactive components were sized using (1), (2), (3), and the following parameters were set: *ri%* = ±6.0%*, rve%* = ±0.2%*, rv%* = ±0.2%. Consequently, the following minimum inductor and capacitor ratings were obtained: *Lmin* = 803 μH, *Ce,min* = 113 μF, *Cmin* = 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, together with their parasitic resistances.

**Table 4.** Reactive components of the Split-pi.


#### **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: *d* = 0.722 and *x* = [*IL*10, *IL*20, *Vc*0, *Ve*0] = [15, 4.167, 180, 180] . No other condition is needed for model B in scenario #5 (SC-GS). As for model A, instead, it is *R* = *Rn* in scenarios #1 (SS-GN) and #2 (SD-GN) and *R=Rn*//*Rd* 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 *Gci*1, *Gcv*2, and *Gci*<sup>2</sup> can be performed with classic techniques that involve imposing suitable values of the crossover frequency ω<sup>c</sup> and phase margin m<sup>ϕ</sup> and ensuring a suitable gain margin mg [23]. For each control loop and scenario, the imposed values and the obtained PI coefficients and gain margins are summarized in Tables 5 and 6. Furthermore, a baseline scenario employing the control scheme of Figure 3 with *Rd* = 0 and without the FF action was also considered for comparison purposes to show the usefulness of such an action.


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

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


It is worth noting that the design of the controllers must be very conservative. By design, the currents *IL*<sup>1</sup> and *IL*<sup>2</sup> have a significant switching ripple compared to the input/output currents and voltages. Thus, the desired crossover frequency for the *IL*<sup>1</sup> loop must be suitably lower than *Fsw* to avoid the switching ripple being processed by the controller. The crossover frequency of the loop for *V*<sup>2</sup> or *I*<sup>2</sup> must be even lower for proper decoupling with respect to the inner loop. When the converter supplies a passive load (*Rload*), 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 *Rload* could determine a significant excursion from the nominal operating point and pronounced under/overshoots; thus, a higher phase margin (i.e., mϕ > 80◦) is required to ensure stability under all the operating conditions. As for the gain margin, the usually adopted criterion (i.e., mg > 12 dB) is enough to ensure robustness against parameter variations.

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


• Using the FF action, the dynamics of *V*<sup>2</sup> 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* = *Rn*//*Rd* or *R* = *Rn*.

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.

$$G\_{add}(\mathbf{s}) = \frac{1}{\left(1 + \frac{s}{533}\right)\left(1 + \frac{s}{606}\right)}\tag{26}$$

#### **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.

**Author Contributions:** Conceptualization, M.L., A.S., M.P. and M.C.D.P.; methodology, M.L., A.S. and A.A.; software, M.L., A.S., A.A. and G.L.T.; validation, M.L., A.A., M.C.D.P. and M.P.; writing original draft preparation, M.L., A.S. and M.P.; writing—review and editing, M.L., A.A., M.C.D.P. and G.L.T.; supervision, M.P. and M.L.; funding acquisition, M.P. and M.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Italian Ministry of University and Research (MUR), program PON R&I 2014/2020—Avviso n. 1735 del 13/07/2017—PNR 2015/2020, project "NAUSICA—NAvi efficienti tramite l'Utilizzo di Soluzioni tecnologiche Innovative e low CArbon," CUP: B45F21000680005.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
