*2.1. Modeling of Distributed Power Unit*

*2.1. Modeling of Distributed Power Unit* The resources are connected to the DC microgrid through power converters, whose control dictates their behavior. Resources under MPPT try to maximize the injection of available power to the DC microgrid regardless of the network status. Under constant weather conditions, the power converter of distributed generation can be modeled as constant power sources (CPSs) when viewed by the bus terminals [5]. This means that despite of the variation on the bus voltage, the current provided by the power converter adapts to keep injecting constant power. Such distributed resources can be considered to be a type of CPLs with a negative power consumption, and both constant power loads and constant power sources can be totally modeled as an ideal constant power model, as illustrated in Figure 2a (taking the photovoltaic as an example). As CPLs and CPSs behave in the same way, they can be modeled as lumped CPLs that demand an equivalent constant power P given by (1). And based on [21], energy storage units based on the droop control are modeled as an ideal voltage source with series resistance, as illustrated in Figure 2b. The resources are connected to the DC microgrid through power converters, whose control dictates their behavior. Resources under MPPT try to maximize the injection of available power to the DC microgrid regardless of the network status. Under constant weather conditions, the power converter of distributed generation can be modeled as constant power sources (CPSs) when viewed by the bus terminals [5]. This means that despite of the variation on the bus voltage, the current provided by the power converter adapts to keep injecting constant power. Such distributed resources can be considered to be a type of CPLs with a negative power consumption, and both constant power loads and constant power sources can be totally modeled as an ideal constant power model, as illustrated in Figure 2a (taking the photovoltaic as an example). As CPLs and CPSs behave in the same way, they can be modeled as lumped CPLs that demand an equivalent constant power P given by (1). And based on [21], energy storage units based on the droop control are modeled as an ideal voltage source with series resistance, as illustrated in Figure 2b. *Appl. Sci.* **2019**, *9*, x FOR PEER REVIEW 4 of 15

$$P = P\_{\rm CPLs} + P\_{\rm CPSs} P\_{\rm CPLs} > 0 \text{ and } P\_{\rm CPSs} < 0,\tag{1}$$

**Figure 2.** Distributed Resources model. (**a**) Maximum Power Point Tracking (MPPT) control. (**b**) Droop control. **Figure 2.** Distributed Resources model. (**a**) Maximum Power Point Tracking (MPPT) control. (**b**) Droop control.

### *2.2. Dimension Reduction Modeling of Multi-Voltage Control Unit under Droop Control 2.2. Dimension Reduction Modeling of Multi-Voltage Control Unit under Droop Control*

*P*=

On the basis of the work done above, the multi-branch combination model of DC microgrids with multiple voltage control units was obtained, as illustrated in Figure 3. On the basis of the work done above, the multi-branch combination model of DC microgrids with multiple voltage control units was obtained, as illustrated in Figure 3.

**Figure 3.** Multi-branch combination model of *n* droop control resources. **Figure 3.** Multi-branch combination model of *n* droop control resources.

*Vi* is the source voltage and *vbus* is the voltage of the dc bus. *Rdi* is the droop coefficient of the *i*t*h* droop controller, *Rti* and *Lti* are the line resistance and inductance from *i*th energy storage unit to the bus, respectively. *C* is the equivalent capacitance on the bus [25]. The droop resistance and line resistance in series of the *i*th control unit can be represented by: *Vi* is the source voltage and *vbus* is the voltage of the dc bus. *Rdi* is the droop coefficient of the *i*th droop controller, *Rti* and *Lti* are the line resistance and inductance from *i*th energy storage unit to the bus, respectively. *C* is the equivalent capacitance on the bus [25]. The droop resistance and line resistance in series of the *i*th control unit can be represented by:

$$\mathbf{R}\_{i} = \mathbf{R}\_{d\bar{l}} + \mathbf{R}\_{l\bar{l}\nu} \tag{2}$$

can be rewritten as: *Ri*≈*Rdi*, (3) Assume that the value of droop resistance is much greater than the line resistance, Equation (2) can be rewritten as:

*Ri*=

*di<sup>2</sup> dt* =

*di<sup>n</sup> dt* =

*dis dt* = (∑ 1 *Lti*

following equation is obtained

1

$$\mathcal{R}\_l \approx \mathcal{R}\_{\text{dif}\prime} \tag{3}$$

system. Each distributed unit can be modeled as a differential equation in (4). *di<sup>1</sup> dt* = 1 *Lt1* (*V1—vbus*) − *Rd1 Lt1 i1 ,* As a complex multi-dimensional system, the DC microgrid power system is not conducive to analysis [26]. It is necessary to simplify the *n*-dimensional equation of representative DC microgrid system. Each distributed unit can be modeled as a differential equation in (4).

$$\begin{aligned} \frac{di\_1}{dt} &= \frac{1}{L\_{l1}}(V\_1 - v\_{\text{bus}}) - \frac{R\_{l1}}{L\_{l1}}i\_{1\prime} \\ \frac{di\_2}{dt} &= \frac{1}{L\_{l2}}(V\_2 - v\_{\text{bus}}) - \frac{R\_{l2}}{L\_{l2}}i\_{2\prime} \\ &\vdots \\ \frac{di\_n}{dt} &= \frac{1}{L\_{ln}}(V\_n - v\_{\text{bus}}) - \frac{R\_{\text{dir}}}{L\_{ln}}i\_{\text{ll}} \end{aligned} \tag{4}$$

Therefore, the sum of the *n* differential equations in (4) can be simplified to *n* Furthermore, the total current *i<sup>s</sup>* provided by the resources can be denoted by

$$i\_s = i\_1 + i\_2 + \dots + i\_{n\nu} \tag{5}$$

In reality, multiple identical energy storage units often exist in the form of clusters [27], and the overall power supply capacity is evenly divided into individual battery unit. Assuming that the Therefore, the sum of the *n* differential equations in (4) can be simplified to

*i=1*

$$\frac{di\_s}{dt} = \left(\sum\_{i=1}^{n} \frac{1}{L\_{ti}}\right) (V\_i - v\_{bus}) - \frac{R\_{di}}{L\_{ti}} i\_{s\_{\!\!\!}} \tag{6}$$

*Rd1*=*Rd2*=⋯=*Rdn*, (7) *Lt1*=*Lt2*=⋯=*Ltn*. Multiplying both sides of (6) by *Lt*= 1 *1 ,* (8) In reality, multiple identical energy storage units often exist in the form of clusters [27], and the overall power supply capacity is evenly divided into individual battery unit. Assuming that the specifications of each energy storage unit and the line length to the bus are the same. Thus, the following equation is obtained

$$V\_{ref} = V\_1 = V\_2 = \dots = V\_{n\nu}$$

$$R\_{d1} = R\_{d2} = \dots = R\_{dn\nu} \tag{7}$$

$$L\_{l1} = L\_{l2} = \dots = L\_{tn}.$$

*Appl. Sci.* **2019**, *9*, 4449

Multiplying both sides of (6) by

$$L\_t = \frac{1}{\sum\_{i=1}^{n} \frac{1}{L\_{tn}}} \, \tag{8}$$

The system is simplified from multiple differential equations to a simple differential equation in (9). Thus, the conversion from multi-dimensional to a low-dimensional is realized *Appl. Sci.* **2019**, *9*, x FOR PEER REVIEW 6 of 15

$$L\_t \frac{di\_s}{dt} = \left(V\_{ref} - v\_{bus}\right) - L\_t \frac{R\_{dn}}{L\_{tn}} i\_{s\prime} \tag{9}$$

And under the condition: And under the condition:

$$\mathcal{R}\_d = L\_t \frac{\mathcal{R}\_{dn}}{L\_{tn}},\tag{10}$$

From (9), the equivalent steady-state model of the DC microgrid system can be obtained, as illustrated in Figure 4. From (9), the equivalent steady-state model of the DC microgrid system can be obtained, as illustrated in Figure 4.

**Figure 4.** Equivalent steady-state model of a DC microgrid. **Figure 4.** Equivalent steady-state model of a DC microgrid.

where *Vref* is the reference voltage of the bus. Resistance *R<sup>d</sup>* is the droop control coefficient. *P*, *Rdc* correspond to the equivalent power of CPLs and the resistive load, respectively. The corresponding whole system parameters are listed in Table 1. where *Vre f* is the reference voltage of the bus. Resistance *R<sup>d</sup>* is the droop control coefficient. *P*, *Rdc* correspond to the equivalent power of CPLs and the resistive load, respectively. The corresponding whole system parameters are listed in Table 1.


**Table 1.** Parameters of DC microgrid system.

### of Voltage Controller **3. Virtual Inductive Control Strategy**

**3. Virtual Inductive Control Strategy**

*3.1. Definition of Instability*

characteristics can be given by

### Voltage Controller *3.1. Definition of Instability*

Filter Coefficient 0.0005 Droop Coefficient 0.5 For CPLs, the input power is equal to the required power of the CPLs. And the power characteristics can be given by

Proportional Coefficient

Integral Coefficient of

$$P = V\_{bus} i\_{load} \tag{11}$$

*Vbusiload*, (11)

1.2

250

*P*=

And (12) can be obtained by small signal analysis.

For CPLs, the input power is equal to the required power of the CPLs. And the power

And (12) can be obtained by small signal analysis.

$$
\Delta \dot{u}\_{\text{load}} = -\frac{P}{V\_{\text{bus}}^2} \Delta v\_{\text{bus}} \tag{12}
$$

where ∆*iload* and ∆*vbus* are the perturbation of load current and bus voltage, respectively. *Vbus* is the steady-state bus voltage value. It can be noted from (12) that the input current of CPLs varies inversely with bus voltage at a multiple of *P*/*V* 2 *bus*. Thus, the rise (decrease) of voltage corresponds to the decrease (rise) of current. That is why the CPLs cause instability, also known as negative impedance characteristics. It is noteworthy in (12) that the negative impedance characteristic effect of the CPLs on the bus voltage can be eliminated by offsetting the current change. The virtual inductive control strategy of this paper will be introduced in detail below. *∆iload=* − *P Vbus <sup>2</sup> ∆vbus*, (12) where *Δiload* and *Δvbus* are the perturbation of load current and bus voltage, respectively. *Vbus* is the steady-state bus voltage value. It can be noted from (12) that the input current of CPLs varies inversely with bus voltage at a multiple of *P Vbus 2* ⁄ . Thus, the rise (decrease) of voltage corresponds to the decrease (rise) of current. That is why the CPLs cause instability, also known as negative impedance characteristics. It is noteworthy in (12) that the negative impedance characteristic effect

### *3.2. Virtual Inductance Control Strategy* of the CPLs on the bus voltage can be eliminated by offsetting the current change. The virtual inductive control strategy of this paper will be introduced in detail below.

o

And *Z<sup>d</sup>* can be rewritten as

where *Ld= 1*⁄*kiv*, and *T= kpv*

expressed as follows

−*L<sup>d</sup>*

(*S*)

> *1*+*ZdGPI*v

⁄

*Zd*= − *L<sup>d</sup> s 1 Ts*+*1*

(*s*)

Voltage and current double closed-loop control is a common control method [12,18]. And transfer function of voltage and current double closed-loop control of DC microgrid given by *3.2. Virtual Inductance Control Strategy*

$$\mathbf{G}\_{\rm CLI}(\mathbf{s}) = \frac{\left(k\_{\rm p\bar{s}}\mathbf{s} + k\_{\bar{u}}\right)V\_0}{L\mathbf{s}^2 + k\_{\bar{p}l}V\_0\mathbf{s} + k\_{\bar{u}}V\_0},\tag{13}$$

$$\mathcal{G}\_{\rm CLv}(s) = \frac{\left(k\_{p\upsilon}s + k\_{\dot{v}\upsilon}\right)(1 - D)}{\rm{Cs}^2 + k\_{p\upsilon}(1 - D)s + k\_{\dot{v}\upsilon}(s)},\tag{14}$$

*kpi*, *kii* are the proportional and integral parameters of current loop, respectively. And *kpv*, *kiv* are the proportional and integral parameters of voltage loop, respectively. According to the selection rules of controller parameters in [16], the specific PI controller parameters of this paper given in Table 1. The dynamic characteristics of the current inner loop can be equivalent to proportional gain one when calculating the voltage outer loop transfer function. The small signal block diagram of the droop control strategy can be derived, as illustrated in Figure 5 [28]. *kpi,kii* are the proportional and integral parameters of current loop, respectively. And *kpv,kiv* are the proportional and integral parameters of voltage loop, respectively. According to the selection rules of controller parameters in [16], the specific PI controller parameters of this paper given in Table 1. The dynamic characteristics of the current inner loop can be equivalent to proportional gain one when calculating the voltage outer loop transfer function. The small signal block diagram of the droop control strategy can be derived, as illustrated in Figure 5 [28].

**Figure 5.** Small signal control block diagram of droop control. **Figure 5.** Small signal control block diagram of droop control.

where *Z<sup>d</sup>* is the droop control, *∆v<sup>o</sup>* , *∆vref* , *∆i*, *∆iref* are the perturbation of the output voltage, the reference voltage, output current, and the output reference current of double closed-loop controller, respectively. *GPIv*(*s*) is the voltage control loop transfer function and *G<sup>c</sup>* (*s*) is the current inner loop controller closed-loop transfer function. The output voltage perturbation *∆v*o can be given by: *GPIvGc1+ZdGPIv*(*s*)*Gc* (*s*)where *Z<sup>d</sup>* is the droop control, ∆*vo*, ∆*vre f* , ∆*io*, ∆*ire f* are the perturbation of the output voltage, the reference voltage, output current, and the output reference current of double closed-loop controller, respectively. *GPIv*(*s*) is the voltage control loop transfer function and *Gc*(*s*) is the current inner loop controller closed-loop transfer function. The output voltage perturbation *varDeltav<sup>o</sup>* can be given by:

$$var\text{Delat}v\_{\upsilon} = \frac{\text{G}\_{\text{Pllr}(\text{S})}\text{G}\_{\text{c}(\text{s})}}{\text{Cs} + \text{G}\_{\text{Pllr}}(\text{s})\text{G}\_{\text{C}}(\text{s})}vartext{Delat}v\_{\text{ref}} - \frac{1 + \text{Z}\_{\text{d}}\text{G}\_{\text{Pllr}}(\text{s})\text{G}\_{\text{c}}(\text{s})}{\text{Cs} + \text{G}\_{\text{Pllr}}(\text{s})\text{G}\_{\text{c}}(\text{s})}vartext{Delat}i\_{\text{lo}} \tag{15}$$

(16)

, (17)

*s,* (18)

determined by the change of the output current. Therefore, reasonable design of control strategy can reduce the influence of the latter. The definition is as follows In (15), the change of *varDeltavre f* and *varDeltai*<sup>0</sup> have an effect on the bus voltage. It is generally considered that the reference voltage of the system is unique and the fluctuation of the output voltage

> (*s*)*Gc* (*s*)=*0*,

*Zd=R<sup>d</sup>* − *L<sup>d</sup>*

And an improved droop controller block diagram is depicted in Figure 6.

is chosen as shown in Table 1. On the basis of the traditional droop control, the control form is

is determined by the change of the output current. Therefore, reasonable design of control strategy can reduce the influence of the latter. The definition is as follows

$$\mathbf{1} + Z\_d \mathbf{G}\_{\rm Plv}(\mathbf{s}) \mathbf{G}\_{\mathbf{c}}(\mathbf{s}) = \mathbf{0},\tag{16}$$

And *Z<sup>d</sup>* can be rewritten as

$$Z\_d = -L\_d \mathbf{s} \frac{1}{\mathbf{T} \mathbf{s} + \mathbf{1}'} \tag{17}$$

where *L<sup>d</sup>* = 1/*kiv*, and *T* = *kpv*/*kiv*. Formula (18) can be considered as a form of multiplication of a coefficient −*L<sup>d</sup>* differential link with a low-pass filter. According to the actual situation, the value of −*L<sup>d</sup>* is chosen as shown in Table 1. On the basis of the traditional droop control, the control form is expressed as follows

$$\mathbf{Z}\_d = \mathbf{R}\_d - \mathbf{L}\_d \mathbf{s}\_\prime \tag{18}$$

*Appl. Sci.*  And an improved droop controller block diagram is depicted in Figure **2019**, *9*, x FOR PEER REVIEW 6. 8 of 15

**Figure 6.** Improved droop controller. **Figure 6.** Improved droop controller.

As shown in Figure 7, with the proposed control strategy, an equivalent virtual control link is added between the source and the transmission line, which is equivalent to a negative inductance link in the system, helps to offset the influence of the line inductance in the system. As shown in Figure 7, with the proposed control strategy, an equivalent virtual control link is added between the source and the transmission line, which is equivalent to a negative inductance link in the system, helps to offset the influence of the line inductance in the system. *Appl. Sci.* **2019**, *9*, x FOR PEER REVIEW 9 of 15

**Figure 7.** Virtual negative inductive reactance control strategy.

**Figure 7.** Virtual negative inductive reactance control strategy.

The control parameters of system are shown in Table 1. Where the value of droop coefficient The control parameters of system are shown in Table 1. Where the value of droop coefficient depends on the maximum voltage deviation

depends on the maximum voltage deviation *∆v0max* and the rated output current *i<sup>0</sup>* . The maximum allowable voltage deviation is ± 5% of the rated bus voltage. *R<sup>d</sup>* can be derived as: *∆v0max varDeltav*0*max* and the rated output current *i*0. The maximum allowable voltage deviation is ±5% of the rated bus voltage. *R<sup>d</sup>* can be derived as:

=

$$R\_d = \frac{varDelta t a v\_{0\text{max}}}{i\_0} \,\tag{19}$$
