*2.2. Control Strategy of the System*

Although the conventional CSF has many advantages for controlling frequency and SOC, it cannot be adopted in the system with multiple diesel generators and BESSs, similarly to isochronous mode in a conventional power system. For applying the concept of previous CSF to multiple generators and BESSs, we sugges<sup>t</sup> a new CSF method for parallel operation. Table 1 shows the comparison between conventional and proposed CSF methods.

**Table 1.** Comparison between conventional and proposed CSF methods.


The proposed control strategy is summarized as follows. (1) RESs are operated by the maximum power point tracking (MPPT) algorithm for maximizing the use of RESs. (2) Diesel generators are operated to manage the energy stored in BESSs by utilizing the proposed hierarchical control scheme. (3) BESSs operate in a hierarchical manner to control the grid frequency. (4) The SOCs of individual BESSs are managed by the proposed self SOC controller (SSC). Figure 3 shows the entire control strategy of the proposed method applied to the Geocha Island microgrid system. The frequency can be regulated from the frequency controllers of BESSs. While the frequency is regulated by BESSs, the individual SOCs of BESSs deviate from their reference values. For restoring the SOCs of all BESSs, the equivalent SOC, *SOCeq*, which represents the SOCs of all BESSs, is controlled by diesel generators. Because only the total energy of all BESSs is considered in diesel generators, an additional controller is adopted in BESSs for restoring the individual SOCs. The detailed scheme of the proposed controllers is presented in the next sections.

**Figure 3.** Proposed control strategy for the Geocha Island microgrid.

### **3. Proposed Control Strategy of Diesel Generators**

In the previous CSF method of [7], a diesel generator regulates the SOC of a single BESS. However, there is no target variable for multiple BESSs because each BESS has a different SOC and capacity. To manage the SOCs of all BESSs, we propose the concept of *SOCeq*, which can be defined from the definition of SOC [13] as follows:

$$\text{SOC}\_{\text{eq}} = \frac{\text{Current Energy}}{\text{Rated Energy Capacity}} = \frac{\sum\_{k=1}^{n} E(k)}{\sum\_{k=1}^{n} E\_{\text{rate}}(k)} = \frac{\sum\_{k=1}^{n} \text{C}\_{\text{rate}}(k) V\_{\text{dc}}(k) \text{SOC}(k)}{\sum\_{k=1}^{n} \text{C}\_{\text{rate}}(k) V\_{\text{dc,rate}}(k)},\tag{1}$$

where *n* is the total number of BESSs, *E*(*k*) is the stored energy (Wh), *Erate*(*k*) is the rated energy (Wh), *Crate*(*k*) is the rated capacity (Ah), *Vdc*(*k*) is the dc voltage of the battery (V), *Vdc,rate*(*k*) is the rated dc voltage of the battery (V), and *SOC*(*k*) is the SOC of the *k*-th BESS. By regulating *SOCeq* with respect to its reference value, *SOC\*eq*, the energy stored in all BESSs can be maintained, and energy imbalance is regulated by diesel generators in the long time scale. Similar to the *SOCeq* in (1), *SOC\*eq* can be derived from the reference SOC values of individual BESSs.

### *3.1. SOCeq Control Scheme for Single Diesel Generator*

Firstly, we verify that *SOCeq* can be regulated by a single diesel generator in an isolated microgrid with multiple BESSs. By differentiating (1), we obtain

$$\mathbb{E}\left(\sum\_{k=1}^{n}\mathbb{C}\_{\text{rate}}(k)V\_{dc}(k)\right)\frac{d(SOC\_{\text{eq}})}{dt} = \sum\_{k=1}^{n}\mathbb{C}\_{\text{rate}}(k)V\_{dc}(k)\frac{d(SOC(k))}{dt}.\tag{2}$$

From [13], the SOC of the *k*-th BESS can be expressed as follows:

$$\text{SOC}(k) = \text{SOC}^0(k) - \int \frac{P\_{\text{BESS}}(k)}{V\_{dc}(k)\mathbb{C}\_{\text{rate}}(k)}dt,\tag{3}$$

where *SOC*0(*k*) is the initial value of SOC and *PBESS*(*k*) is the active power of the *k*-th BESS. By taking the derivative of (3), we obtain

$$\frac{dSOC(k)}{dt} = -\frac{P\_{BESS}(k)}{C\_{\text{rate}}(k)V\_{dc}(k)}.\tag{4}$$

By substituting (4) into (2) and under the assumption that *Vdc*(*k*) is almost constant and equal to the rated value within the normal SOC region [7], the derivative of *SOCeq* can be expressed as follows:

$$\frac{d(SOC\_{\text{eq}})}{dt} = -\frac{(P\_{BESS}(1) + \dots + P\_{BESS}(n))}{\left(\sum\_{k=1}^{n} \mathbb{C}\_{\text{rate}}(k) V\_{dc, \text{rate}}(k)\right)}.\tag{5}$$

To satisfy the power-balance equation of an isolated microgrid, the summation of the total active power outputs of the BESSs and diesel generator should be equal to the active power of net load.

$$P\_{BESS}(1) + \dots + P\_{BESS}(n) + P\_d = P\_{net, \text{load}} \tag{6}$$

where *Pd* is the active power output of the diesel generator and *Pnet,load* is the net load including the uncontrollable outputs of RESs, loads, and system losses.

For a single diesel generator, the isochronous control mode can be adopted for frequency regulation in a conventional power system [14,15]. Likewise, in the proposed method, a single diesel generator operates in the isochronous mode for *SOCeq* control. Figure 4 represents the proposed *SOCeq* control structure for a single diesel generator, including a corresponding plant model from (5) and (6). Since the plant model between the diesel output and *SOCeq* is a first-order system as shown in Figure 4, a proportional integral (PI) controller can be adopted for *SOCeq* control [16].

**Figure 4.** Plant model of the isochronous mode of the diesel generator.

### *3.2. SOCeq Control Scheme for Multiple Diesel Generators*

In a power system with multiple synchronous generators, a hierarchical frequency-control structure is utilized to prevent hunting effects on the frequency and inaccurate power sharing between generators at the steady-state [15]. Similarly, we propose a hierarchical control structure for multiple diesel generators to regulate *SOCeq*. Owing to the proposed hierarchical control, *SOCeq* can be regulated

stably, and accurate power sharing between diesel generators at the steady-state is possible. Figure 5 shows the concept of the proposed hierarchical control structure for multiple diesel generators.

**Figure 5.** Hierarchical *SOCeq* control of diesel generators: (**a**) droop; (**b**) supplementary control.

Firstly, we introduce the *SOCeq*–*<sup>P</sup>* droop-control strategy for primary responses of *SOCeq* regulation and power sharing between diesel generators. As shown in Figure 5a, when an unexpected system variation occurs, *SOCeq* deviates from the reference value, *SOC\*eq*. Based on the droop characteristic, the active power outputs of diesel generators increase (or decrease) from the reference value of the active power output of the *i*-th diesel generator, *P\*d*(*i*). Eventually, *SOCeq* can be saturated at the point where the active power is balanced (red dot in Figure 5), and the BESSs make zero active power. The droop coefficient of the *i*-th diesel generator can be defined based on the slope in Figure 5a and represented by *Rd*(*i*) (*i* = 1, 2, and 3 for the target network).

Owing to the innate characteristics of the droop controller, steady-state error exists between the saturated value and the target reference value, <sup>Δ</sup>*SOCeq*. To restore *SOCeq* to its corresponding reference *SOC\*eq*, supplementary control for secondary responses is provided as shown in Figure 5b. By integrating the concepts of the droop controller in Figure 5a and the supplementary controller in Figure 5b, we develop the proposed hierarchical control structure including the plant model between the active power of diesel generators and *SOCeq*, as shown in Figure 6, where *Pf*(*i*) is the participation factor, *Tv*(*i*) is the time constant of the valve actuator, and *Td*(*i*) is the time constant of the diesel engine for the *i*-th diesel generator. The supplementary controllers are implemented with a PI controller for eliminating the steady-state error and participation factor for determining the sharing ratio. Similar to the conventional load frequency control structure, the supplementary controller must be operated to respond slower than the droop controllers [15] in diesel generators.

**Figure 6.** Plant model for the hierarchical control scheme of multiple diesel generators.

### **4. Control Strategy of Battery Energy Storage Systems (BESSs)**

### *4.1. Frequency Control Scheme for Multiple BESSs*

When a single BESS controls the frequency of the system, a converter is operated in the grid-forming mode to control the frequency directly. The detailed scheme of the grid-forming mode is shown in [16,17]. To regulate the grid frequency when using multiple BESSs with grid-forming converters, hierarchical control for frequency regulation can be utilized [18,19]. For the primary response, the *<sup>P</sup>*-*f* droop control method is adopted for the parallel operation of BESSs, as shown in Figure 7a. Additionally, for secondary response, as shown in Figure 7b, a supplementary controller including an integrator is exploited to eliminate the steady-state error of frequency generated from droop control.

**Figure 7.** Hierarchical frequency control of BESSs. (**a**) Droop and (**b**) supplementary control.

In Figure 7, *f* 0 is the nominal frequency, *P*\**BESS*(*k*) is the reference active power, and *Rb*(*k*) is the droop constant of *k*-th BESS.

### *4.2. Self State of Charge (SOC) Controller for Individual BESSs*

Frequency can be regulated almost perfectly by utilizing BESSs, and *SOCeq* can be maintained at the desired reference value with diesel generators. However, the managemen<sup>t</sup> of individual SOCs cannot be guaranteed by exploiting the controllers presented in the previous sections, because only the total energy of BESSs is considered in diesel generators. In addition, from the integral of supplementary frequency controllers, the power sharing between BESSs is not guaranteed from the desired value [18,19]. Therefore, the individual SOCs of BESSs should be controlled at their reference values by themselves with an additional controller. To regulate the individual SOCs of BESSs at the reference values, we develop SSC for restoring the individual SOCs. Since the BESSs are operated by grid-forming converters, which control the frequency of their terminals, BESSs should regulate their SOCs by adjusting their terminal frequencies. To validate the necessity of SSC, a simplified circuit for BESSs, as shown in Figure 8, is first investigated.

$$\boxed{\begin{array}{c} \text{kth} \\ \text{BESS} \end{array}} \xrightarrow{\begin{subarray}{c} \|V(k)\| \angle \ \Theta(k) \end{array}} \xrightarrow{P\_{\text{BESS}\{k\}}} \underbrace{\begin{subarray}{c} P\_{\text{BESS}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \end{subarray}} \begin{split} \text{BESS} \end{split} \xrightarrow{\begin{subarray}{c} \|V\|\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \mid \angle \ \Theta\_{\text{bus}}(k) \end{split}$$

**Figure 8.** Simplified single line diagram between a terminal node and the main bus of the *k*-th BESS.

In Figure 8, *X*(*k*) is the reactance component of the transformer, ∠θ(*k*) and |*V*(*k*)| are the phase angle and magnitude of ac voltage at the terminal node of the *k*-th BESS, and <sup>∠</sup>θ*bus*(*k*) and |*Vbus*(*k*)| are the phase angle and magnitude of ac voltage at the bus connected to the *k*-th BESS. As reactance is much larger than the resistance in the transformer [20], the transformer is modeled by a single reactance. From Figure 8, the active power output of the *k*-th BESS can be approximated as follows [21]:

$$P\_{BESS}(k) = \frac{\left| V(k) \right| \left| V\_{bus}(k) \right| \left( \angle \theta(k) - \angle \theta\_{bus}(k) \right)}{X(k)}. \tag{7}$$

From (4) and (7), the SOC of the *k*-th BESS can be expressed as follows:

$$\frac{dSOC(k)}{dt} = -\frac{1}{C\_{\rm rdte}(k)V\_{dc,\rm rtc}(k)} \frac{\left| V(k) \right| \left| V\_{bus}(k) \right| \left( \angle \theta(k) - \angle \theta\_{bus}(k) \right)}{X(k)},\tag{8}$$

with the assumption that the entire system except the *k*-th BESS is constant, the derivative of the phase angle for bus *k* is zero. By differentiating (8), we can obtain:

$$\frac{d^2SOC(k)}{dt^2} = -\frac{2\pi \|V(k)\| \|V\_{\text{bus}}(k)\|}{C\_{\text{rate}}(k)V\_{\text{dc,rate}}(k)X(k)} f(k),\tag{9}$$

where *f*(*k*) is the frequency output of the *k*-th BESS. As the plant model between the frequency and SOC of the *k*-th BESS is a second-order system, each BESS can regulate its SOC by itself with a PI controller [16].

The purpose of SSC is to restore the SOC of the *k*-th BESS to its corresponding reference value, *SOC\**(*k*). However, as diesel generators respond slowly compared to BESSs, the transient difference between *SOCeq* and its reference value, *SOC\*eq*, should be considered in SSC. Therefore, in SSC, the reference value for the SOC of the *k*-th BESS must be modified as follows:

$$\text{SOC}\_{nf}(k) = \text{SOC}^\*(k) + \text{SOC}\_{eq} - \text{SOC}\_{eq}^\*. \tag{10}$$

where *SOCref*(*k*) is the adjusted reference value of *SOC*(*k*) considering the transient state. Note that *SOCref*(*k*) and *SOC\**(*k*) eventually become equal in the steady state because *SOCeq* is regulated to *SOC\*eq* by diesel generators in the long time scale. Figure 9 shows the total control structure of the *k*-th BESS including inner control loops [14,17], the hierarchical frequency controller, and SSC.

**Figure 9.** Proposed controller of BESS for frequency and self state of charge (SOC) recovery.

In Figure 9, *I*(*k*) is the line current from the *k*th BESS, *V*(*k*) is the terminal voltage, *Iout*(*k*) is the line current to the terminal node, *L*(*k*) and *Cf*(*k*) are the filter components, θ*\**(*k*) is the reference phase angle for the modified reference frequency value, and *u*(*k*) is the modulating signal of the *k*-th BESS. Superscripts *d* and *q* indicate the *dq* components of corresponding variables.

The reference frequency value and θ*\**(*k*) are determined from the hierarchical frequency controller and SSC. The *dq* components of the reference voltage are derived from the reference value of voltage magnitude, *Vd\**(*k*), and θ*\**(*k*). Finally, the terminal voltage of the *k*-th BESS is regulated to the reference value via a nested voltage and current control loop, as shown in Figure 9. Through the entire control loop, the frequency and SOC of each BESS are regulated.

### *4.3. Maintaining Desired Active Power Outputs of BESSs by the Linear Time-Varying SOC Control*

By regulating SOCs as the linear time varying value, the BESSs can be controlled as the desired level of active power outputs [7]. In other words, while BESSs control frequency, they can maintain the scheduled or dispatched active power. From integrating (4), to make the active power of *k*-th BESS as the reference value *P\*BESS*(*k*), SOC reference value can be determined as:

$$SOC\*(k) = \int -\frac{P\*\_{BESS}(k)}{\mathbb{C}\_{nte}(k)V\_{dc}(k)}dt,\tag{11}$$

As *P*\**BESS*(*k*) is desired constant value, SOC reference has the linear time varying value from (11). Through controlling SOCs as the linear time varying reference values as (11), BESSs can maintain the desired active power outputs.
