# The State of the Art of the Control Strategies for Single-Phase Electric Springs
## 5 Control Strategy for Distributed ESs  
The various control methods currently available can explain that a single ES plays a role in stabilizing the CL voltage and improving power quality, but the ES design was originally intended to support CL voltage at various installation points of the grid. As shown in Figure 8, the microgrid contains many ESs in a distributed manner. The limited range of NCL voltages will limit the operating range of a single ES. If the control of a large number of ESs can be coordinated, these ESs can provide effective support for the microgrid. In order to implement the plug-and-play functionality of ESs, distributed control strategies are necessary so that they can support the voltage at their location while taking voltage variations along the distribution line into account. In response to this situation, [PERSON] et al. [PERSON] et al. (2018) introduced the idea of a droop control in the ES motor and tested it on a distributed line using three 1 KVA ESs, demonstrating that distributed ESs can be coordinated through a droop control to minimize the total reactive power injection.  
Figure 8: Schematic of microgrid with distributed ESs.  
Figure 7: Power decoupling control scheme with BMS ([PERSON] et al., 2016).  
The advantage of a droop control is that only local information is needed, and the mathematical expression of the Q-V droop control of the generator is provided as follows.  
\[V=V^{*}-nQ \tag{10}\]  
where \(V\), \(V^{*}\), \(n\) and \(Q\) are the output voltage, the standard voltage, the droop coefficient, and the output reactive power, respectively.  
In order to realize the droop control of the SL, an improved droop control has been proposed by [PERSON] et al. [26], which can be derived from Equations (11) and (12), as follows:  
\[V_{S\_ref}^{*}=V_{S\_ref}-KV_{ES} \tag{11}\]  
\[V_{ES}=(K_{P}+\int K_{I})(V_{S\_ref}-V_{S}) \tag{12}\]  
where \(K_{P}\), \(K_{I}\) and \(K\) are proportional gain, integral gain, and the droop coefficient, respectively. The droop coefficient \(K\) is selected based on the maximum reactive power of the ES. Figure 9 shows the droop curve of Equation (11), which has been adopted from [PERSON] et al. [27]. As the ES voltage changes, the reference value of the CL voltage will move around the standard voltage point.  
Figure 10 is a complete control block diagram of the droop control, which has also been adopted from [PERSON] et al. [27]. The modified reference voltage \(V_{S\_ref}^{*}\) is used as a new voltage set-point. The SL is then operated to adjust the CL voltage \(V_{S}\), at the voltage set point of the offset.  
Figure 10: The droop control of ES [27].  
Figure 9: Droop curve of the ES reference voltage [27].  
Since the PI controller has no steady-state error for the DC signals, for all ESs under steady-state conditions,  
\[V_{S\_ref}^{*}=V_{S} \tag{13}\]  
substituting Equation (13) into Equation (11), yields,  
\[V_{S\_ref}-V_{S}=KV_{ES} \tag{14}\]  
It is revealed by Equation (14) that, at the steady state, the CL voltage error is a linear function of the ES voltage. Excessive compensation or insufficient compensation of \(V_{ES}\) can cause local CL voltage errors. When an ES has a positive voltage error and nearby ESs have a negative voltage error, they might operate in the opposite way to regulate the CL voltage. For instance, one SL operates at an inductive mode (reactive power consumption) and the other SL operates at a capacitive mode (reactive power injection). At the same time, a high droop gain \(K\) is usually required to ensure a reactive power sharing [28]. However, as shown by Equation (14), a high value of \(KV_{ES}\) will result in a large CL voltage error. In order to eliminate the steady-state error, \(K\) is limited to a small range, which in turn affects the sharing of reactive power between the ESs.