# The State of the Art of the Control Strategies for Single-Phase Electric Springs
## 3 Control Strategies of ES-2
### \(\delta\) Control  
The \(\delta\) control proposed by [PERSON] et al. [14] is shown in Figure 4. It has a proportional resonant (PR) controller for the external grid voltage loop and a proportional (P) controller for the internal inductor current loop. \(\delta\) control is performed by controlling the instantaneous phase angle of the PR controller to achieve four typical compensations that the ES can support, namely, (1) pure reactive power compensation, (2) specified power factor compensation, (3) constant power compensation, and (4) constant reactive power compensation.  
Figure 3: Radial-chordal decomposition control [12].  
In Figure 4, \(v_{S}\) is the instantaneous CL voltage, and \(v_{S\_fb}\) is the feedback voltage detected from \(v_{S}\), multiplied by the coefficient \(k_{\_{\_{\mathit{eff}}}}\) (e.g., 0.1). The signal \(v_{S\_ref}\) is a predefined sinusoidal reference voltage whose amplitude is given as needed and whose phase is lagging \(v_{G}\) (power generation side voltage) by \(\delta\). The voltage error is used to generate a reference value for the current loop through the PR controller. \(i_{\_{\_{\mathit{L}\_fb}}}\) is obtained by multiplying the current \(i_{\_{\mathit{L}}}\) detected from the inductance by the coefficient \(k_{\_{\mathit{if}}}\) (e.g., 0.1). The current error is used to generate a modulated signal \(v_{\_{\_{\mathit{comp}}}}\) through the P controller and the limiter. The PWM generator generates four drive signals.  
The four typical compensations supported by the ES under the \(\delta\) control are essentially the different phase relationships between the ES voltage and the ES current. A vector diagram for different situations has been depicted by [PERSON] et al. [14] and then steady state algorithm for \(\delta\) calculation is derived. With the \(\delta\) control, the operating mode of the ES can be automatically determined according to a different point of common coupling (PCC) voltages, and the same algorithm can be used for all compensation when the reactance in the NCL changes while the real part remains unchanged. Regardless of the nature of the CL and NCL, ES can operate in capacitive, resistive, and inductive modes. Obviously, the most important part of the control loop is the \(\delta\) control, which directly affects the reference value and the control targets.  
The \(\delta\) control is based on the vector diagram. The circuit parameters that need to be known in advance are impedance of the CL, impedance of the NCL, the resistance and inductance of the distribution line, the effective value of the voltage on the demand side, the effective value of the voltage on the power generation side, and the phase information. It is difficult to adopt \(\delta\) control in actual applications since the power generation side is far away from the demand side. As a result, a communication technology is required between two adjacent ESs, which inevitably leads to an increase in cost. In addition, the mathematical expressions of the parameters \(a\), \(b\), and \(m\) in the \(\delta\) control scheme are different in the case when CL and NCL are pure resistance, resistive-inductive, and inductive-capacitive, which increases the complexity of the controller design. Additionally, as the PR controller only keeps an infinite gain at the resonant frequency, the control accuracy is sensitive to the grid frequency deviations [15].  
Figure 4: \(\delta\) control scheme of ES-2 [14].