# The State of the Art of the Control Strategies for Single-Phase Electric Springs
## 3 Control Strategies of ES-2
### Radial-Chord Decomposition Control Strategy  
From the perspective of power flow, the control schemes above either directly controls the power flow of the converter itself or indirectly controls the power of connected circuits and loads. For example, when applying the _dq_ transformation on the _d_-axis and the _q_-axis current control loops, the ES can directly and independently control the active and reactive powers of the ES, which are hereafter designated as \(P_{ES}\) and \(Q_{ES}\), but it cannot directly control the power flow through the entire SL [9]. Therefore, it is unable to realize the achievement of two control targets completely (such as the CL  
Figure 2: _dq_ decoupling control with current controller [12].  
voltage and the power angle of the SL by a single ES. The radial-chordal decomposition (RCD) control proposed by [PERSON] et al. [12] can directly control the active and reactive powers of the SL, by means of the control block diagram shown in Figure 3.  
The idea behind the RCD control is to decompose the ES voltage into a chord component and a radial component in phase or anti-phase with the NCL voltage. The influence of the two components on the system has been well-studied by [PERSON] et al. [12]. When analyzing the influence of the chord component, it is assumed that (i) there is no radial component, and (ii) the magnitude of the grid voltage is always equal to the nominal reference value (e.g., 220 V). By the chord component control, the NCL voltage and the grid voltage are always equal in magnitude. When analyzing the influence of the radial component, it is assumed that (i) there is no chord component, and (ii) the radial component is in phase or in anti-phase with the NCL voltage. It is concluded that the chord component is related to the power angle of the SL, whilst the radial component is related to the apparent power of the SL, both of them being completely decoupled in mathematical relationship, thereby, achieving simultaneous control of the two targets. In the control process, when the grid voltage fluctuates, the apparent power of the SL is indirectly controlled by adjusting the radial component to realize power balance between the power generation and the load demand. When the grid voltage rises and the CL voltage is higher than the reference value, the total apparent power tends to decrease; otherwise it increases.  
RCD control requires the a priori knowledge of the internal parameters of the SL, such as the impedance angle of the NCL. This facilitates the embedding of the RCD controller in a separate device, such as a heater and a refrigerator. Furthermore, as shown in Figure 3, an advantage of the RCD strategy is that the grid voltage regulation can be performed using only one PI control loop, while the power factor can be controlled in an open-loop manner. This is not possible by the \(dq\) transformation method shown in Figure 2. However, the RCD control strategy has its shortcomings. For instance, the algorithm is more complicated, and the control of ES-2 is difficult under a pure reactive power compensation mode.