# Dead-Beat Control Cooperating with State Observer for Single-Phase Electric Springs
## 1 Introduction  
Electric springs (ESs) have been proposed five years ago as a new solution to fully exploit the unpredictable power generated from intermittent renewable energy sources (RESs) [1]. Compared to the traditional operating way of the power system, they carry out the paradigm that load demand matches power generation automatically [2; 3]. From the topology point of view, an ES is built up by connecting a circuit which behaves like a voltage source inverter (VSI), in series to the non-critical loads (NCLs) of a user with the purpose of keeping constant the magnitude of the voltage across its critical loads (CLs). Consequently, the control approaches for VSI have been initially introduced to drive the ES systems [4; 5]. Subsequently, the control approaches for power converters [6; 7] and microgrids [8; 9] have been adopted. With the increasing power generation from RESs [10], the ESs have gained an increasing interest and many papers have appeared, reporting on system modeling [11], reactive power compensation [2], power decoupling [12; 13], voltage and frequency control [14], power balance control for a three-phase system [15], and so on. Among the compensation functions that the control of an ES can implement, the pure reactive power compensation is the key one because it stabilizes the voltage across the CLs without exchanging (absorbing or delivering) active power. The basic control of an ES implementing such compensation was reported together with the ES concept in Reference [1]. After that, the \(\delta\) control was proposed in Reference [2]; it imposes the instantaneous phase angle \(\delta\) by which the CL voltage lags the line voltage to ensure that ES does not exchange any active power at the steady state. Once calculated theangle \(\delta\), the control scheme uses a proportional resonant (PR) controller for the outer-loop regulation of the CL voltage and a proportional (P) controller for the inner-loop adjustment of the ES current. This setup of the \(\delta\) control exhibits some shortcomings. For instance, its application requires the design and tuning of the three parameters of the PR and P controllers. Another shortcoming is that the PR controller is not suitable for nonlinear systems so that the CL voltage may be severely distorted when the line voltage is affected by harmonics. What is more, system modeling in [2] is not enough accurate since the line voltage is regarded as a disturbance which results in unexpected dynamic performance. Considering the benefits of the \(\delta\) control and side effects of the PR controller, the goal of this paper is to find out a more practical controller that facilitates the application of the \(\delta\) control and, at the same time, improves the performance of the existing setup. The dead-beat control uses the discrete-time model of a system to predict the amplitude of the controlled variable one or more sampling times in advance. By forcing this amplitude to track that one of the reference, the error between the controlled variable and the reference is zeroed [16]. The characteristics of the relevant controller are the low complexity and a high-performance system [17]. Therefore, it is utilized as the control tool that allows the achievement of the goal of this paper. As explained in Reference [2], an ES is a multi-input and multi-output system where only two control objectives can be achieved at the same time. The main objective is the regulation of the CL voltage that is expected to have a sinusoidal shape with a preset magnitude. The other objective cis to implement a specific compensation function by imposing the phase angle \(\delta\). In this paper, the two most significant functions, i.e., the pure reactive power compensation and the power factor correction (PFC), are chosen as objective. The effectiveness of the resulting control system is highlighted by comparing its performance to that obtained with the existing PR and P controllers. In detail, this paper is organized as follows.