UDE Based Robust Control of Grid Tied Inverters

In this paper, a modified (proportional-integral) PI control is suggested to improve current tracking performance of three-phase grid-tied inverters (GTI). Presence of the L filter between inverter and grid, makes complex to design a controller with proper parameters, due to characteristics of the filter. Clasical PI control depends on an accurate dynamical model, thus its performance is deteriorated by parametric uncertainties, unmodelled dynamics and external disturbances, when operating conditions affect the filter parameters. To solve this problem, uncertainty and disturbance estimator based PI current control approach is proposed for grid-tied inverters, which provides robustness against to parametric perturbations. An UDE based observer that has been adopted into the PI current loop is used to eliminate lumped disturbances and the steady-state tracking error of current states, which can enhance the robustness of the control performance. Then, parameter design method, stability and robustness analysis are explored and presented. Performance comparison among the clasical PI and proposed control scheme. Efficacy and performance of the proposed approach are carried out by simulations and experiments. Experimental results show that effectiveness of the suggested control method against parametric uncertainties and disturbances are succesfully validated. Besides, the precise current tracking performance with zero steady state error has been reached.


Introduction
Grid-tied inverter (GTI) has a very important role in ensuring high quality current injected into the grid, have progressively been adopted in renewable energy systems [1], distributed generation, battery storage systems, and uninterruptible power supplies (UPSs), hybrid electric mobility, smart grids, etc. To obtain pure sine wave, an L grid filter is usually placed as an interface between inverter and power grid to reduce high frequency current harmonics [1]. However, when the grid filter varies, due to temperature, saturation etc., closed loop control performance may be adversely affected and output current is contaminated with harmonics caused by parametric uncertainties. In addition to, the unmodeled dynamics like disturbances and dead-time, can deteriorate the control performance and stability. For the current control of GTI, many strategies have been applied, such as proportional-integral (PI) control [2], proportional-resonant (PR) control [3], and repetitive control (RC) [4], etc. Although these controllers are good to eliminate steady state error, they have lack about robustness.
For robustness concern of the GTI, there have been a number of conventional strategies, such as adaptive control [5], sliding mode control (SMC) [6], disturbance observer based control (DOBC) [7], active disturbance rejection control (ADRC) [8] and uncertainty and disturbance estimator (UDE) based control [9]. Among those methods, UDE based control has become very attractive research point because of ıt gives a new solution for disturbance rejection and also its good reference tracking performance [10].
The main principle of the UDE based control scheme is estimating the lumped terms, including parametric uncertainties and disturbances by using state measurements and a low-pass filter with a appropriate bandwidth. Then, estimated lumped disturbance could be adopted to the control action to reject against to disturbances. UDE based control doesn't require an accurate model of the system and provides the decoupled control design for desired model an filter bandwidth [11]. Due to its superior performance, UDE based control scheme was applied to control of piezoelectric stages [12], wind turbines [13], motor drives [14] and power converters [15]. However, to our best knowledge, UDEbased control of GTI has been represented on few studies and it should be developed [16], [17].
In this paper, a UDE-based control scheme is forming PI controller for GTI with L filter is proposed. By using desired model and low-pass filter parameters, a simple and practical PI control approach converted from UDE based control is composed.

Fig.1 Grid Tied Inverter System
The schematic diagram of the grid-tied inverter system studied for this work is depicted in Fig. 1. The control of grid side currents and dc-link voltage of inverter represents the main purpose of this paper. Following [18], differential equations that describe the nonlinear affine form of the dynamics of dc-link voltage and grid-side inverter currents, in the synchronous rotating reference frame (SRF), can be defined as follows: Where, R and L are grid side filter inductance and resistance respectively and C represents dc-link capacitor. Angular frequency of the grid voltage is denoted as , which is obtained by phase locked loop (PLL) scheme. Grid voltage ( , ), control functions ( , ), and grid side currents ( , ), are all obtained in dq synchronous rotating frame Before designing the proposed UDE-based PI current control strategy, parameter uncertainties of grid currents channels of the dynamic model given in (1)-(3) must be derived as follows Where the parameters R, L and C are nominal values, and represents lumped disturbances and parametric variations

Besed PI Control Law
Dynamic equations of GTI system given in (4)-(5) can be rewritten as the following state space compact form: control input functions, and = [ ] is parametric uncertainty and disturbance vector Assume, desired closed-loop dynamics of GTI system can be defined with reference model as Where, ] is the reference command input vector. is desired state matrix, and is desired control vector. In order to get desired specifications of the closed loop system, and matrix coefficients are selected to meet desired bandwith. Decoupled first order system with grid current bandwith and can be described as Control law ( ) is to ensure that ( ) asymptotically tracks the desired state and, idealy tracking error, i.e., converges to zero. One method to design a control law ( ) is to satisfy the condition for the error dynamics as ̇= By equating (6) and (9)  Then, the control law can be obtained as Based on (6), uncertainty terms and external disturbances , which can be defined as Following the control guidelines in [11], can be approximated by where " * " is the convolution operator and ( ) is the strictly proper low-pass filter which is used to estimate lumped term . By replacing ( ) with ̂( ) , the equations (11) becomes Then by substituting (12) into (14) results in Rearranging control function u(t) in (15) and taking the Laplace transform, the control law in s-domain can be derived in equation (16). Where ( ) is represented as the Laplace transform of ( ). Frequency characteristic of ( ) filter needs to have large enough bandwith, unity steady state gain and zero phase shift over the spectrum of uncertainty term . An accurate estimation of filter bandwidth is difficult due to deadtime of the inverter and control delay. Hence, first order low-pass filter is often preferred as follows Overall control daigram is shown in Figure 2. Substituting (16) into the laplace transform of (6), decoupled structure of the system response can be derived as follows: It is clear that, the parameter design problem can be considered as adjustment of the two bandwidths, which are and . For fixed value of , which is designed according to desired transient specifications, closed loop performance can be adjusted with until to meet desired disturbance attenuation. For larger values of damping , the dominant poles move far from the real axis, which can cause to underdamped response with large overshoot.

Simulation Studies
In order to validate the proposed UDE based PI current control method in Fig 4,     Dynamic response of the proposed control strategy was tested against to reactive power change applied to grid. Figure 6 shows active and reactive power references, besides figure 7 shows injected grid current and voltage.Active power is kept 1kw whereas reactive power is suddenly cahnged from 500 Var to 0. As shown in Figure 7, proposed control strategy can response to the reactive power change with fast and zero steady state error.

Conclusion
This paper proposed a proportional-Integral (PI) control strategy based uncertainty and disturbance estimator (UDE) for three phase GTI systems. With the poposed control method, an improved disturbance attenuation was achieved. Deviations of grid filter parameters and operating conditions make hard to choose control parameter with satisfied performance. This paper presents a simplified parameter selection approach for PI control by using two parameters which are associated with bandwidths of desired closed loop and lumped uncertainty repectively. Also this stategy enable two degree of freedom control. Simulation studies demonstrate that proposed controller performed better performance than conventional PI controller in terms of robustness and dynamic performance

Author's Note
Abstract version of this paper was presented at 9th International Conference on Advanced Technologies (ICAT'20), 10-12 August 2020, Istanbul, Turkey with the title of "UDE Based Robust Control of Grid Tied Inverters".