Sensorless Robust Flatness-Based Control with Nonlinear Observer for Non-Ideal Parallel DC-AC Inverters

Parallel operation of multiple AC/DC inverters is favorable in hybrid AC/DC microgrids to avert consecutive conversion stages and increase the system’s efficacy. Yet, several emerging technical challenges impede the expansion of such layout such as the circulating currents, sensitiveness to the input voltage disturbances, load variations and complexity of the control structures caused by using several required measurements. This paper proposes a one-loop sensorless controller which is based on the flatness technique for a non-isolated power supply consisting of n-parallel inverters. The proposed control scheme primarily relies on employing a nonlinear online observer to estimate the line inductor currents and the dc link voltage via information from the input voltage, output voltage, and load conditions to avoid using excessive sensors. In this way, the system reliability is improved by reducing burdens of the communication delays and/or the failures, signal noise, thus the system is featured by simple control. Besides, the system entire losses are modeled by equivalent voltage sources and one current source which implicitly represent all types of the losses by using an online nonlinear estimator for the control purposes. The proposed controller not only has high dynamic performance, wide-bandwidth, low voltage THD but also robust to the abrupt variations in the load and the input voltage. To validate the applicability of the proposed control method and the observer, both simulations and experimental investigations are performed for two paralleled three-phase inverters setup. The obtained results assure the effectiveness of the proposed control method in regulating the output voltage of the parallel DC/AC inverters with fewer number of the sensors against the fluctuations of the input dc voltage and the load perturbations.


INDEX TERMS
T HE recent advance in DC/AC inverters are gaining remarkable attraction in various fields such as the hybrid microgrids (MGs) [1], electric transportation systems, and machine drive applications, owing to their high energyconversion efficiency that allow it transfer bulk power [2]- [6]. The control structures which are dedicated for parallel DC/AC inverters entail robust performance against variations in the power supply, load, system parameters. Meanwhile, they ensure adequate stability under conceivable operating conditions [7]- [11]. Modern control theories like the FBC necessitates more sensors than the linear controllers such as PI compensators [12], [13]. However, employing many sensors results in increasing the system cost, weight, volume, and complexity of the system. Thus, worsening the reliability, increasing the installation difficulty, increasing the need for proper wiring, and high sensor-breakdown incidence [14], [15]. The sensorless-based control architectures are implemented in diverse power electronic converters such as the boost, buck, three-phase inverters, and full-bridge DC/DC or DC/AC converters [16]- [19]. Accordingly, various strategies are proposed to dispense multiple sensors in the hybrid MGs, power converters, electric machines, and electric transportation. Instead, the observers are used to estimate the state variables and build an accurate model to render the system as sensorless-based control structure [20]- [25]. The sensorless controller which is based on a state observer is employed in [26], [27] for a three-phase inverter by using Lyapunov method. In [25], a current observer is presented to replace the current sensor and estimate the inductor current despite disturbances and uncertainties. This observer can estimate the transformer current in a full-bridge isolated converter. This method is based on the transient dc bias characterization and the load current feedforward. However, these approaches rely on complicated computations which need a high speed controller with high processing capability. Therefore, emerging delays and complicated control system leading to inadvertent performance issues and thus, the DC link voltage transient oscillations can't be completely mitigated using this method.
In [26], a Lyapunov-based control is also used to eliminate the observer error under the load changes and uncertainty conditions. Where many computational vectors are handled every sampling step for the real time control and high capability of a digital processor is needed.
In [28], the flatness theory is employed to ascertain efficient performance of the FL-VSIs under different loading conditions. The application of the flatness control enhances the PMSM drive performance as a model-based estimation [29]. The flatness controllers ensure that, the state variables can accurately follow a prescribed behavior either during normal or transient conditions for the AC/DC systems. Besides, the flatness technique is robust against the changes in the system parameters and the load conditions, thus the flatness control allows high communication among the units [30].
Therefore, the flatness-based control approach is preferable for the nonlinear controllers which are devoted to the control of the parallel DC/AC inverters [31]. For example, in [30], [32] the flatness control is implemented to the paralleled VSIs to achieve reliable operation during the normal as well as abnormal scenarios due to sudden loss of any inverter. On the other hand, the flatness techniques planning features which are based on ESSs are investigated in [33]- [36]. The choice of one loop control structure which is based on the flatness guarantees higher bandwidth controller, low noise, lower output voltage THD and better damped transient performance [30], [32]- [36]. Moreover, the inher-ent system enables estimating the state variables behavior without integration, by the instant conduct of the controlled flat output. The FBC for parallel DC/AC inverters is realized in a master-slave manner where, the first inverter performs as a master dictating the output voltage whereas, the other inverters behave as slave, where, either output currents or load power is adjusted between the parallel inverters. The control schemes which are based on master-slave are extensively employed to share the load power, suppress circulating currents and enhancing the system reliability [37], [38]. This paper suggests a one-loop FBC for a non-isolated power supply composed of n-parallel DC/AC inverters. Additionally, an online observer is augmented to estimate the inductor line currents and the dc link voltage via the information from the input voltage, output voltage, and load conditions. In this manner, the number of the sensors that are used in the FBC is minimized to overcome the complexity of the control structure, signal noise and potential communication failures and/or delays. Table 1 compares the FBC to other typical control structures which are usually dedicated for DC/DC and/or DC/AC converters. The comparison shows that the FBC demonstrates high dynamic performance compared to the other existing control techniques. Moreover, the FBC provides a good tracking performance, robustness to the parameters variation, high capability for disturbance rejection, and easy for the implementation in the real time [39]. Furthermore, due to its superior properties, the FBC can be integrated with conventional PI controllers. A loopshaping problem is formulated where a combination of FBC with PI regulator is incorporated with the control system [40]. With the FBC, the measured trajectory is planned to follow a predefined reference trajectory with a high grade of accuracy and reliability. Accordingly, owing to its high dynamic performance, the FBC responds quickly in case of abrupt load changes which makes it also preferable to assist in the fault-tolerant control design of the power electronic converters, such as the open-switch faults [32].
Major contributions of this paper are: (1) Using FBC scheme to not only regulate output voltage of the parallel inverters but also increase system robustness against abrupt variations of supply voltage and load. The FBC properties would enhance reliability of the parallel inverters operation in any conceivable application such as that in hybrid AC/DC MGs. (2) A nonlinear observer is suggested to minimize the number of required sensors in the entire system in order to simplify the control scheme and counteract controller sensitivity to noise of measured signals and/or failures. (3) Proposed losses estimation approach for the DC/AC inverters is augmented to estimate the entire system losses by equivalent voltage sources and one current source..

II. SYSTEM STRUCTURE AND ANALYSIS
The layout of a typical n-parallel three-phase DC/AC inverters is illustrated in Fig. 1. To estimate the losses of the parallel inverters system, three serial equivalent voltage sources V tabcn and current source I P at the input dc-link are employed. During the balanced conditions, sum of the load side currents is zero, as in (1). Likewise, sum of the inverter output line currents is zero at the PCC, as given by (2).
Consequently, sum of i cabcs is zero and sum of V cabcs is constant. Then, the system has 3n + 5 variables and two special linked relations. The derivative of the voltages V cdq at the PCC with V co = 0 are defined as: Fig. 1, and as implied from (2), sum of the homopolar currents at the PCC is zero i.e., n k=1 i 0k = 0. Thus, the current state variables are of the order n − 1.
Arbitrarily, the homopolar current of the first inverter (the master one), can be found as i 01 = − n k=1 i 0k (any another inverter can be behave as master). Consequently, the homopolar currents of the remaining (n−1) inverters become independent variables. Then, the master inverter output line current dq components can be formulated as: The inductive currents of any k th module, with k ∈ [2, .., n] can be written as: Voltages V t0k , V tdk , V q0k for k ∈ 1, .., n represent the voltages drops through the system which result from the losses. In this model, the parameters V tdqk , indirectly represent the losses through the entire system, and as these parameters vary slowly, their time-derivatives are neglected. The dynamics of the DC link can be described as: Inverter #2 Inverter #n Power supply with input filter Circulating current

III. OBSERVABILITY AND ESTIMATOR A. PARAMETER OBSERVABILITY OF NONLINEAR SYSTEM
There are several ways to prove the observability of a system where observability conditions and methods are discussed in [41]- [47]. The observability of the estimated parameters V tdq and I p must be verified before the derivatives of the estimator. By proving the observability of the system, the estimated variables exponentially converge to the real ones with the proposed state observer. The proposed state observer is inspired from the subclass of the state observers which is designated in [48], [49]. The considered state vector X = [x, p] T where p represents the estimated parameters as detailed later, is given as follows: The measured state variables y (in our case y=x) is given by: The observability vector θ of such system can be considered as following: According to the criterion in [48], [49] rank(Jacob(θ)) = dim(X) = 13 (12) For the proposed system, there are three currents for each inverter and two voltage values for the ac bus. In addition, one current source I p , and two voltages V tdq to be estimated, besides the input variables V dc and I dc . For the given system, two inverters in parallel are considered (n = 2), it can be easily proved that rank (Jacob(θ)) = 13. Thus, in the considered case, the system is observable.

B. DEFINITION OF PROPOSED STATE-OBSERVER
In this section, a state observer is proposed. This observer is applied to a specific subclass of nonlinear systems which is based on nonlinear FBC. The state variables are required to estimate the system parameters to obtain the differential flatness property. The losses through the whole system are estimated and modelled which are considered as a disturbance which is compared to the ideal system. The estimated parameters are linked to the state variables of the model. Then, an online state observer (or parameter estimator) is essential to enhance the system performance.
Several observation techniques exist in the literature, contrary to the Extended Kalman Filter (EKF) or Luenberger observers, a thorough comparison bethween the EKF and Luenberger observers is elaborated in [48]. The proposed observer is well-adapted to the nonlinear systems to estimate the line currents of each inverter and to estimate the unknown parameters V tdqk and I p . The parallel inverters model can be represented by (13) where x represents the state vector (all variables are supposed to be measured), p is the vector of the unknown parameters whose dynamics are too slow. So, they are considered as constant in the steady state. The differential system which is used to estimate these parameters can be firstly described as: where the two functions G, H are nonlinear and u is the input vector. The estimated state variables, parameters, and their dynamics are given by: for n = 2 and identical filter line inductances, the system becomes: wherex ∈ R 2n+4 andp ∈ R 2n+1 . S ∈ R (2n+4)(2n+4) a positive definite matrix. The estimator dynamic parameters are given by: Determining of parameters k p and k i depends on inverse of the matrix H(x, u), where H is not square. Parameter k p (consequently k i ) is calculated by considering a pseudoinverse of H(x, u).
The following error variables are introduced: whereṗ = 0. The proposed observer is expressed by:

C. ESTIMATION STABILITY VERIFICATION
To validate the stability of the suggested estimator, a candidate Lyapunov function is selected. The exponential stability can be readily verified via classical Lyapunov method. Accordingly, the proposed estimator stability and dynamics are dictated by coefficients k i and H as given by (19) and (20). Hence, to explore their stability, the candidate Lyapunov function is expressed by: This function is globally positive-definite over the whole state space. The time-derivative of (21) is defined as: Assumingx = x, so, the system dynamics is slow. By substitute from (19) and (20) into (22), the derivative of V become as following: The exponential stability of the estimator can be guaranteed since S is a positive-definite matrix. The tuning of the S matrix is determined to ensure that, the dynamics of ϵ x quite faster than that of ϵ p . Then, by employing the value of kp.H(x, u) = −S and k i = k p .S, it results in: The candidate Lyapunov function demonstrated that the estimation errors exponentially converge to zero, then, the system is exponentially stable.

A. IMPLEMENTATION OF CONTROLLER BASED ON FLATNESS THEORY
Differential flat systems are a specific class of the control systems. Under this control system, the structure of the desired trajectories and their dynamics can be completely characterized, Martin et. al [31]. Accordingly, all state and input variables can be expressed in terms of a set of particular variables (i.e., the flat outputs) and their derivatives. More specifically, a system with a state vector x ∈ R n , and input vector u ∈ R m , is termed differentially flat if x and u can be found as a function of the chosen flat output w ∈ R m (the number of input variables equals the number of the flat outputs) as the following form: where the rank of (φ) = n, (ψ) = m and (ϕ) = m. Thanks to the FBC approach, the prediction of the state variables behavior in both steady-state and transient conditions is feasible. The FBC can planify the trajectory of the flat output and its derivatives [30].

B. APPLICATION OF FBC
Finding a candidate flat output as well as input and state vectors as a function of the candidate flat output without solving the differential equations system is challenging for optimal FBC design.
To realize the control objectives of the proposed parallel inverters system, the energy of the capacitive ac bus and the errors of the circulating currents are picked as the candidate flat outputs. According to the control objectives, the candidate flat outputs vector is selected as the electrostatic energy w c which is stored in the output ac filtering capacitance and the error of the circulating currents w zk between the parallel inverters. The flat output vector becomes w = [w c , w zk ] T , where w c = [w d , w q ] T . The control of w c to its respective reference w c−ref ascertains voltage constancy of the ac capacitive bus at the PCC, where w c is defined by (26). The voltage vector [V cd , V cq ] T of the ac capacitance can be rewritten as given by (27).
The flat output components w zk = [w z2 , . . . , w zn ] T represent the errors of the circulating currents. Each component of w zk allows reducing the circulating currents between the first/master inverter and the k th inverter, with w zk ∈ R 3 ∀ k ∈ {2, .., n}.
Controlling w zk to accurately follow its respective reference w zk−ref ensures not only, that the circulating currents between the parallel inverters are minimized but also, equal distribution of the demanded power among the parallel units. For this purpose, the homopolar currents of (n − 1) modules are controlled to cancel the zero-sequence currentcomponent of the first/master inverter homopolar current (i 01 ).
Derivatives of the voltage vector [V cd , V cq ] T and line currents [i d1 , i q1 ] T are given in terms of w c , w zk and their respective derivatives by (26), (27) and (28) as following: The line current i 0dqk : k ∈ {2, ..., n} of the remnant k th inverters which is given by using (28) is: The current vector vector is expressed as a function of w c = [w d , w q ] T which is given by (26) and derivatives of the components of the voltage vector [V cd , V cq ] T which are given by (29). Derivatives of dq current components which is given by (30) becomes: For the other (n − 1) th inverters, derivatives of the inverter output line currents are reformed by (33). Based on the above model, the input/control vector of the master inverter is given by (34).
The input/control vector of the (n − 1) th inverters can be rewritten as following: The references of the stored energy are determined as following (the corresponding reference of the current-error is controlled to zero, i.e., w z−ref = 0): Eventually, the input vector u = [V d1 , V q1 , . . . V 0k , V dk , V qk ] T is to be formulated in terms of w c−ref , w z−ref and their corresponding derivatives. Hence, relations (34) and (35), can be written as following: As implied from (26)- (38) and according to flat system formula (25), the system states and input vector verify the flatness conditions and they are found by the candidate flat output w.

C. LINEARIZATION OF CONTROL LAW
When a nonlinear system possesses the flatness property, it can be converted into a linear controllable system by feedback linearization. After linearization, stabilization, the controller becomes easier, robust to parameters variations [30]. To assure that, the trajectories w c , w zk perfectly follow their set values, i.e., w c−ref , w zk−ref , traditional inputoutput linearization is employed. Such technique introduces fictitious control variables β c = [β d , β q ] T for output energy vector w c and β z = [β z2 , . . . , β zn ] T for the current error vector w z . For the energy output vector, derivatives of the energy components are set to: These controllers are obtained via (40) and (41), which permit asymptotic convergence to their respective reference trajectories.
As inferred from the control polynomial in (40) and (41) for the flat output, the constraints of the control system are then expressed as equalities. The derivative term of the control law is employed to avert introducing discontinuities. Likewise, an integral part is augmented to realize a zero steady-error and counteract other modeling mismatches. In this way, stable trajectory tracking with prescribed tracking error dynamics is guaranteed.
Fictitious control variable β zk = [β z0k , β zdk , β zqk ] T for current vector error can be similarly identified as used with energy vector: The second-order law is augmented with the components w zk to assure that w zk = [w zok , w zdk , w zqk ] T perfectly follows its planned reference w zk−ref . So, the current error control law is given as following: To express the dynamics of the flat output vector, a variable δ is introduced to simplify the model which is given by (40), (41), and (43).
p wc (s) = (s + p 1 ) s 2 + 2ζ wc ω wc s + ω 2 wc p wz (s) = s 2 + 2ζ wz ω wz s + ω 2 wz (46) By matching (44), (45) and (46), to obtain the rewarded specified root locations, the operational behavoir of the proposed control method is set stable for: k 11 , k 12 , k 13 , k 21 , k 22 > 0. Accordingly, the system dynamics will be influenced by these coefficients which are associated with control laws which listed in Table 2. The input vector u = [V di , V qi , . . . V 0k , V dk , V qk ] T which is described by (34) to (35) can be expressed in terms of the candidate flat output components and its derivatives following the I/O linearization technique. Then, u can be formulated by (47) and (48) in terms of w c−ref , w zk−ref , δ c and δ z :

D. REFERENCE TRAJECTORY PLANNING
As an advantage of the FBC, limiting the maximum current of the inductor or the input current can be readily achieved by incorporating a current limiting function in the reference of the inductor current into the controller by using a secondorder filter which is introduced to the trajectories. A lowpass second-order filter is used to planify the requested path of the flat output vector w = [w c , w z ] T and their sub-respective components. This is to restrain power transient variation owing to variations of V rms−ref . In addition, the trajectories follow their references without steadystate errors. Therefore, the new planned reference trajectory By applying the flatness control, all constraints are reported through the candidate flat output (the energy). Also, the constraints can be given on the state and/or on the control in terms of the flat output and its derivatives. On the other hand, the presence of the observation and the current error controller system can lead to non-negligible time delays, hence, the choice of the system dynamics is crucial. The dynamics of the current error trajectories, the observer and β z0k  Table 2 to guarantee high convergence without time delay.

V. SIMULATION VERIFICATION
A detailed system is structured in Matlab environment to explore the proposed control strategy by using two-parallel inverters. The system parameters and control gains are identical to those used for the experimental investigations as listed in Table 2. Fig. 2 depicts schematic diagram of the proposed observer-based control system. Fig. 3.a illustrates response of w cd and w cq for the ac output filter with a step of the RM S output voltage from 80 V to 120 V at t = 20ms. The components w c0 and V C0 equal zero. The measured components perfectly follow their respective references. Figs. 3b,c demonstrate response of the output three-phase voltages and their components V cd and V cq with the control strategy which are pure sine waves for P L = 3 kW. The proposed FBC adjusts the output ac bus voltage to its reference set value under the normal conditions. Figs. 3-d,e show the estimated voltages for the first and the second inverters for given values for the voltages V td/qn and the current I p . These voltages and current artificially represent the losses of the system. The online estimator gives the same value of the inserted voltage and current sources. Figs. 3-f,g shows the dc link voltage and the input dc current. Fig. 3.h shows the load power with a step of the RM S output voltage from 80 V to 120 V at t = 20 ms. Figs. 4-a,b show the response V c0dq and the instantaneous voltages V cabc which are set to RM S value of 110 V with a step of the load power from 3 kW to 5 kW at t = 20 ms. The corresponding load currents are given in Figs. 4-c,d with P L changes from 3 kW to 5 kW. The load is set to a balanced load with a step from 3 kW to 5 kW at t = 20 ms as indicated in Fig. 4.e. Fig. 4.f shows the result of the total losses P Loss under balanced load with a step from 3 kW to 5 kW at t = 20 ms and ac bus voltage equals 110 V. As the two inverters are identical, the circulating currents are absent. Furthermore, the impact of circulating currents is insignificant as the switching actions are strictly synchronous.

VI. EXPERIMENTAL VALIDATION
For further validation of the simulation tests, experimental validations are executed for 5 kW test system comprising two parallel inverters as depicted in Fig. 5. The proposed FBC method is realized by MATLAB/Simulink-RTW software. The control method is implemented owing to the dSPACE-1005 real-time control card. All the measurements are attained by digital oscilloscope: Tektronix TDS5104B and Yokogawa WT1800E. Fig. 6 shows the performance of the tested system by using the proposed FBC while initial operating point is set to V dc = 500 V, V rms = 110 V, P L = 3.2 kW. Fig. 6.a illustrates the three-phase ac voltages and load currents for P L = 3.2 kW and the output voltage of the ac bus V abc is fixed at 110 V. The reported voltage THD is equal to 1.8%. Figs. 6 7.b shows the experimental results of V cd , and V cq when voltage changes suddenly from 80 V to 120 V and back to 80 V. Experimental results of V dc , I dc , and their respective estimated values under output ac bus voltage step from 80 V to 120 V and back to 80 V is depicted in Fig. 7.c. Fig.  7.d shows the experimental behaviors of V td1/2 , V tq1/2 for output ac bus voltage steps from 80 V to 120 V. Fig. 7.e shows the behaviors of Z d1/2 , Z q1/2 , these values are forced to zero to minimize the circulating currents between the parallel inverters. These results emphasize the efficacy of the proposed current balancing scheme which ensures balanced power sharing among parallel inverters. Fig. 7.f demonstrates I dc and I P for voltage change from 55 V to 120 V. Fig. 7.g depicts the shared power between the inverters and the corresponding losses P 1 /2, P Loss1 /2 for each inverter under unbalanced operation cause by inserting 1.5 Ω per phase to the second inverter. As seen, P 1 , P 2 are maintained balanced though the second inverter encounters higher losses. From the simulation and experimental results, the proposed FBC is effective for controlling the parallel inverters. The controlled system is robust against the uncertainties of the system parameters and the results validate the proposed estimator-based control method. The system efficiency is about 94% for the assumed load. In addition, in virtue of the proposed FBC, V dc and I dc , I abcn sensors are dispensed. It is evident that, the proposed estimators can provide information to the controller to obtain the corresponding duty cycle that maintains the output voltage equal to its reference. As seen, the number of voltage and current sensors is about 5 sensors only. In this paper, a sensorless flatness-based controller (FBC) for non-ideal paralleled DC-AC inverters is presented. A nonlinear online observer is suggested to estimate the line inductor currents and the dc link voltage to avoid using excessive sensors and simplify the control structure. Thereby increasing the system robustness against sudden variations of the input voltage and the load changes. A proposed method for modelling of the losses of DC/AC inverters is presented where the losses are modeled by voltage sources for each inverter and one current source. The observation system is exponentially stable, and its stability is validated by the candidate Lyapunov function which ensures that the estimation errors exponentially converge to zero. The system performance is explored by both simulation and experimental investigations, where the simulation verification is executed for two 5 kW parallel DC/AC inverters, whereas experimental validation is performed for 2 kW parallel inverters. The obtained results reveal that the control system is robust against various variations of the loads and the supply voltage.