Stability analysis and nonlinear currentlimiting control design for DC microgrids with CPLs
Abstract
In this study, a DC microgrid consisting of multiple paralleled energy resources interfaced by both bidirectional AC/DC and DC/DC boost converters and loaded by a constant power load (CPL) is investigated. By considering the generic dq transformation of the AC/DC converters' dynamics and the accurate nonlinear model of the DC/DC converters, two novel control schemes are presented for each converterinterfaced unit to guarantee load voltage regulation, power sharing and closedloop system stability. This novel framework incorporates the widely adopted droop control and using inputtostate stability theory, it is proven that each converter guarantees a desired current limitation without the need for cascaded control and saturation blocks. Sufficient conditions to ensure closedloop system stability are analytically obtained and tested for different operation scenarios. The system stability is further analysed from a graphical perspective, providing valuable insights of the CPL's influence onto the system performance and stability. The proposed control performance and the theoretical analysis are first validated by simulating a threephase AC/DC converter in parallel with a bidirectional DC/DC boost converter feeding a CPL in comparison with the cascaded PI control technique. Finally, experimental results are also provided to demonstrate the effectiveness of the proposed control approach on a real testbed.
1 Introduction
Driven by the energy crisis, environmental pollution and greenhouse gas emissions [[1]–[3]], the seamless integration of renewable energy sources (RESs) has been actively pursued worldwide, over the past decades. With the uninterrupted growth of RES, the smart grid and microgrid concepts have been proposed as a benchmark of the future grid to enable efficient utilisation of renewable resources and distributed generations (DGs). The centrepiece of these frameworks is represented by the power converters [[4]] which are the interface devices of RES to the microgrid system.
In DC microgrids, DG units are connected to a common DC bus through AC/DC and/or DC/DC converters, often operating in parallel leading to a series of nontrivial issues such as voltage regulation and accurate distribution of the load power. A widely used technique to accomplish these tasks, implemented in a fully decentralised way, that does not require communication between each DG, is to introduce a virtual resistance at the output of each converter, a method also referred to as ‘droop control’ [[5]–[9]]. The main disadvantages of the conventional droop control consist of significant load voltage drop and inaccurate power sharing due to mismatches at the line impedances. Therefore, several methods have been proposed to tackle and improve its existing performance, such as the robust droop control [[10], [11]] where the line impedances are not considered, the nonlinear droop control [[12]] where each DG unit is optimised against hypothetical DGs, or the quadratic droop control [[13]], implemented as a special case of the general feedback controller. However, in the majority of these works, the stability of the parallel operated power converters has been insufficiently addressed mainly due to the complexity of the dynamics that increases with the nonlinear characteristics of the AC/DC and DC/DC converters and their nonlinear loads. Power converters fed by the main bus create unique dynamic characteristics and have been a research subject for years. As shown in [[14], [15]], under tightspeed regulation, the motor drive exhibits constant power behaviour at the DC bus, similar to tight regulated downstream converters [[16]–[18]]. The dynamic behaviour of constant power loads is equivalent to a dynamic negative impedance which can produce instability at the DC bus and, consequently, in the system [[16]]. Limitations of practical constant power loads (CPLs) in realworld applications have been assessed in [[17]], and there is an increased interest in designing droop controllers that guarantee closedloop system stability for DC microgrids loaded by CPLs [[19], [20]].
The existing stability methods for investigating DC microgrids are based on the smallsignal model of the power devices and linear approximation approaches, mostly employing the Middlebrook and Cuk criterion [[21]]. Whilst smallsignal modelling is useful to obtain the system's openloop gain by considering only the input impedance of the loads and output impedance of the sources [[22], [23]], the nonlinear dynamics of the power converters are not taken into account. Stability of reducedorder models has been investigated in [[24], [25]] and stable operating regions have been obtained, but they ignore the dynamic performances of the DCDC converters. Global stability results can be obtained using nonlinear control techniques, such as passivitybased control (PBC) methods, which have been successfully applied to power converter systems applications [[26], [27]]. However, these control schemes require the knowledge of the system and load parameters, which may not be available in practice. To overcome this issue, advanced control techniques such as adaptive PBC [[28]] or the interconnection and damping assignment PBC (IDAPBC) [[29]] have been designed. Particularly, the IDAPBC guarantees closedloop stability with enhanced system robustness as it is parameter free. However, its main shortcoming is that it needs the solution of a partial differential equation (PDE) system of order equal to the system order. Thus, in a DC microgrid application with multiple DC/DC and AC/DC converters, the PDE solution cannot be analytically obtained.
Apart from achieving stability in the microgrid, other control issues that relate to the technical requirements of each DG unit should be taken into account in the control design such as the capability of the power converters to be protected at all times, particularly during transients, faults and unrealistic power demands. The overcurrent protection as presented in [[30], [31]], guarantees the converter operation and protection of the equipment without violating its technical limitations. Existing strategies are based on protection units such as using additional fuses, circuit breakers or relays [[32]–[34]]; however, it still represents a challenge to design control methods that ensure an inherent currentlimiting property [[35]–[37]]. Although currentlimiting control methods based on saturated PI controllers are often used to guarantee a given upper limit for the current, the shortcomings of these methods have not been completely overcome, e.g.: (i) only the reference value of the converter's input current is limited, i.e. overcurrent protection is not achieved during transients as shown in [[31]] and (ii) closedloop stability cannot be analytically guaranteed since the controller can suffer from integrator windup problems that could potentially yield instability in the system [[38]].
For this reason, in this paper, two novel nonlinear droop control strategies are proposed for parallel operated bidirectional threephase AC/DC and DC/DC boost converters feeding a CPL in a DC microgrid architecture to ensure accurate distribution of the load power among the paralleled units in proportion to their power ratings and inherent overcurrent protection. Based on the nonlinear dynamics of the converters and using inputtostate stability (ISS) theory, it is proven that the proposed controllers guarantee an inherent currentlimiting property for each converter independently from each other or the load. In addition, accurate power sharing and load voltage regulation close to the rated value are accomplished and the stability of the closedloop system is proven when connected to a CPL using singular perturbation theory. The effectiveness of the proposed controllers and the stability conditions are verified through simulation testing and they are compared to the cascaded PI technique to highlight its superiority.

(i) the parallel operation of both bidirectional threephase AC/DC and DC/DC boost converters is investigated here, which are inherently nonlinear systems, opposed to only unidirectional boost converter [[31]], or only buck converters, as studied in [[20]] which have linear dynamics;

(ii) compared to [[20]], a new droop control structure that achieves improved power sharing and output voltage regulation closer to the rated value is proposed and analysed;

(iii) an inherent current limitation is introduced via the proposed control design for all power converters;

(iv) in contrast to [[40]] where a linear resistive load was considered, in this paper closedloop stability is analytically guaranteed for the CPL case.
The structure of this paper is divided as follows. In Section 2 the nonlinear model of a DC microgrid consisting of multiple paralleled bidirectional threephase AC/DC and DC/DC boost converters is presented. The control framework of the currentlimiting droop controller is explained and analysed in Section 3. In Section 4, the closedloop system stability analysis is presented and then analysed from a graphical perspective in Section 5. In Section 6, simulation results are displayed to test the controller performance, which is further validated in Section 7 on a real experimental testbed. Finally, in Section 8 some conclusions are drawn.
2 Nonlinear model of the DC microgrid
2.1 Notation
Let be defined as the diagonal matrix whose diagonal entries are the elements of the n dimensional vector . Let and be the n dimensional vector and square matrix, respectively, with all elements zero, be the identity matrix and let and be the n dimensional vector and square matrix, respectively, with all elements equal to one.
2.2 Dynamic model
A typical topology of a DC microgrid is shown in Fig. 1 consisting of several types of energy sources, power converters and loads connected to a common bus. The configuration of the DC microgrid under investigation is shown in Fig. 2, containing n bidirectional threephase rectifiers and m bidirectional DC/DC boost converters feeding a constant power load, where is the inductor at the input, a DC output capacitor with a line resistance and six controllable switching elements that operate using PWM and capable of conducting current and power in both directions. The input voltages and currents of the rectifier are expressed as and , while output dc voltage is denoted as with . The bidirectional DC/DC converters have two switching elements, an inductor at the input and a capacitor with a line resistance at the output, while is the output voltage, where . At the input, the voltage and the current of the converter are represented as , and , respectively, with the latter being either positive or negative to allow a bidirectional powerflow.
Assumption 1.It holds that
Assumption 2.Let be the maximum input current of each converter (maximum RMS current for AC/DC converters and maximum inductor current for DC/DC converters). Since for threephase rectifiers and for boost converters , let
3 Nonlinear control design and analysis
3.1 Proposed controller
The purpose of the designed controller is to achieve accurate distribution of the load power and tight load voltage regulation close to the rated value, ensuring that the current of each converter does not violate certain bounds. The proposed concept is based on the idea of partially decoupling the inductor current dynamics, introducing a constant virtual resistance with a bounded controllable voltage for both the bidirectional threephase AC/DC and the DC/DC boost converters. In both cases, the dynamics of the controllable virtual voltage will guarantee the desired upper bound for the converters’ currents regardless of the direction of the power flow.
3.1.1 Threephase rectifier
3.1.2 Bidirectional DC/DC boost converter
Assumption 3.For every constant and , satisfying
Assumption 4.For it holds that , with when and when .
3.2 Current limitation
3.2.1 Threephase rectifier
3.2.2 Bidirectional boost converter
It is underlined that compared to existing conventional overcurrent protection control strategies, here it has been mathematically proven according to the nonlinear ISS theory that the proposed controller maintains the current limited during transients and does not require limiters or saturation units which are prone to yield instability in the system. At the same time, it maintains the continuous time structure of the closedloop system that facilitates the stability analysis that follows.
4 Stability analysis
where , , , , , , , , , , , , , .
Therefore matrix is Hurwitz. Hence, there exist and a domain where such that (33) is exponentially stable at the origin uniformly in x.
In the literature, the above model is referred to as quasisteadystate model, since and introduce a velocity that is very large when is small and , leading to fast convergence to a root , which also represents the equilibrium of the boundarylayer.
According to Theorem 11.4 in [[44]], there exists such that for all (or equivalently and ), the equilibrium point of (32)–(33) with , and is exponentially stable; thus completing the stability analysis of the entire DC microgrid.
5 Validation of closedloop system stability
To validate the theoretical stability analysis presented in Section 4 and demonstrate how conditions (38)–(39) can be tested, let us consider the system in Section 6 with parameters given in Table 1. Although (38)–(39) might seem difficult to verify, by taking into account that , and , which is guaranteed by the proposed control design, the procedure to verify whether the system is stable is the following: One can start by selecting a virtual voltage , inside its defined range, for the rectifier. Then the values of the equilibrium points of the inductor current and load voltage are computed. Based on these obtained values, the remaining virtual voltages of the DC/DC converter can be calculated. Thereafter, critical points of the output voltages are calculated, followed by the eigenvalues of matrix . Finally, the two conditions can be tested for each converter.
Parameters  Values  Parameters  Values 

P  k  
Hence, following this procedure for different values of the set power of the battery, , corresponding to the battery operation, charging and discharging, respectively, one can observe in Fig. 4 that for any in the bounded range , the expressions (38)–(39) for each converter are positive, thus guaranteeing closedloop stability.
To further validate the stability analysis, in Fig. 5, a graphical interpretation of the stability conditions is provided for the entire range of the set power, , to visually confirm that the two stability conditions always take positive values in the entire operating range of the particular DC microgrid (Fig. 6).
6 Simulation results
To test the proposed controller and compare it to the cascaded PI approach, a DC microgrid consisting of a bidirectional boost converter and a threephase rectifier feeding a CPL is considered having the parameters specified in Table 1. The aim is to achieve tight voltage regulation around the reference value , accurate power sharing in a 2:1 ratio among the paralleled AC/DC and DC/DC converters at the load bus while also assuring protection against overcurrents. But first the conditions for stability must hold.
The model has been implemented in Matlab Simulink and simulated for 45 s considering a full testing scenario.
During the first 5 s, the power requested by the load is 200 W and it can be observed in Fig. 7 b that the load voltage is kept close to the reference value of 400 V, at ∼398 V in both cases. However, the power sharing is only accurately guaranteed (Fig. 7 c) in a 2:1 manner with the proposed controller having and , unlike the case with cascaded PI s where and . The input currents haven't reached their imposed limits yet as shown in Fig. 7 a.
For the next 20 s the operation principle of the battery is simulated. The direction of the power flow is reversed to allow the battery to charge and discharge. At the power set by the battery controller becomes negative , thus leaving the battery to be supplied by the threephase rectifier. The input current of the battery becomes negative, while the rectifier's input current increases to satisfy the new amount of power requested in the network (Fig. 7 a). The power sharing ratio between the battery and the rectifier disappears since the current of the battery changes its direction, and becomes negative as shown in Fig. 7 a. The load voltage remains closely regulated to the desired 400 V value, at around 396.5 V in both cases. After 10 s the set value of the power returns to its initial 0 value, allowing the battery to return to its former discharging state. The power sharing ratio comes back to 2:1 as displayed in Fig. 7 c.
At the power requested by the load increases and, thus, more power is needed from the battery and the threephase rectifier to be injected in the microgrid. The load voltage drops down to 396 V according to Fig. 7 b when using the proposed controller and when having cascaded PI s. At the same time, the input currents increase and, therefore, the power injected increases at the common bus (Fig. 7 a). One can see that the sharing is kept between the two sources, the battery and rectifier, to the desired proportion of 2:1 having and with the proposed controller, and and with the cascaded PI technique, as presented in Fig. 7 c, given the fact that none of the inductor currents have reached their maximum allowed current. To test the input current protection capability, the power demanded by the load is further increased. Thus, at the power requested by the load reaches a higher value than before, , forcing the battery and the threephase rectifier to increase their power injection at the load bus. As noticed in Fig. 7 a, the input current of the battery reaches its limit without violating it when using the proposed controller, but in the case of the cascaded PIs the transient current exceeds the upper limit prior reaching to steadystate. The power sharing is sacrificed (Fig. 7 c) to ensure uninterruptible power supply to the load. The load voltage remains within the desired range, with a voltage drop of 6.5 V, which is about 1.5% when having the proposed controller and about with the cascaded PI approach.
Consequently, to further verify the theory presented, the controller states and are presented in Figs. 8 a and b. When the input current of the battery reaches its maximum, the virtual voltage of battery also arrives at its imposed limit . One can notice in Fig. 8 b that the corresponding control state goes to zero when reaches maximum.
It is noted that for the particular DC microgrid scenario and the parameters used, the closedloop performance with the cascaded PI control remains stable. However, this might not be true for a different system since there is no rigorous proof of stability. On the other hand, the proposed control approach provides a strong theoretic framework, as proven in Section 4, that can be easily tested for different systems as well.
7 Experimental results
A DC microgrid, with the parameters given in Table 2, consisting of two parallel Texas Instruments DC/DC boost converters connected to a common DC bus and feeding an ETPS ELP3362F electronic load, operated in CPL mode, is experimentally tested. A switching frequency of 60 kHz was used for the pulsewidthmodulation of both converters. The aim is to experimentally validate the proposed nonlinear currentlimiting control scheme. The main tasks are to regulate the output voltage to and regulate the power in a 2:1 ratio, whilst ensuring overcurrent protection.
Parameters  Values  Parameters  Values 

As one can see in Fig. 9 a, when the power changes from 40 to 60 W, the voltage is kept close to the reference value of 48 V, while the output currents are accurately shared proportionally to the sources rating, in a 2:1 manner, having and , provided the input currents, and , have not reached their upper limit.
In Fig. 9 b, the load power demand decreases from 60 to 40 W. The output currents are accurately shared, having and , and the load voltage is kept fixed at 48 V.
To test the currentlimiting capability, the power increases from 40 to 80 W, as displayed in Fig. 9 c. One converter reaches to its imposed limit (), the power sharing is sacrificed to ensure the uninterrupted power supply of the load. The load voltage is still fairly close to the rated value of 48 V. As it can be seen, the current limitation is not exactly at the 1.5 A limit. This is due to the fact that the parasitic resistance, , of the converter's inductance is ignored, in the experiment and the analysis, which in turn causes a slightly lower bound of the input current. If the parasitic resistance is considered, then based on the ISS analysis in Section 3, one can easily obtain that the controller parameters and should satisfy in order to reach the upper limit of the converter. Nevertheless, it is clear that by ignoring this resistance, the current still remains below as desired.
8 Conclusions
In this paper, a detailed control design was presented for multiple parallel operated threephase AC/DC and bidirectional DC/DC boost converters in a DC microgrid framework, loaded by a CPL. The nonlinear dynamic control scheme was developed to ensure load power sharing and output voltage regulation, with an inherent input current limitation. The stability of the entire DC microgrid was analytically proven when the system supplies a CPL using singular perturbation theory. Introducing a constant virtual resistance with a bounded dynamic virtual voltage for the threephase AC/DC and for the bidirectional DC/DC boost converter, it has been shown that the input currents of each converter will never violate a maximum given value. This feature is guaranteed without any knowledge of the system parameters and without any extra measures such as limiters or saturators, thus, addressing the issue of integrator windup and instability problems that can occur with the traditional overcurrent controllers’ design. The effectiveness of the proposed scheme and its overcurrent capability are verified by simulating a DC microgrid considering different load power variations and battery operations (charging, discharging), and by experimentally testing a parallel converter microgrid configuration feeding an electronic load, acting as a CPL.
9 Acknowledgment
This work was supported by Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/S001107/1 and Grant EP/S031863/1.