Volume 3, Issue 3 p. 355-366
Research Article
Open Access

Stability analysis and nonlinear current-limiting control design for DC micro-grids with CPLs

Andrei-Constantin Braitor

Corresponding Author

Andrei-Constantin Braitor

Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, S1 3JD UK

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George C. Konstantopoulos

George C. Konstantopoulos

Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, S1 3JD UK

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Visakan Kadirkamanathan

Visakan Kadirkamanathan

Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, S1 3JD UK

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First published: 21 May 2020
Citations: 11


In this study, a DC micro-grid 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 converter-interfaced unit to guarantee load voltage regulation, power sharing and closed-loop system stability. This novel framework incorporates the widely adopted droop control and using input-to-state 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 closed-loop 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 three-phase 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 micro-grid 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 micro-grid system.

In DC micro-grids, 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 non-trivial 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 tight-speed 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 real-world applications have been assessed in [[17]], and there is an increased interest in designing droop controllers that guarantee closed-loop system stability for DC micro-grids loaded by CPLs [[19], [20]].

The existing stability methods for investigating DC micro-grids are based on the small-signal model of the power devices and linear approximation approaches, mostly employing the Middlebrook and Cuk criterion [[21]]. Whilst small-signal modelling is useful to obtain the system's open-loop 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 reduced-order models has been investigated in [[24], [25]] and stable operating regions have been obtained, but they ignore the dynamic performances of the DC-DC converters. Global stability results can be obtained using nonlinear control techniques, such as passivity-based 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 (IDA-PBC) [[29]] have been designed. Particularly, the IDA-PBC guarantees closed-loop 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 micro-grid application with multiple DC/DC and AC/DC converters, the PDE solution cannot be analytically obtained.

Apart from achieving stability in the micro-grid, 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 current-limiting property [[35][37]]. Although current-limiting 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) closed-loop stability cannot be analytically guaranteed since the controller can suffer from integrator wind-up 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 three-phase AC/DC and DC/DC boost converters feeding a CPL in a DC micro-grid 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 input-to-state stability (ISS) theory, it is proven that the proposed controllers guarantee an inherent current-limiting 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 closed-loop 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.

One distinctive fact is that compared to the cascaded PI approach or when a linear resistive load has been used [[31], [39], [40]], in this paper a new control structure is proposed that does not require the measurement of the converter output currents and additionally guarantees closed-loop system stability with a CPL. Moreover, in contrast to the control methods and stability analysis of the DC micro-grid presented in [[20]], the proposed approach not only guarantees stability but also has a better performance in achieving its control tasks whilst ensuring overcurrent protection at all times. The novel contributions of the proposed work are highlighted by the following aspects:
  • (i) the parallel operation of both bidirectional three-phase 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 closed-loop stability is analytically guaranteed for the CPL case.

Therefore, proving closed-loop system stability of a DC micro-grid with a CPL using the nonlinear model of the bidirectional three-phase AC/DC and the DC/DC boost converters together, while guaranteeing improved power sharing accuracy, load voltage regulation and an inherent current-limiting property is to the best of our knowledge novel.

The structure of this paper is divided as follows. In Section 2 the nonlinear model of a DC micro-grid consisting of multiple paralleled bidirectional three-phase AC/DC and DC/DC boost converters is presented. The control framework of the current-limiting droop controller is explained and analysed in Section 3. In Section 4, the closed-loop 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 micro-grid

2.1 Notation

Let urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0002 be defined as the diagonal matrix whose diagonal entries are the elements of the n -dimensional vector urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0004 . Let urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0006 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0008 be the n -dimensional vector and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0010 square matrix, respectively, with all elements zero, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0012 be the identity matrix and let urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0014 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0016 be the n -dimensional vector and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0018 square matrix, respectively, with all elements equal to one.

2.2 Dynamic model

A typical topology of a DC micro-grid 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 micro-grid under investigation is shown in Fig. 2, containing n bidirectional three-phase rectifiers and m bidirectional DC/DC boost converters feeding a constant power load, where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0020 is the inductor at the input, a DC output capacitor urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0022 with a line resistance urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0024 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0026 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0028 , while output dc voltage is denoted as urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0030 with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0032 . The bidirectional DC/DC converters have two switching elements, an inductor urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0034 at the input and a capacitor urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0036 with a line resistance urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0038 at the output, while urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0040 is the output voltage, where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0042 . At the input, the voltage and the current of the converter are represented as urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0044 , and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0046 , respectively, with the latter being either positive or negative to allow a bidirectional power-flow.

Details are in the caption following the image

Typical configuration of a DC micro-grid

Details are in the caption following the image

Parallel operated three-phase AC/DC and bidirectional DC/DC boost converters feeding a common constant power load

To obtain the dynamic model of the rectifier, the average system analysis and the dq transformation can be used for three-phase voltages and currents, using Clarke and Park transformations [[39]]. Following [[41]], the mathematical model of the rectifiers in the dq coordinates is set up, in matrix form as
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0054 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0056 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0058 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0060 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0062 is the rotating speed, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0064 is the amplitude of the three-phase AC voltage source when voltage orientation on the d axis is considered and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0066 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0068 are the d and q components of the AC source currents, respectively, and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0070 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0072 are the duty-ratio control inputs of the rectifier with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0074 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0076 being the d and q components of the rectifier voltage urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0078 , respectively.
Using Kirchhoff laws and average analysis [[42]], the dynamic model, in matrix form, of the bidirectional DC/DC boost converter becomes
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0084 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0086 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0088 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0090 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0092 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0094 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0096 . One can observe that system (1)–(3), (4)–(5) is nonlinear, since the control inputs urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0098 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0100 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0102 are multiplied with the system states, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0104 , and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0106 respectively.
As the AC/DC and DC/DC converters supply a CPL, the power balance equation becomes
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0112 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0114 represent the output voltages and currents, respectively, with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0116 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0118 is the load voltage, and P is constant and represents the power of the CPL. Consider now the following assumptions:

Assumption 1.It holds that

Thus, substituting the output current urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0123 from (7) into (6), one can obtain the following expression for the load voltage given by the real solutions of the second order polynomial as

Assumption 2.Let urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0128 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 three-phase rectifiers urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0130 and for boost converters urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0132 , let

hold, for every urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0136

The load voltage has two solutions, a high voltage and a low voltage, with the high voltage representing the feasible solution because of Assumption 2, which gives urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0138 . Therefore, the voltage of the load can be described as
Considering an equilibrium point urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0142 for constant control inputs urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0144 , by taking the partial derivative of the output current urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0146 from (7) with respect to the capacitor voltage urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0148 , we obtain the admittance matrix
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0151 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0153 with the following expression
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0156 according to Assumption 1. Since urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0158 is a diagonal positive-definite matrix, then it is clear that matrix urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0160 is a positive-definite diagonal matrix, with eigenvalues of the form

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 three-phase 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 Three-phase rectifier

Although a current-limiting controller was recently proposed in [[39]], it only allows unidirectional power flow, which is a significant limitation when storage units are introduced or the AC/DC converter represents an interface between a DC and an AC micro-grid. To overcome this problem, here the control inputs urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0164 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0166 , with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0168 are proposed to take the following form
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0174 is a constant virtual resistance and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0176 a virtual voltage that change according to the following nonlinear dynamics:
with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0182 representing an additional control state, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0184 the load voltage reference, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0186 the set output power, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0188 the droop coefficient, and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0190 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0192 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0194 being positive constants. The proposed controller introduces the desired droop expression via the input urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0196 , while it forces the current urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0198 to zero through urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0200 in order to guarantee unity power factor operation, since urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0202.

3.1.2 Bidirectional DC/DC boost converter

Following a similar control framework with the AC/DC converter, for the DC/DC boost converter the control input urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0204 , with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0206 , becomes
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0210 represents a constant virtual resistance and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0212 a virtual controllable voltage
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0218 being an additional control state, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0220 the set output power, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0222 the droop coefficient, and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0224 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0226 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0228 positive constants. Compared to the robust droop controller [[11]], the proposed strategy does not require the measurement of the output current urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0230 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0232 of each converter, thus leading to a simpler implementation. It is highlighted that a second controller state urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0234 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0236 is based on the bounded integral controller concept [[43]]. For more details on the bounded dynamics of the control states the reader is referred to [[43]] where it is shown that the control states are guaranteed to stay within their imposed bounds urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0238 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0240 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0242 for all urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0244 , given typical initial conditions urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0246 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0248 . The block diagram depicting the controller implementation, measurement and actuation parts is presented in Fig. 3. Having introduced the proposed control schemes, consider the additional assumptions for the system:
Details are in the caption following the image

Block diagram with the control implementation of the proposed controllers

(a) 3-phase bidirectional rectifier controller, (b) DC/DC bidirectional boost converter controller

Assumption 3.For every constant urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0251 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0253 , satisfying

there exists a unique steady-state equilibrium point urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0257 corresponding to a load voltage regulation, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0259 , where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0261.

Assumption 4.For urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0264 it holds that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0266 , with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0268 when urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0270 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0272 when urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0274.

For the selection of urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0276 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0278 the following condition should hold
The desired current-limiting property for each converter can be now investigated in the next subsection.

3.2 Current limitation

3.2.1 Three-phase rectifier

For system (1)–(2), consider the following continuously differentiable function:
Substituting urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0284 from (12) and (13) into (1) and (2), and taking into account that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0286 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0288 , the time derivative of urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0290 becomes
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0294 . Consider now that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0296 for an arbitrarily small urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0298 . Then
which means that system (1)–(2) is ISS [[44]] with respect to the virtual voltage urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0302 . Since urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0304 is bounded below the chosen maximum virtual voltage value urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0306 , then both the d and q currents, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0308 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0310 will remain bounded at all times.
Since urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0312 then taking into account the dq transformation, it results in
it is proven from the ISS property (21) that if initially the RMS AC/DC converter current is below the maximum allowed value urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0318 , i.e. urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0320 , then
Hence, the input current of each rectifier separately is always limited below urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0324 with the appropriate choice of urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0326 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0328 given in (23), ensuring protection at all times. It is shown that the current-limiting property of each converter is guaranteed independently from the power sharing expression urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0330 that has to be regulated to zero. This means that each converter has as first priority to protect itself from high currents that can damage the device. When the current is below the maximum value, the converter contributes to the desired power sharing within the DC micro-grid.

3.2.2 Bidirectional boost converter

By applying the proposed controller expression (16) into the bidirectional converters dynamics (4), the closed-loop system equation for the inductor current urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0332 takes the following form
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0336 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0338 . One can clearly see that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0340 represents a constant virtual resistance in series with the converter inductor L.
To investigate how the selection of the virtual resistance and the bounded controller dynamics of E are related to the desired overcurrent protection, let the following continuously differentiable function:
for closed-loop current dynamics (24). The time derivative of urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0344 yields
Considering urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0348 for arbitrarily small urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0350 , then
which means that system (24) is ISS with respect to the bounded virtual voltage, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0354 . Similar to the rectifier case, since urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0356 then
holds true if initially the inductor current is below the same value, i.e. urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0360.
Hence one can clearly select the parameters urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0362 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0364 in the proposed controller in order to satisfy
and guarantee that
Any selection of the constant and positive parameters urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0370 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0372 that satisfy (27) results in the desired overcurrent protection (28) of the converter's inductor current regardless the load magnitude or system parameters.

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 closed-loop system that facilitates the stability analysis that follows.

4 Stability analysis

By applying the proposed controller (12)–(15), (16)–(18) into the DC micro-grid dynamics (1)–(3), (4)–(5) the closed-loop system can be written in the following matrix form:

where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0377 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0379 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0381 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0383 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0385 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0387 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0389 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0391 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0393 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0395 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0397 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0399 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0401 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0403.

Consider an equilibrium point urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0405 calculated from (29)–(30) at the steady-state, satisfying Assumption 3. By setting urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0407 , there exists urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0409 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0411 such that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0413 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0415 . Thus (30) becomes
where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0419 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0421 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0423 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0425 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0427 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0429 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0431 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0433 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0435.
Hence, the closed-loop system equations and can be written as
For arbitrarily large values of the controller gains cd, cb the value of urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0443 is small and therefore (32)–(33) can be investigated as a singularly perturbed system using two-time-scale analysis [[44]]. The controller's system (31) is also referred to as the boundary layer since it represents the immediate vicinity of a bounding surface where the effects of stability are significant.
Considering f, g being continuously differentiable in the domain urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0445 , when the controller gains cd, cb are selected sufficiently large, then urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0447 and, based on singular perturbation theory, g will have an algebraic form of urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0449
The roots of the above system can be computed as shown below
These roots can also be written as urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0455 with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0457 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0459 , and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0461 , such that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0463 . Thus, the roots also represent the equilibrium points of the nonlinear system. Exponential stability at the origin can be investigated via system's (33) corresponding Jacobian matrix
where it is obvious that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0466 has negative eigenvalues since it is lower triangular and the diagonal elements
are diagonal and negative definite matrices.

Therefore matrix urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0470 is Hurwitz. Hence, there exist urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0472 and a domain urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0474 where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0476 such that (33) is exponentially stable at the origin uniformly in x.

To obtain the reduced model, the roots urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0478 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0480 are substituted from (35) into (29), yielding
with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0484 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0486 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0488 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0490 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0492.

In the literature, the above model is referred to as quasi-steady-state model, since urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0494 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0496 introduce a velocity urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0498 that is very large when urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0500 is small and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0502 , leading to fast convergence to a root urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0504 , which also represents the equilibrium of the boundary-layer.

The second equation of (29) is independent, thus there are n eigenvalues where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0506 . The corresponding Jacobian matrix of the reduced system (37) that remains to be investigated will have the form of J2
with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0510 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0512 being
The characteristic polynomial can be calculated from
with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0520 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0522 expressed
Following factorisation the matrices urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0526 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0528 become as presented
Let the characteristic polynomial be
Considering urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0536 , the characteristic polynomial becomes
with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0540 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0542 . As the determinants urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0544 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0546 are positive, the polynomial reduces to
which is a quadratic eigenvalue problem urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0550 with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0552 symmetrical, and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0554 , according to Lemma 2 in [[20]], diagonalisable whose eigenvalues are all real, since it is a product of a positive-definite diagonal and a symmetrical matrix.
The characteristic equation then becomes
Note that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0558 is a diagonal matrix with the eigenvalues of urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0560 as main entries and the similarity transformations urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0562 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0564 are symmetrical, as urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0566 is orthogonal urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0568 , and they share the same spectrum as urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0570 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0572 , respectively. If urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0574 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0576 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0578 are positive-definite, then urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0580 which means that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0582 is Hurwitz. Hence, since urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0584 is already positive-definite, it is sufficient to show that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0586 , or equivalently that urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0588 has positive eigenvalues, and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0590 . Since matrix urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0592 is represented by a multiplication where one term is the diagonal matrix urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0594 , according to the same Lemma 2 in [[20]], the remaining symmetrical term, denoted urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0596 , will have the same index of inertia as urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0598 . The condition urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0600 becomes
which represents a sum between a diagonal positive-definite real matrix and the real symmetric matrix urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0604 . According to Lemma 1 in [[20]], if
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0608 holds, then urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0610 is satisfied. When urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0612 , then urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0614 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0616 , whereas when urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0618 , then urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0620 . Regarding condition urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0622 , taking into account Assumption 4, and according to the same Lemma 1 if
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0626 holds, then urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0628 is satisfied. Hence, if the two conditions (38)–(39) are satisfied for each converter then there exist urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0630 and a domain urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0632 where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0634 such that the reduced model is exponentially stable at the origin.

According to Theorem 11.4 in [[44]], there exists urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0636 such that for all urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0638 (or equivalently urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0640 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0642), the equilibrium point urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0644 of (32)–(33) with urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0646 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0648 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0650 is exponentially stable; thus completing the stability analysis of the entire DC micro-grid.

5 Validation of closed-loop 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0652 , urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0654 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0656 , 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0658 , 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0660 of the DC/DC converter can be calculated. Thereafter, critical points of the output voltages are calculated, followed by the eigenvalues of matrix urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0662 . Finally, the two conditions can be tested for each converter.

Table 1. Controller and system parameters
Parameters Values Parameters Values
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0664 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0666 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0668 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0670
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0672 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0674 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0676 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0678
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0680 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0682 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0684 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0686
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0688 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0690 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0692 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0694
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0696 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0698 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0700 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0702
P urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0704 k urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0706
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0708 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0710 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0712 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0714
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0716 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0718 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0720 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0722
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0724 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0726 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0728 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0730

Hence, following this procedure for different values of the set power of the battery, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0732 , corresponding to the battery operation, charging and discharging, respectively, one can observe in Fig. 4 that for any urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0734 in the bounded range urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0736 , the expressions (38)–(39) for each converter are positive, thus guaranteeing closed-loop stability.

Details are in the caption following the image

Checking stability conditions (38)–(39)

(a) urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0738 , (b) urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0740 , (c) urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0742

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, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0744 , to visually confirm that the two stability conditions always take positive values in the entire operating range of the particular DC micro-grid (Fig. 6).

Details are in the caption following the image

Graphical representation of the stability conditions (38)–(39)

(a) 3D visualisation of condition (38), (b) 3D visualisation of condition (39)

Details are in the caption following the image

DC micro-grid considered for testing, containing a three-phase AC/DC converter connected to the grid, a bidirectional DC/DC boost converter interfacing a battery, and a CPL connected to the main bus and fed by the two converters

6 Simulation results

To test the proposed controller and compare it to the cascaded PI approach, a DC micro-grid consisting of a bidirectional boost converter and a three-phase 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0746 , 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0748 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0750 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0752 , unlike the case with cascaded PI s where urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0754 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0756 . The input currents haven't reached their imposed limits yet as shown in Fig. 7 a.

Details are in the caption following the image

Simulation results of the DC micro-grid system with PI cascaded control (left) and the proposed controller (right)

(a) Inductor currents, (b) Output voltages, (c) Output currents

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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0758 the power set by the battery controller becomes negative urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0760 , thus leaving the battery to be supplied by the three-phase 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0762 the power requested by the load increases urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0764 and, thus, more power is needed from the battery and the three-phase rectifier to be injected in the micro-grid. The load voltage drops down to 396 V according to Fig. 7 b when using the proposed controller and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0766 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0768 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0770 with the proposed controller, and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0772 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0774 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0776 the power requested by the load reaches a higher value than before, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0778 , forcing the battery and the three-phase 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0780 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 steady-state. The power sharing is sacrificed (Fig. 7 c) to ensure uninterruptible power supply to the load. The load voltage remains within the desired range, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0782 with a voltage drop of 6.5 V, which is about 1.5% when having the proposed controller and about urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0784 with the cascaded PI approach.

Consequently, to further verify the theory presented, the controller states urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0786 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0788 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0790 . One can notice in Fig. 8 b that the corresponding control state urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0792 goes to zero when urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0794 reaches maximum.

Details are in the caption following the image

Dynamic response of the control states

(a) Virtual voltages, (b) Additional control states

It is noted that for the particular DC micro-grid scenario and the parameters used, the closed-loop 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 micro-grid, 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 ELP-3362F electronic load, operated in CPL mode, is experimentally tested. A switching frequency of 60 kHz was used for the pulse-width-modulation of both converters. The aim is to experimentally validate the proposed nonlinear current-limiting control scheme. The main tasks are to regulate the output voltage to urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0796 and regulate the power in a 2:1 ratio, whilst ensuring overcurrent protection.

Table 2. Controller and system parameters
Parameters Values Parameters Values
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0798 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0800 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0802 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0804
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0806 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0808 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0810 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0812
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0814 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0816 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0818 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0820
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0822 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0824 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0826 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0828
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0830 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0832 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0834 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0836
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0838 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0840 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0842 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0844
urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0846 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0848 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0850 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0852

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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0854 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0856 , provided the input currents, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0858 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0860 , have not reached their upper limit.

Details are in the caption following the image

Experimental results under the proposed controller

(a) Load demand increases from 40 to 60 W, (b) Load demand decreases from 60 to 40 W, (c) Load demand increases from 40 to 80 W

In Fig. 9 b, the load power demand decreases from 60 to 40 W. The output currents are accurately shared, having urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0862 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0864 , and the load voltage is kept fixed at 48 V.

To test the current-limiting capability, the power increases from 40 to 80 W, as displayed in Fig. 9 c. One converter reaches to its imposed limit (urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0866), 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, urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0868 , 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 urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0870 and urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0872 should satisfy urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0874 in order to reach the upper limit of the converter. Nevertheless, it is clear that by ignoring this resistance, the current still remains below urn:x-wiley:25152947:media:stg2bf00132:stg2bf00132-math-0876 as desired.

8 Conclusions

In this paper, a detailed control design was presented for multiple parallel operated three-phase AC/DC and bidirectional DC/DC boost converters in a DC micro-grid 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 micro-grid 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 three-phase 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 wind-up 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 micro-grid considering different load power variations and battery operations (charging, discharging), and by experimentally testing a parallel converter micro-grid 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.