Decentralized Voltage Balancing and Power Sharing in Islanded Bipolar DC Microgrids

— This paper presents a decentralized control scheme for voltage balancing and power sharing in bipolar dc microgrids. This relies on utilizing a converter topology which offers three levels of output voltage availability with the key features of boosting the input voltage and balancing the output voltages. This converter makes it possible to further improve the structure of bipolar dc microgrids as it does not require a central voltage balancer. Small-signal analysis is done and system transfer functions are derived. Based on the RGA concept the highly coupled input-output pairs are found which helps with replacing the MIMO control system of the converter by two SISO systems. The appropriate voltage and current controllers are designed based on SISO principles. Moreover, a double droop control method is proposed which fulfills the simultaneous power sharing and voltage regulation of DG units in the host microgrid. The effectiveness of the proposed control strategy is demonstrated through simulation studies conducted on an islanded bipolar dc microgrid involving unbalanced loads, while the voltage balancing of the bipolar dc microgrid and the power sharing accuracy are evaluated.


I. INTRODUCTION
DC microgrids are becoming more popular mainly due to the increase of power-electronic loads and the dc nature of the majority of distributed energy resources (DERs) such as PV, fuel cell, and wind generation systems [1], [2]. In fact, while microgrids are replete with dc loads, it is technically and economically preferred to feed these loads directly by dc DERs avoiding multiple conversion of voltage from dc to ac and vice versa. Moreover, dc microgrids are simpler to control as they need no frequency and reactive power regulations [3] - [5].
Traditionally, the main focus of researches on dc power systems has been in the area of high-voltage dc power transmission systems (HVDC), where the continuous reduction in the price of power electronic devices has paved the way for further development in this area [6], [7]. In distribution level, data centres and navy shipboard are two areas where the replacement of conventional ac distribution networks with the zonal dc microgrids have been investigated. In both applications, many advantages, including higher efficiency and improved reliability are associated with the use of dc microgrids [3], [8], [9].
Recently, bipolar dc microgrids have been proposed offering additional benefits compared to the conventional dc microgrids [10]. Some of the challenges associated with bipolar dc microgrid have been extensively investigated as they are common between conventional dc microgrids and bipolar dc microgrids. However, there are some issues including power sharing, voltage balancing, and voltage stability which need to be specifically investigated for bipolar dc microgrids.
Within this context, this paper focuses on the voltage balancing of bipolar dc microgrids while proposing a doubledroop power sharing method. To this aim, we use a bipolar dc/dc boost converter topology which provides the possibility of performing the voltage balancing in a decentralized fashion. Although a decentralised voltage-balancing scheme has been proposed in [11], the main focus was given to the control system of z-source converter. In this paper, small-signal analysis is done to obtain the transfer functions. From the Relative Gain Array (RGA) analysis, the Multi-Input Multi-Output (MIMO) converter model is replaced by two distinct Single-Input Single-Output (SISO) systems. Hence, the voltage and current controllers are designed based on the control principles of SISO systems. The bipolar boost converter provides the possibility of adopting a double droop control method to fulfill the simultaneous power sharing and voltage regulation of DG units in the host microgrid.
The paper is organized as follows: in section 2, we provide a cursory review about the bipolar dc microgrids and their operation principles, while the voltage balancing issue in this type of microgrids is discussed. In section 3, the small-signal analysis is carried out. The design of voltage and current controllers and the power-sharing scheme for bipolar boost converter are explained in section 4. Finally, an islanded bipolar dc microgrid is simulated and studied in section 5.

BALANCING CHALLENGE
The typical topology of a bipolar dc microgrid is presented in Fig.1(a) whose salient feature is the availability of two voltage levels including dc V and 2 dc V , distributed by three wires throughout the microgrid. The bipolar dc microgrid can be considered as an effective solution to the applications where a high level of reliability and efficiency is required. In this type of microgrid, in case of loss of one wire, the microgrid continues to supply the loads connected between two remaining wires. Furthermore, it is possible to connect each of loads to both terminals; while one terminal feeds the load, the other is kept as the spare terminal. Hence, if one terminal is lost, the load supply can be switched to the other terminal. Providing two voltage levels, heavy loads can be connected to 2 dc V which in turn reduces the ohmic power losses to the half.
The original topology of bipolar dc microgrids requires a central voltage balancer aiming to keep the voltages of upper and lower terminals equal, in particular, under unbalanced loading. Hence, any malfunction in the central voltage balancer may affect the overall functionality of the whole microgrid. To cope with this problem, this paper proposes a distributed voltage-balancing strategy in which all DERs play their own role as shown in Fig.1(b). It is obvious that the conventional dc/dc boost converters, interfacing dc sources to the microgrid, are not capable of performing the voltage balancing duty, as they have only two wires at the output terminals.
So far, several topologies have been proposed to perform voltage-balancing function, some of them are able to buck their input voltages as well [12], [13]. However, to the best of our knowledge, no small-signal analysis is available for these topologies. Moreover, rather than buck converters, integrating the voltage balancing function with the boost converters is essentially preferred as the majority of DERs require boost converters for interfacing with the grid. Technically, a variety of dc/dc high-step-up (boost) converter topologies can be employed to incorporate voltage-balancing function into their controllers. However, those converters with two capacitors at their output terminals, such as three-level boost converter [14], [15], two-inductor boost converter [16], or interleaved boost converter with coupled inductors [17], are easier to be modified to fulfill the voltage-balancing function.
Three-level boost converter has originally been proposed to reduce the voltage stress on power switches by 50 percent as compared with the conventional boost converter [14]. In [18], authors developed an interleaved version of the threelevel boost converter which balances the neutral-point potential mainly to cope with the neutral-point drifting problem. In this paper, for the integration in bipolar dc microgrids, the design of the conventional three-level boost converter is changed so that it fulfills the proposed voltagebalancing strategy, in particular, in the presence of unbalanced loads.

III. MODELLING AND SMALL-SIGNAL ANALYSIS
The bipolar boost converter is shown in Fig.2(a). As known, in the original topology only two wires are available at the output terminal. However, for the application in bipolar dc microgrid, we extend the neutral point in order to serve as the zero-potential wire in the grid. Referring to the switching  times of S 1 and S 2 , as they are shown in Fig.2 V , respectively. To design the voltage and current controllers, small-signal analysis is required in order to obtain the transfer functions [19]. The dc input-to-output transfer functions can be achieved by conducting simple circuit analysis on the dc equivalent circuit of the converter, which is shown in Fig.3(a). The inputto-output transfer functions are where 1 D and 2 D are the nominal duty cycles of S 1 and S 2 respectively. The loads connected to upper and lower terminals are, respectively, denoted by 1 R and 2 R . Fig.3(b) shows the small-signal equivalent circuit based on which the small-signal transfer functions are derived. In this circuit, lower case letters represent the small-signal values, while capital letters correspond to dc values.
The converter's output impedance, seen from the upper terminal, and the input-to-output transfer functions are 1 1 . Equivalent circuit of the bipolar boost converter, (a) dc equivalent circuit, and (b) small-signal equivalent circuit.
Following the same procedure, the transfer functions relating 2 o v to 2 d and 1 d are obtained The input-to-output and output impedance transfer functions, seen from the lower terminal, are respectively given in (4) and The transfer function model of the converter including the voltage controllers is shown in Fig.4. The small-signal value of the reference voltage is r v , which is treated as disturbance, along with 1

Parameter
Operating-point Value Values of the capacitors and inductor are given in Table 1, which are chosen to satisfy the design criteria for keeping the converter in the continuous conduction mode and limiting the output voltage ripple below one percent of the nominal voltage. It should be noted that at the operating point, 1 D and 2 D are equal, since the converter is supposed to have the equal output voltages in the steady state condition. By β β substituting the values given in Table 1 in (3), (4), and (8) where [ (0)] ij G and 1 [ (0)] ji G − are the th ij element of the converter dc gain matrix and the th ji element of the inverse of the dc gain matrix respectively. The detailed procedure of calculating RGA is presented in [20]. The calculated RGA matrix is given in (21).
Rows and columns of the obtained RGA matrix are associated with the outputs and inputs, respectively. The small diagonal values imply weak interactions between the two input-output pairs of

B. current controller
In the grid-connected mode, the converter's terminal voltages are imposed by the grid. So, it is favourable to have the converter work in the current-control mode. In this mode, the converter should inject certain currents dictated by the references. In order to design an appropriate current controller, we need to derive control-to-output transfer functions which reveal the interactions between the duty cycles (i.e., 1 D and 2 D ) and each terminal of the converter. Performing circuit analysis on the equivalent circuit shown in Fig.3(b) leads to the following transfer functions: Substituting the operating-point values given in Table 1 in (27) and (25) and ignoring the effect of (24) and (26) on 1 i and 2 i respectively, the control-to-output transfer functions can be written as,

C. power sharing scheme
One of the interesting features of the bipolar boost converter is the possibility of implementing two droop controllers for each converter. As the upper and lower terminals of bipolar boost converter can be considered decoupled, as described by (21), one droop controller can be used to modify the reference voltage of the upper terminal, and the other to modify the lower terminal's reference voltage. Droop gains can be obtained where 0 V is the no-load voltage, while the full-load voltage (or the lowest allowable voltage) is fl V . fl I is the full-load current of each terminal (for example, if the rated power of the converter is 10 kW, the power of each terminal will be 5 kW, which will have the full-load current of 12.5 A). The two droop gains 1 k and 2 k will then be used to calculate voltage reference for the upper and lower terminals of the converter, are the reference voltages of control loops. 1 I and 2 I are, respectively, the upper and lower terminals' currents (see Fig.6) passing through low-pass filters with the cutoff frequency of 5 Hz in order to remove highfrequency ripples.

V. SIMULATION RESULTS
An islanded bipolar dc microgrid consisting of three DGs and five dc loads is shown in Fig.7. The rated power of both DG#1 and DG#2 is 10 kW which are connected to the microgrid by means of bipolar boost converters. So, each terminal of their respective converters can supply 5 kW to the grid. DG#3 is also connected to the microgrid by the bipolar boost converter;  however, it works in the current-control mode. The control loops of the upper and lower terminals are shown, respectively, in Fig.8(a) and Fig.8(b), where the constatnt currents of 8 A and 3 A are injected to the upper and lower terminals, respectively. The design parameters of DG#1, DG#2, and DG#3 are given in Table 1. All three DGs are simulated considering their detailed switching models. The microgrid accommodates five dc loads whose rated powers, rated currents, and switching times are given in Table 2. It should be noted that the nominal voltage of Load#5 is 800 V (i.e., connected between -400 V and +400 V) while Load#1 Load#2 are constant power loads connected to the upper and lower terminals, respectively. The impedances of line#1 and line#2 are assumed to be 1 = 0.5 R Ω and 1 = 200 L H µ , and  The no-load voltage is considered to be 400 V, while the lower limit of the nominal voltage, when the converter supplies the full-load current of 12.5 A, is 380 V. The two droop gains 1 k and 2 k will then be used to calculate references for the voltage control loops of DG#1 and DG#2 (each DG has two droop gains, see Fig.6). The loading profile of the upper and lower terminals are depicted in Fig.9. Subsequent to the considered loading, the main busbar voltages are shown in Fig.10. As it can be seen from this figure, the main busbar voltages (i.e., 1 V and 2 V in Fig.7) are not exactly regulated to 400 V due to the voltage drift imposed by the droop controller. However, voltages are kept balanced while the upper and lower terminals supply different load currents. The output currents of DGs are shown in Fig.11. The upper terminal loads are changed at t =0.55, 0.7 and 0.85 sec. For instance, during the time interval t =[0.55-0.7] sec, the load current is 31.25 A, which ideally, DG#1 and DG#2 should supply the equal portion of 11.6 A, as DG#3 injects the remaining 8 A. However, the output currents of DG#1 and DG#2 are slightly different from the ideal value, which can be attributed to the effect of line impedances. The lower terminal currents 12 I , 22 I , and 32 I supplied by DG#1, DG#2, and DG#3 respectively, are shown in Fig.11(b). DG#1 and DG#2 share the load current while DG#3 supplies constant current of 3 A to this terminal.
To further evaluate the performance of the bipolar boost converter in voltage balancing, the voltages and currents of the upper and lower terminals of DG#1 are shown in Fig.12(a) and (b), respectively. It can be seen that despite different loading profiles for the upper and lower terminals (for instance at t =0.8 sec), the output voltages are kept within the allowable margin (i.e., 380-400 V).

VI. CONCLUSION
In this paper a decentralized control strategy for voltage balancing and power sharing in bipolar dc microgrids was proposed. The microgrid DGs are interfaced with the grid by means of bipolar boost converters with the ability of boosting the input voltage and balancing the voltage of its two output terminals, particularly under unbalanced loading. Having done the small-signal analysis, the converter's transfer functions were derived. Using the RGA analysis, the highly coupled input-output pairs were identified, which helped to replace the MIMO control system of the converter by two SISO systems. To obtain a proper power sharing and voltage regulation in bipolar dc microgrid, the control system of each DG was modified by adopting a double droop control method. The effectiveness of the proposed topology and the double droop control strategy was evaluated through simulation studies conducted on an islanded bipolar dc microgrid involving unbalanced loads. From the simulation results, the proposed approach was found successful in providing a balanced voltage throughout the microgrid. By virtue of the proposed double droop control strategy, an accurate power sharing among the DGs in the host microgrid was obtained.