Logarithmic droop-based decentralized control of parallel converters for accurate current sharing in islanded DC microgrid applications

Remarkable progress in distributed power generation invigorated research into DC micro-grids using a controllable DC–DC converter contingent to the functionalities therein. The study explores the circulating current issue in a parallel-connected DC–DC converter and its associated current-sharing capability. Despite, the conventional droop being a promising solution with less control complexity, the current sharing is achieved at an expense of voltage drop. The prime objective of this research work is to design a control strategy that guarantees a minimized circulating current with proper current sharing for a parallel-operated DC–DC converter. This paper puts forth a dual reference-based control to alleviate the droop and load effects of the system. The primary reference is generated by considering the effect of a marginal change in the input power of the converters. The virtual resistance-based droop technique utilizes the prime reference for secondary droop reference generation. A current sharing algorithm is designed to assimilate adaptability to the scheme running under a variable line and load parameters. The comprehensive approach to the proposed control strategy has the advantages of improved current sharing and voltage regulation. A parallel DC–DC converter with the proposed control mechanism was investigated using MATLAB/Simulink and validated experimentally.

liberalization of the economic markets, remarkable progress has been made in the field of microgrids during the past decade.DC microgrids were identified as an appealing choice owing to their flexible integration with renewable energy resources.
DC microgrids also hold the merits of high efficiency and better compliance with consumer electronics requirements [5].The tapered expense of power electronics, non-existent reactive power issues, and frequency regulation problems have made DC microgrids reliable and resilient at the domestic level [6].The demand for self-sufficiency at an affordable price in residential distribution paved the way not only for extensive research into DC-DC converters but also the development of sustainable cities and societies [7,8].Different architectures of lowvoltage DC (LVDC) distribution systems have been discussed in the literature [9]. Figure 1 depicts the structure of autonomous low-voltage DC microgrid, which constitute source system, PEI and load system.The review of the literature in the area of DC microgrids suggests that a voltage range of 48 to 120 V in a DC system has gained significant attention because the energy distribution is efficient when compared to a 380-V DC system [10].
With diversified penetration of sources like solar, battery fuel cell, bidirectional power flow exists between the generating sources and the prosumers transforming the DC microgrid model as hybrid [11].During the islanded operation of a DC microgrid, the parallel connection of converters in PEI is important owing to its extended power level at the output, its modularity, and easy maintenance.
These converters as power electronic interface need to cope with the uncertainties while satisfying the key power prospects.In DC microgrid, converters and inverters have low inertia which reduces stability of the system [12].Emulating the inertia of inverters through the regulation of DC bus voltage, equips the inverter to operate as virtual synchronous generator [13,14].In autonomous microgrids, the virtual synchronous generator fails to regulate the DC bus voltage due to negligible grid support.Hence the DC bus voltage regulation need to be carried out at the parallel connection of the PEI especially in DC microgrids.
Hence, the current research work focuses on parallel operation of converters and its control for voltage regulation.The control of Dc microgrid includes voltage regulation of the common DC network in the parallel connection, the distribution of the load, energy storage, and monitoring [15][16].
The intermittent nature of renewable energy sources tends to cause sudden changes at the output of parallel-connected converters [17,18].Reasons for the flow of circulating current through the system include unequal output voltage at the common DC bus, parametric variations in terms of the line resistance, and quantitative error processing.The current research work focus on achieving accurate current sharing between the parallel converters and minimizing the circulating currents through efficient voltage regulation.Hence, simplified model of parallel DC converters feeding a common load is used for research study as shown in Figure 1(b).
Different control strategies like centralized current control, master slave technique, droop method have been proposed in the literature to address the present research problems [19][20][21][22].The droop control strategy is widely employed since it is decentralized, wireless, and simple.The conventional droop control for power sharing is achieved at the expense of voltage deviation.The droop control method has proven to be simple and efficient method that allows for linearization of the voltage in connection with increments in the output current [23].A generalized droop approach for current sharing proposed in [24] requires a prerequisite of line resistance parameters.Although a droop scheme based on the average value of the DC output could restore the nominal DC bus voltage, this was proven ineffective in achieving accurate load sharing during low-speed communication and its failure [25,26].Load sharing tends to be inaccurate owing to the effects of line impedance.Although an adaptive voltage positing system with virtual resistance was load independent and robust, it was unable to satisfy the objective of accurate load sharing when the complex line impedances were taken into account [27].In [28], the inclination of droop resistance was varied to achieve proper power sharing, but the deviation owing to the droop was not addressed.The exchange of information between neighbouring modules of a distributed system requires a sparse communication network to tune the droop curve [29].Most droop techniques employ a static droop value irrespective of instantaneous voltages and currents.An estimation of droop values based on a figure of merit called the droop index was proposed in [30], which needs extensive tuning of droop parameters.A compensation-based virtual impedance droop control introduced in [31,32] regulates the power at the output based on the load applied and concentrates on power loss reduction.The reference [33] inoculates similar kind of control that compensates the DC voltage at common bus through signal injection.
An agent-based coordinated control and power management system proposed in [34,35] also requires a communication network for load sharing.Although, the communication-based control using state estimation of microgrid through log mapping technique and least mean square algorithm was inoculated in [36,37], the programming involved makes the process quite tedious.An approach to achieving optimal efficiency in the application of multi DC conversion systems was put forth in [38].In [39], a control that tunes the droop curve according to the load was introduced.However, it offers poor regulation under light-load conditions owing to improper selection of the droop variable.Other studies suggested in the literature [40,41] robust non-linear control of low-voltage DC microgrids, which still suffer from large current-sharing errors under specific load conditions.Despite the extensive research on the linear and non-linear droop control of DC microgrids, the tradeoff between load sharing and voltage regulation still needs to be addressed to improve the load sharing.Reference [42] proved that the linear approximation of droop characteristics achieves high and balanced voltage regulation and load sharing.
Given these developments, this paper focuses on achieving improved current sharing with the linear approximation of a droop scheme in combination with a current-sharing algorithm in a parallel-operated DC converter.
The main contributions of the paper are as follows: • Design and implementation of novel logarithmic droop technique to estimate linearly approximated droop coefficient for nullifying the droop effect at the source side.• Implementing the droop compensation to transform the output voltage set point to nullify the droop effect at the load side.• Implementing the current sharing algorithm to achieve equal current sharing among the converters and to reduce the circulating current.• Hardware realization and detailed analysis of current sharing among the converters with the proposed control strategy, when the line resistance are equal and unequal.
The key advantage of the proposed technique, is the advantage of logarithmic function.In boundless algebraic perspective, using a logarithmic function, the marginal effect of a concerned variable that has an effect on another variable is made linear because it interprets the variable change as a percentage change.This helps to estimate the linearly approximated droop coefficients, thereby facilitating a fundamental change in the variables if any non-linearity persists.
The combination of droop control and an instantaneous current-sharing algorithm addressed in the present work improves the current-sharing capability in parallel DC converters.In addition to enhanced load sharing, the circulating current proportional to the current-sharing error is drastically minimized by means of a formulation of correction factor in the algorithm.
The paper is organized as follows.Section 2 analyses the circulating current and load sharing.Section 3 discusses the proposed logarithmic-based droop control in combination with current-sharing control.The simulation and experimental validation analysis are carried out in Section 4. Section 5 concludes the paper with a discussion of future directions.

CURRENT SHARING ANALYSIS
A parallel DC-DC converter connected to a common load fed from renewable energy based controlled source is shown in Figure 2. A circulating current analysis is carried out assuming that the voltage fed from the renewable source is a constant DC supply.
V DC1 and V DC2 and I 1 and I 2 are the corresponding input voltages and currents to the converters.Let V a and V b be the output voltages, I La and I Lb be the output currents, and R a and R b be the line resistances of converter-A (Conv-A) and converter-B (Conv-B), respectively.V L and I L are the corresponding load voltage and current.Figure 3 illustrates an equivalent circuit of the parallel converter feeding the load.By applying a superposition theorem to the above circuit, currents I La and I Lb are obtained considering the voltage drop due to the line resistances as negligible, the equation follows as depicted below: Practically, the voltage drops owing to line resistances R a and R b is dominant.Hence, we obtain the currents: Substituting Equation ( 4) into Equation (3), we get, Similarly considering the voltage drop across R b , the current I Lb is given by: The current sharing accuracy can be expressed as: The generalized expression for circulating current that flows from converter-a is as follows.When the line resistance connected are equal, i.e. for " When the line resistance connected are not equal, i.e. for The expression for the circulating current suggests that currents I La and I Lb should be equal in order to achieve minimum circulating currents.This is possible by regulating the voltages by adjusting the equivalent resistances R a and R b of Conv-A and Conv-B, respectively.

Basic concept of droop control method
Figure 3 shows the equivalent circuit of a simple microgrid configuration.The current-sharing error is minimized by the inclusion of a virtual resistance at the output of the converter.Thus, the purpose of load sharing is fulfilled.This is termed a linear droop or virtual resistance-based droop control method [28].If R Da and R Db are the droop resistances, added to R a and R b of converter-A and converter-B, respectively, then the droop control is mathematically expressed as: where V ref , i , V nom , R Di , and I i represent the voltage reference, nominal bus voltage, virtual droop resistance, and output current of the ith converter, where i is a and b respectively.The virtual droop resistance can be computed by assuming the voltage deviation (ΔV i ) and the maximum current rating of the converter (I max ).The virtual droop resistance is given by: The proper selection of the droop resistance minimizes the current-sharing error, which is given by: percentage of current sharing error = Current error Total current × 100

PROPOSED CONTROL STRATEGY
Different control strategies have been proposed in the literature, which consider only local measurements [9,13].The proposed control strategy in the current work combines a logarithmic droop mechanism and an instantaneous current-sharing algorithm.This control mechanism generates a dual reference based on regulation of the power and droop voltage.A correction factor is incorporated to further nullify the current-sharing error, and thus the load sharing is improved.

Novel logarithmic droop control
The proposed logarithmic droop strategy utilizes the output power capacity of the converter and the power at the load as prime attributes in the design of the droop equation.Utilizing the nuances of the logarithmic function, the designed droop control expresses variables in percentage errors.This facilitates the linearization of the droop characteristics in case of nonlinearity.In particular, when converters are loaded with a constant power load, the system is prone to instability owing to the negative resistance.The logarithmic function incorporated in the equation stabilizes the system by identifying the exponential rise in the voltage deviation.The dual reference generation is depicted in Figure 4 and overall control diagram is depicted in Figure 5.
The difference between the rated and observed power is estimated and expressed as a percentage error through the logarithmic function.This helps to linearize and approximate the marginal effect of the power difference on the voltage, which tends to nullify the droop effect at the source side The V ref reference generated can be expressed as: where P_conv i is power at the ith converter; P_ref i is the designed power capacity of ith converter, m i is the droop coefficient.

Droop compensation
Although the droop equation formulated in Equation ( 13) nullifies the droop effect owing to the source-side parameters.
Owing to the load effect, the voltage drop at the common bus is at stake.Hence, dual reference generation is put forth in this research work.The droop compensation utilizes the reference generated via the logarithmic droop as a prime reference, and the virtual resistance R D is calibrated based on the novel droop method established in [28].A new reference is generated based on the virtual resistance compensation, as shown in Equation ( 14).
where V * ref is the new generated reference.The virtual resistance R D is also estimated based on a preassumed value of the error deviation method.The accuracy in determining the virtual droop resistance is indefinite.This leads to the most possible approximation of the generated reference, and the accuracy of load sharing under a step change in load is at stake.The droop resistances of converter-A and converter-B remain the same since the converters are identical.The estimated virtual resistance R D of 0.13 is shown in Figure 6.

Current-sharing algorithm
In the current scenario, to drastically nullify the current-sharing error, a correction factor (CF) is required to be incorporated in the controlling phenomenon.The correction factor depends solely on the loading factor K. This loading factor K is the ratio of the converter currents, which is proportional to their respective resistances and their equivalent loading.
The correction factor is also derived based on the load currents.For accurate power sharing, the converter currents will be equal.
For inaccurate current sharing, the converter current equation is as follows: The deviation due to the intermittent variations at the supply side, is regulated by the algorithm.The algorithm was designed such that it also enables accurate power sharing.The flowchart in Figure 7 depicts the control algorithm that facilitates the accurate current sharing.
The flowchart in Figure 7 is represented in pseudocode as follows:  If the sum of converter currents equal to the load current, proceed to step 6, else proceed to step 4. 6.If the ratio of converter currents is unity, approximate the correction factor to 1, and stop the process.7. Else, calculate the loading factor K. 8. Calculate the correction factor using the loading factor and load current and process the current error.9. Stop the process.

Cascade control design and stability analysis
The dual droop reference generated through the droop algorithm utilized to process the voltage and current error by means of proportional integral (PI) controller.Based on the generalized transfer function model of the boost converter, steady state frequency analysis is carried out to design the gain constants of the proportional integral controller.The control loop is as shown in Figure 8.The K pv , K iv are the control constants of the outer voltage loop and K pi , K ii are the control constants of inner current loop.Since the power stage of the converter, the reference and the control constants are the same, the control signal also remains identical.The identical control signals tend to synchronize the pulses through which the impact of the time delay is negligible.The Figure 9 depicts the synchronized pulse pat-FIGURE 10 Step response of inner current loop tern of the parallel converters.The inner current loop utilizes the output current of the converter as feedback to process the current ratio and the outer voltage loop is used to regulate the voltage and the mismatch is overcome by the droop algorithm.The average small signal model of DC converter [43] whose closed loop transfer function model are given in Equations ( 21) and (22).The transfer function model of the DC converter is utilized to determine the control coefficients of Proportional-Integral control of the current and voltage control.The closed loop transfer function of inner current control (G ic ) is: The closed loop transfer function model for outer voltage control (G vc ) is: The cascade control loop is tuned using the frequency analysis plots.The step response tuning of the open loop current control is depicted in Figure 10.The rise time of the system step response is 0.0261 s (s) and the setting time of the response is 0.0465 s.The Bode analysis of the system is also carried out The control coefficients of both the converters are designed identical since the converters are identical and voltage regulation for any change is load is completely taken care by the proposed algorithm through adjustment of reference.
The impact of communication time delay in the control algorithm is assumed less significant due to the fact that small change in delay will not affect the stability of the system [44].
Although the communication delay prospect in the proposed algorithm is not considered, the delay time based on the step response of the system is incorporated in the pulse width modulation block of the control circuit.This reduces the error between the time to error signal sampling and the time to duty update at the trial of the forthcoming switching cycle.The delay time of both the converters are designed identical which is 6.5 micro seconds, ensure the converters to operate synchronously.

SIMULATION RESULTS AND DISCUSSION
The combined logarithmic droop and instantaneous-currentalgorithm control of parallel-operated DC-DC converters proposed in the research work are simulated using MAT-LAB/Simulink.The simulation study is carried out considering that the output voltages at the converters are different with variations in the supply voltage.The simulation parameters for the research study are depicted in Table 1.

Without droop control
The parallel operation of converters without droop control is investigated with the line resistance being equal in one case and different in the other.For the same line resistance, the individual converters in parallel are connected to an output cable resistance of 100 mΩ.For different line resistances, the parallel converters are connected to resistances of 100 and 150 mΩ.The simulation results clearly illustrate improper current sharing between the parallel converters.The results are listed in Table 2.
Case (1): Same line resistance Figure 13(a) shows the output voltages of converter-A, converter-B, and the load.

With proposed droop control
The proposed droop control regulates the voltage under variable supply conditions for the same cable resistance of 100 mΩ.Although the droop reference generation based on droop coefficient calculations does not nullify the current error, it helps to regulate the voltage to a satisfactory level.The droop control coefficients for converter-A and converter-B are m a = m b = 0.016259.Figure 15 illustrates the droop curve for the converter.The droop reference generated is less adaptive.Hence, the online calculation of the droop resistance according to the change in irradiation is estimated, and a new improvised reference is generated.The voltage and current errors are processed through a linear PI controller.The correction factor K is calibrated based on the current ratio to further assimilate the adaptability to the scheme and to succinct the converter currents for significant modifications on the either source or load side.
Case (1): Same line resistance Figure 16(a) shows the output voltages of converter-A, converter-B, and the load.The current spectrum during the change in load is depicted in Figure 18, in which the total load is increased by 25%.
The percentage of the current-sharing error for the same resistance is 0.6%, and the current-sharing error for different resistances is 1.6%.The simulation results illustrate that accurate load sharing is achieved even for a step change in the load.
Irrespective of the choice of converter and the controller, the percentage of the current-sharing error remains the prime attribute in determining the power quality through the circulating current.Table 3 depicts the better performance of the pro-  posed control algorithm in terms of the current-sharing error when a simulation analysis is carried out.

EXPERIMENTAL VALIDATION
To validate the proposed control strategy, a laboratory prototype of a parallel DC-DC converter is built, as shown in Figure 19.Real-time control is carried out using the real-time interface (RTI) DSPACE RTI 1202 [MicroLabBox].Table 4 lists the specifications of the boost converter used in the experimental validation.
The regulated power supply at the supply side of the boost converter is assumed to be an irradiation variation of PV.A maximum 15% of variation is maintained between the boost converters so as to create terminal voltage difference at the input side of the converters.A constant power load is connected as a common load at the parallel connection.In a closed-loop environment, the control signal is appropriately applied to PWM block set to generate the switching pattern.Since maximum current drawn from Dspace board is 13 mA, the opto coupler-based gate driver is used to fire the MOSFET.An experimental validation is carried out for a maximum load current of 0-3 A for each converter, considering safety.Table 4 lists the parameters for the experimental laboratory prototype.

Without droop control
A hardware-in-the-loop simulation of parallel converters connected to a common lamp load without any control is carried out.The currents at converter-A and converter-B are measured using a current clamp and are 0.95 and 1.75 A, respectively.The output voltages of converter-A and converter-B are measured using a differential probe and are 48 and 47.9 V, respectively.These values are listed in Table 5.
Equal load sharing can take place only when the individual converter currents are equal to 1.35 A. An experimental validation of the voltage waveform, current-sharing pattern, and circulating current waveform without a controller is shown in Figure 20(a) and Figure 20(b), respectively.

With proposed control
The output voltage and current waveforms for the proposed control strategy are illustrated in the figures below.Irrespective of the ratings of the converters, the results clearly indicate that the proposed control strategy exhibits superior performance in terms of accurate load sharing.In the novel droop method, current sharing is achieved at the expense of a voltage drop owing to the increased droop resistance.In the proposed method, the trade-off between the output voltage and the load sharing is  6.The robust control of the system can be analysed by varying the load.The total load current at the common bus is varied from 1.8 to 3.5 A, and a current-sharing analysis is carried out.Figure 23 depicts the step increase in load, voltage at the common bus, and current profile of the converters.Table 7 lists the voltage and current of the converters and load after a sudden change in load.Since the scope of the current work is limited to the control of the converter for accurate current sharing, this paper illustrates the current-sharing profile, and a robustness analysis will be carried out in future work.
Irrespective of the system rating and supply parameters, the percentage of current-sharing error in the power-sharing phenomenon remains the same analytically.From this perspective, traditional DC-DC converters exhibit 18.1% of current-sharing error [28] with the novel droop method and 2.1% with the DI method [27].The current-sharing error is only 1.4% to 1.81% with the proposed controller.The comparative analysis of the current-sharing error put forth the superior performance of the proposed method, as shown in Figure 24.Table 8 illustrates  the comparative analysis of the current sharing among different methods.

CONCLUSION
The paper provides an insight into the current-sharing capability of the parallel operation of a DC-DC converter for DC microgrid applications.The scope of the current research work was limited to minimization of the circulating current and currentsharing error.In this paper, the proposed combination of the droop method and an instantaneous current-sharing algorithm drastically reduced the current-sharing error.
The proposed logarithmic droop linearizes the marginal effect of power on voltage, through which the load-sharing accuracy is improved, and the algorithm implemented in the current work offers adaptability in the control for a variable load.This feature is quite feasible when the converters are integrated with renewable energy sources such as solar PV.The proposed method is simple since the control is carried out with the measurement of local parameters.
The proposed droop control strategy proved to be efficient in terms of current sharing and minimizing the circulating current.Simulation and experimental results suggest the superior performance of the proposed droop technique for parallel DC-DC converter.Since the proposed droop scheme focuses on current sharing, the introspection of robustness analysis will be future direction of research.

FIGURE 1
FIGURE 1 (a) Structure of autonomous LVDC microgrid.(b) Single line diagram of microgrid model for current research

FIGURE 2
FIGURE 2 Parallel connection of DC-DC converter with circulating current flow

FIGURE 3
FIGURE 3 Single-line diagram of parallel DC converter

FIGURE 4
FIGURE 4 Pictorial representation of dual reference generation

FIGURE 5 FIGURE 6
FIGURE 5 Proposed overall control loop block diagram of converter

FIGURE 7
FIGURE 7 Flowchart of proposed algorithm implemented in control

FIGURE 8 FIGURE 9
FIGURE 8 Cascade control loop of converter

FIGURE 11 (
FIGURE 11 (a) Open loop Bode plot of inner current control.(b) Closed loop Bode plot of current control

FIGURE 12 ( 1
FIGURE 12 (a) Open loop Bode plot of outer voltage control.(b) Closed loop Bode plot of voltage control

TABLE 2 69 FIGURE 13
FIGURE 13 (a) Output voltages of converter-A, converter-B, and load.(b) Output currents of converter-A, converter-B, and load Figure 13(b) depicts the output currents of converter-A, converter-B, and the load for the same line resistance of 100 mΩ connected to the output terminals of the converters.Case (2): Different line resistances of 100 and 150 mΩ Figure 14(a) shows the output voltages of converter-A, converter-B, and the load.
Figure 14(b) depicts the output currents of converter-A, converter-B, and the load for different line resistances of 100 and 150 mΩ connected to the output

FIGURE 14 (FIGURE 15
FIGURE 14 (a) Output voltages of converter-A, converter-B, and load.(b) Output currents of converter-A, converter-B, and load

FIGURE 16 (
FIGURE 16 (a) Output voltages of converter-A, converter-B, and load.(b) Output currents of converter-A, converter-B, and load Figure 16(b) shows the output currents of converter-A, converter-B, and the load.Case (2): Different line resistances The simulation analysis is carried out with different line resistance.Figure 17(a) depicts the output voltage waveforms of converter-A, converter-B and load.Figure 17(b) shows the output current waveforms of converter-A, converter-B and load.Case (3): For step change in load

FIGURE 17 (
FIGURE 17 (a) Output voltages of converter-A, converter-B, and load.(b) Output currents of converter-A, converter-B, and load The hardware-inloop simulation in DSPACE-1202 facilitates data acquisition and the control.The model-based input-output (I/O) integration and control desk software helps evaluation during run time.The Analog input and output voltage range of the Microlab box is −10 V to +10 V and has digital I/O channels for pulse generation measurement and connection to sensors.The voltage transducer LEM-LV-25P and current transducer LEM-LA-55P shown in Figure 19 are used to sense and scale down the voltage to match the input and output range of Microlab box.

FIGURE 19 TABLE 5 4 FIGURE 20
FIGURE19 Hardware setup of DC-DC converter

FIGURE 21 (
FIGURE 21 (a) Voltage waveforms with controller of same resistance.(b) Current waveforms with controller of same resistance

FIGURE 22 ( 6 Case ( 3 )
FIGURE 22 (a) Voltage waveform with controller of different line resistance.(b) Current waveform with controller of different line resistance

FIGURE 23 ( 7 FIGURE 24
FIGURE 23 (a) Voltage and current waveform for same resistance.(b) Voltage and current waveform for different resistance

TABLE 3
Simulated results with proposed droop controller

TABLE 4
Parameters for laboratory prototype

TABLE 8
Analysis of current-sharing error through different methods CF Correction factor DC Direct current DG Distributed generation G ic Closed loop transfer function of current loop G vc Closed loop transfer function of voltage loop I 1 Input current to the converter-A I 2 Input current to the converter-A I ca Circulating current from converter-A I i Output current of ith converter I La Output current of converter-A I Lb Output current of converter-B I max,i Maximum current from ith converter K Loading factor K ii Integral coefficient of current loop K iv Integral coefficient of voltage loop K pi Proportional coefficient of current loop K pv Proportional coefficient of voltage loop LVDC Low voltage direct current m i Droop coefficient of ith converter N Number of converters P_convi Power output of ith converter P_refi Designed power capacity of converter PEI Power electronic Interface R a Line resistance connected to converter-A R b Line resistance connected to converter-B R Di Droop resistance of ith converter R L Load resistance connected to common point V a Output voltage of converter-A V b Output voltage of converter-B V da Drop due to line resistance V DC1 Input voltage to converter-A V DC2 Input voltage to converter-B V nom Nominal voltage of common bus V ref * New reference generated V ref,,i Reference voltage fed to the ith Converter ΔV i Voltage deviation of ith converter