Weighted DC Virtual Generator Control Scheme for Interlinking Converters in DC Microgrids

This article proposes a new control strategy for power electronic converters interfacing two dc networks. The proposed control, based on a modification of the dc virtual generator concept, has grid-forming capability in both sides of the converter simultaneously without intermediate energy storage device. Departing from the dc virtual generator control concept, capable of controlling voltage in one of the converter sides while withdrawing the power from the other side, an adaptation is done to make it reversible. A priority weight is assigned to each side voltage control. A weight calculation method, proportional to the voltage deviation from its rated value, is proposed. This weight calculation strategy makes the control system to contribute to the side most in need (using voltage deviation with respect to rated value as an indicator of the neediness). The interlinking converter with the proposed weighted strategy is capable of stable work in grid-forming mode at both sides simultaneously, just requiring a grid-feeding converter with $P/V$ droop to be connected in either side. No control architecture was found in the literature with this capability for dc–dc interlinking converters. The theoretical discussion has been supported with small-signal analysis and hardware-in-the-loop validation.


I. INTRODUCTION
T HE increasing importance of distributed energy resources (DERs) interfaced by power electronic converters (PECs) has led to the appearance of microgrids [1].Although ac was traditionally more used, the benefits of dc has caused a gradual move toward dc distribution.These benefits include the easiness of integration of intermittent DER due to renewable sources (which normally requires extra dc-ac conversion stages in ac microgrids) and the reduced losses, due to the absence of reactive current flow, skin effect, and the reduced number of conversion stages [2], [3], [4], [5].
Despite of their advantages, microgrids present some challenges, such as their reduced inertia [6], [7], due to the substitution of rotating generators directly connected to the grid with PEC interfaced elements or the significant presence of renewable energy sources, which are often controlled to produce the maximum available power.This led to the appearance of the virtual generator concept [8], emulating the inertial characteristic of the rotating machines with the control strategy.
Interlinking converters play a key role to enhance reliability.Interlinking dc-dc converters can be used for connecting more than one dc microgrid to increase stability by having the capacity of exchanging power [4] or to have different voltage levels in the same microgrid [5].These interlinking converters can also contribute to voltage regulation.A deep review of the capability to contribute to voltage regulation in both sides of the interlinking converters depending on the control architecture is presented in [9], distinguishing between grid-supporting and grid-forming capabilities for the converter.In [9] and [10], converters are considered to have grid-forming capability if they behave as a voltage source.This is a change in the traditional naming of droop-controlled converters as grid-supporting units regardless they were current-source or voltage source-based [11].As stated in [10], this previous classification is less useful since it is not based on a fundamental difference between the sources.The definition used in [9] and [10] will be used throughout this article, especially for highlighting the necessity of at least one grid-forming converter in each network.Some solutions can be found in the literature for dc-ac converters interfacing an ac microgrid with a dc-link capacitor at the output of a converter connected to an energy source.This is done with a droop relating ac and dc voltage (V g /V dc droop) [12] or a dual droop, adding a dc voltage term to ac droop equations [13].
However, apart from being dc-ac converter cases, grid-forming capability is only supported in one output, having dual gridsupporting and single grid-forming capability [9].
Similarly, previous research is reported both for ac-dc [14] and dc-dc [15] interlinking converters that consider additional grid-forming units in either side.In the event an interconnected network losses its grid-forming unit, the interlinking converter will give voltage control capability to that network by power transfer from the grid-forming unit in the other side.This can be done without control scheme switching and operation mode detection.However, this method needs a grid-forming unit in one of the networks to have a reversible grid-forming capability with seamless transition.
The strategy proposed in [8] for ac-dc converters has dual grid-forming capability, i.e., the capability of controlling voltage in both sides of the converter to reasonable levels, just requiring some type of grid-feeding converter with grid-supporting strategies in one of the networks.No control architecture has been found in the literature with dual grid-forming capability for dc-dc converters [9].
This article proposes a new control strategy based on the well-known dc virtual generator (DCVG) concept [8], [16], [17].From the DCVG scheme, an adaptation is done to make the control reversible, i.e., to use the same voltage control scheme for controlling the voltage in either side.The proposed modification considers a weighted approach, in which a priority can be given to each side.By considering a weight calculation strategy proportional to the deviation with respect to the rated voltage, dual grid-forming capability is achieved: the proposed method provides grid-forming capability to both outputs at the same time.It only requires a grid-feeding converter with a P/V droop in any side to keep the voltage in both grids at reasonable levels and without requiring communication between the converters.As compared to the DCVG control, the proposal is able to: 1) adapt to change in configurations of the network, for example, automatically adapt to islanded mode (reversible grid-forming capability) and 2) provide grid-forming capability to both outputs simultaneously, just requiring an appropriate grid-supporting element in any side (dual grid-forming capability).
The rest of this article is organized as follows.Section II presents the grid used for this study.Section III describes the proposed control, whose stability is analyzed in Section IV.Section V explains the case study, presenting the different grid configurations that are considered for the analysis.Section VI presents the results obtained with hardware-in-the-loop (HIL) for the validation of the proposed control.Finally, Section VII concludes this article.

II. PROPOSED DC MICROGRID TOPOLOGY
The proposed study uses a subgrid from the one explained in [18].The schematic representation is shown in Fig. 1.As it can be seen, it includes a 375 Vdc bus and a 48 Vdc network connected to it using an interlinking PEC.The 48 Vdc network will be referred as converter side 1, while the 375 Vdc bus connection is named side 2.   The converter with the proposed control strategy is labeled as DCVG, meanwhile in each of the dc networks there is another dc-dc converter contributing to voltage regulation.PEC2 represents a connection to the mains, simplified here as a dc-dc converter, always operating as a grid-forming unit.PEC1 can represent any DER operating in grid forming or grid feeding with P/V droop.PEC1 and PEC2 control is shown in Fig. 2, meanwhile interlinking control (marked as DCVG in Fig. 1) is explained in the following section.

A. Weighted Dc Virtual Generator (WDCVG)
The proposed control, shown in Fig. 3, is obtained from the conventional DCVG scheme [8], [16], [17], with some modifications in order to be able to control the voltage in either side of the converter (V dc1 or V dc2 ) when the DCVG is acting as an interlinking converter.Current control, not included to keep the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
figure readable, has the same structure as shown in Fig. 2 for PEC1 and PEC2.
The proposed control is based on combining a voltage control strategy focused on each side of the converter by applying a weighted average.This is shown in (1a) and (1b), being w 1 and w 2 the weight corresponding to sides 1 and 2, respectively, with 0 ≤ w x ≤ 1 and w 1 + w 2 = 1.V dc and I ff represent the averaged voltage and feedforward current, obtained from output voltage and current in side 1 (V dc1 , I o1 ) and side 2, but referring the latter ones to side 1 (V dc2 , I o2 ') For adapting the voltage in side 2 (V dc2 ) to side 1 (V dc2 ) a change in voltage base is applied, but also changing the sign of voltage deviation.For example, dc1 , being ΔV dc2 the per unit voltage deviation in side 2. The change in sign of the voltage deviation is due to the fact that for increasing the voltage in one of the sides, the required power has opposite direction to the one required for increasing the voltage in the other side.The required transformation is shown in the following equation: For adapting side 2 current to side 1, I o2 is calculated as the current in side 1 output which will produce the same power output I o2 produces in side 2. This is done by multiplying the output current I o2 by V dc2 /V dc1 and changing its sign.The corresponding expression is shown in the following equation: Weights w 1 and w 2 can be fixed according to different criteria.An upper control level could set it depending on the grid situation.For example, if one of the grids has another converter with grid-forming capability, the weight corresponding to that side can be set to 0. By doing this, the DCVG will only control the voltage in the other side, acting as a grid-forming unit for that network.If both grids have grid-forming units, the weights can be set dynamically, taking into account which grid is closer to reach any saturation limit or has more critical loads, lower DER participation, or lower inertia.
Another option is to do a complete switch in the side whose voltage is being controlled attending to some voltage threshold.For example, initially w 1 can be set to 1 (and w 2 = 0) meanwhile voltage in side 2 is within some range.If voltage in side 2 exits the predefined range, the interlinking control starts controlling that output (w 1 = 0 and w 2 = 1).This is similar to the voltage margin control found in [20].

B. Proportionally Weighted DCVG (PWDCVG)
This article proposes a weight calculation strategy proportional to the voltage deviation with respect to the rated value in each converter side.This is shown in (4), where ΔV dc1 and ΔV dc2 are each side per unit voltage deviation and w x is the weight of side 1 or 2 By doing this, the control focuses more on the side, which is further away from nominal voltage, without requiring any upper layer control to infer it from the grid conditions.Additionally, the proposed system avoids any sharp transition between the side whose voltage is being controlled produced by sudden changes of weights w 1 and w 2 .Instead, a smooth and continuous change in the priority for each side is achieved, resulting in smoother transients.
By following this strategy, equal per unit voltage deviation is achieved in both sides of the converter in steady state, as it will be shown in Section VI.This can be proven by finding the value, which makes V dc in (1a) equal to V * dc1 , because due to the virtual resistance decoupling, equilibrium point fulfills 5) is obtained.V dc1 and V dc2 are expressed in terms of V * dc1 , taking into account V dc2 is obtained by changing the sign of the per unit voltage deviation measured in side 2 as shown in (2) dc1 and with the cancelation of all terms independent of ΔV dc1 and ΔV dc2 due to the fact that w 1 + w 2 = 1, the expression in ( 6) is obtained Finally, w 1 and w 2 can be substituted by the formula in (4).After simplifying the denominator, ( 7) is obtained.The only solution for this equation is ΔV dc1 = ΔV dc2 , obtaining, in steady state, w 1 = w 2 = 0.5 As a final remark about the proposed mechanism, it has to be highlighted that with the weighted strategy the interlinking converter can behave as a grid-forming converter for both sides simultaneously, just requiring an additional grid-feeding unit with P/V droop in either side.This will be demonstrated in Section VI.

IV. STABILITY ANALYSIS
The stability of the proposed control is analyzed by using the state-space model.The state-space matrix, together with the input and state variables, are shown in (8) to (13).The matrices and terms in red are time varying.Thus, linearization is required to obtain the small-signal model for the analysis.I o1 and I o2 were considered as system inputs.However, this will generally not be the case when looking to the complete system.One or both sides of the converter will normally be connected to a voltage source through a line impedance, which provides the required power.In that case, the corresponding input I ox should be replaced by , thus affecting both A and B matrices Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. x The proposed PWDCVG is compared with the traditional DCVG scheme applied to side 1 (w 1 = 1) or to side 2 (w 2 = 1).Since w 1 = w 2 = 0.5 in steady state for PWDCVG (as shown in Section III-B) the application of the WDCVG with a fixed value of 0.5 for the weights is also considered.The system parameters are shown in Table I.
To prove the reversible and simultaneous grid-forming capability of the proposed control, two different scenarios are considered.Case V s1 − I o2 : side 1 is connected to a voltage source that provides the power demand by a current source in side 2. Case I o1 − V s2 is the opposite scenario.The eigenvalues for both scenarios are shown in Fig. 4, where it can be seen that only PWDCVG and w x = 0.5 are stable in both scenarios, with w 2 = 0 having an eigenvalue at the origin and w 1 = 0 at the right half-plane in the case where the voltage source is in the opposite side of the one with weight equal to 0.
The step responses of the same cases for voltage outputs V dcx are shown in Fig. 5, where a variation of a step of magnitude 0.1V * dcx and I nx (nominal current in side x) is applied to the corresponding input.It can be seen that only PWDCVG and w x = 0.5 are stable in all scenarios, but PWDCVG is faster and has less overshoot.

V. CASE STUDY
The microgrid used as the case study is shown in Fig. 1, with all the grid and converter parameters included in Table I.For checking the operation of the converter in different scenarios, transitions between different configuration modes of the microgrid are introduced by connecting/disconnecting PEC1 and PEC2 units.The different operation modes of the microgrid are shown in Table II.

TABLE II DIFFERENT OPERATING MODES OF THE MICROGRID
Four different scenarios are considered.For the first three, PEC1 operates as grid-forming unit.Each of the 4 scenarios listed there correspond to one subsection in the same order they appear.So scenario 1 is presented in Subsection VI.A, scenario 2 is presented in VI.B and so on.1) Step in weights setting different values to show how sending weight references can change the way the WDCVG contributes to both networks.Converters in mode A1.
2) The strategy based on voltage margin control is followed.
Initially, w 1 is fixed to 1 (w 2 = 0), making the WDCVG only contribute to control in side 1 (48 Vdc network).
Starting in mode A1, there is a transition to mode B1, leading to an uncontrolled voltage in side 2 (375 Vdc network).When the voltage in that network goes outside some limits, w 1 is changed to 0 and w 2 to 1 so that the WDCVG starts to control the voltage in side 2. 3) Proportionally WDCVG is used in the same scenario shown for voltage margin control case in order to compare both scenarios.A longer HIL simulation for proportionally WDCVG is also shown to demonstrate how transition between different scenarios affects the grid situation, with transitions between A1, B1, and C modes.4) Apart from that, a HIL simulation equal to the last mentioned one is done but changing PEC1 to be a grid-feeding unit with droop control (transitions between A2, B2, and C modes).

VI. HIL RESULTS
The proposed control has been validated by using a HIL setup, composed of a Typhoon HIL404 platform and a TMS320F28335 TI control card.The HIL time step is 1 μs, the sampling time of the converters control is 100 μs and their switching frequency is 10 kHz.The data shown in this article have been captured with a rate of 10 kS/s.

A. Steps in Weights
The first scenario is done for the weighted DCVG generator working with different values of weights which are step-wise varied.The results are shown in Fig. 6.Power is expressed using active sign convention: positive for generation, negative for consumption.
The initial situation is with w 1 = 1 (w 2 = 0).Starting from no load, at t = 0.1 s a step of 2 kW in the power demanded by the load in 48 Vdc network.At t = 0.3 s, a step of 2.4 kW in the power demanded by the load in 375 Vdc network.At t = 0.5 s, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the power demanded by the loads in both networks is set back to 0 kW.w 1 (w 2 ) is changed to 0.5 (0.5) at t = 0.6 s and to 1 (0) at t = 1.2 s.After these two changes, the same steps in demand mentioned for w 1 = 1 are used.
When w 1 = 1 (w 2 = 0), the control is focusing only in the voltage control in 48 Vdc network.It can be seen that in that interval the control of the DCVG is not affected by the power changes in the other side (t = 0.3 s).
When w 1 = w 2 = 0.5 (after t = 0.6 s), the control is contributing in the same proportion to both sides.It can be seen that, in this case, the control reacts to power changes in both grids (t = 0.7 s and t = 0.9 s).
After the second vertical line, w 1 is set to 0 (w 2 = 1), having the opposite of the initial situation.The control is only focusing in the 375 Vdc network and not affected by power changes in the other network (t = 1.3 s)

B. Weighted DCVG With Voltage Margin Control
In order to check the capability of autonomous voltage control, when any one side is lacking a grid-forming converter, a weight calculation strategy similar to voltage margin control is studied.The weight of the DCVG is set to 1 in normal operation, thus focusing on voltage control in the 48 Vdc network.When the voltage in the 375 Vdc network goes outside a pre-set dead-band (370-380 Vdc in this case), the weight is changed to 0, switching to voltage control in the other side.
If the margin control strategy is applied to the weighted DCVG, the results shown in Fig. 7 are obtained.Starting in mode A1, with PEC1 and PEC2 operating as grid forming, load references in each side are changed.At t = 0.1 s, the load in the 48 Vdc network is set to −2 kW (demand) and changed to 3 kW (production) at t = 0.7 s.At t = 0.3 s, the load in the 375 Vdc network is set to −2.4 kW and changed to 1.6 kW at t = 0.5 s.It can be seen that load changes in the 375 Vdc network are totally absorbed by PEC2, since the DCVG converter is only focused in side 1 when the voltage in side 2 is within the limits.
After the vertical line, the only converter controlling the voltage in 375 Vdc network (PEC2) is disconnected.When the voltage goes out of the specified limits, the DCVG starts to control the voltage in that side, setting the dc voltage reference to 380 V.This allows the converter to be able to work in both directions.However, sudden changes in the control mode also causes fast voltage variation when there is no converter in voltage control in a given grid.It should also be remarked that, when similar conditions occur at the two sides, it would be difficult to select which one to give priority.

C. PWDCVG (With Grid-Forming Converters in Both Networks)
To solve the previous two problems, the proposed weight calculation strategy obtains each weight proportional to the voltage deviation with respect to the rated values in each converter side.In that way, the voltage deviation from the reference point is used as a measure of the neediness of each side and the priority is set proportional to it.This leads to weights within the range 0 to 1 and results in a smooth operation during grid situation changes.This can be seen in Fig. 8, which is the same situation (configuration and load sequence) of Fig. 7 but with the explained weight calculation strategy.The proposed strategy reacts smoothly to PEC2 disconnection, reducing the rate of change of the voltage in the uncontrolled grid.
Fig. 9 shows a longer sequence of changes for a complete demonstration of the operation of the converter in different grid configurations.Starting in mode C (PEC1 off, PEC2 acting as grid forming), the sequence of changes is the following.
1) t = 0.1 s: load in 48 Vdc network is set to −2 kW (demand).2) t = 0.3 s: load in 375 Vdc network is set to −2.4 kW (demand).3) t = 0.4 s: PEC1 is connected, acting as grid forming, changing operating mode to A1. 4) t = 0.5 s: load in 375 Vdc network is set to 1.6 kW (production).5) t = 0.7 s: load in 48 Vdc network is set to 3 kW (production).6) t = 0.8 s: PEC2 is disconnected, changing operating mode to B1. 7) t = 1.0 s: load in 375 Vdc network is set to −2.4 kW (demand).8) t = 1.4 s: PEC2 is connected, acting as grid forming, changing operating mode to A1. 9) t = 1.6 s: PEC1 is disconnected, changing operating mode to C. It can be seen that the control is able to act as grid-forming converter in any of the sides keeping the voltage stable in both sides, even if the other grid-forming converter in the corresponding network is disconnected.Apart from that, when both PEC1 and PEC2 are operating, the DCVG reacts to load changes in both sides, making it possible to each PEC to contribute to the load in the other network.
Current outputs are shown too, where it can be seen that the ripple in the current is not very significant.The ripple is higher in 375 Vdc network (side 2) because side 1 is the one with the LC filter (see Fig. 1).
Finally, a comparison in the per unit voltage at each side of the converter is shown.As it was demonstrated mathematically, it can be seen that, in steady state, the voltage deviation is equal in both sides.This means that it is able to reach a balanced situation in which both networks are in similar neediness as seen from both converter output voltages.

D. PWDCVG (With Grid Feeding)
The proportionally WDCVG has the capability of acting as grid forming in both networks at the same time, with no need of Fig. 9. HIL results for PWDCVG with PEC1 as grid forming.From top to bottom: power in side 1 (48 Vdc network); current in side 1; voltage in side 1; power in side 2 (375 Vdc network); current in side 2; voltage in side 2; per unit voltage in each side; each side weight value.
other grid-forming unit in either side.Only one converter acting as grid feeding with droop control is required for the proposed control to be able to control voltage in both sides at the same time.This is shown in Fig. 10, with exactly the same scenario shown in Fig. 9, but changing the grid-forming unit in 48 Vdc network by a grid-feeding unit with P/V droop.Thus, the sequence is the one presented before, but changing modes A1 and B1 for A2 and B2.It can be seen that even when the only grid-forming unit in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 10.HIL results for PWDCVG with PEC1 as grid feeding with P/V droop.From top to bottom: power in side 1 (48 Vdc network); current in side 1; voltage in side 1; power in side 2 (375 Vdc network); current in side 2; voltage in side 2; per unit voltage in each side; each side weight value.
the system apart from the DCVG interlinking converter (PEC2) is disconnected after the second vertical line, the interlinking converter is able to reach stable situation forming both grids at the same time.
In this case, the voltage deviation is higher than for the case with grid-forming units in both networks due to the P/V droop required for the grid-feeding unit to contribute.However, this situation will only happen due to contingencies and a secondary control can recover the voltage level.Besides that, voltage oscillations are more significant due to the lower capability of grid-feeding elements to contribute to voltage support.

VII. CONCLUSION
This article proposes a new control strategy based on a modification of the DCVG to adapt the control to make it able to control voltage in either side of the converter, providing grid-forming capability.A weight is given to each side so that the control focuses more on that side.
The proposed weighted DCVG is designed using weights proportional to the deviation of each converter side voltages from rated point.This makes the control to prioritize the grid, which is more in need.Small-signal analysis was performed to study the stability of the proposed method.
The performance was validated with Typhoon HIL404 platform.The results show the capability of the converter of operating in any scenario, provided that at least in one of the grids there is a converter providing the required power, either with grid-forming units or grid feeding with P/V droop.No control architecture was found in the literature with this capability for dc-dc interlinking converters.It was shown that, in steady state, the proposed control achieves the same voltage deviation in both sides, obtaining a good balance between the neediness of both.

Fig. 1 .
Fig.1.Grid used for HIL validation.Shadowed boxes show filter and converter configuration and control variables for each subgrid converter (PEC1 and PEC2) and for interlinking converter (DCVG).All converters have a synchronous buck configuration, being the inductor filter in the low voltage side (V dc1 in the case of the DCVG).

Fig. 5 .
Fig. 5. Step response for PWDCVG and different values for w 1 and w 2 .Left column: case V s1 − I o2 ; right column: case I o1 − V s2 .First row: response of V dc1 to change in V sx .Second row: response of V dc2 to change in V sx .Third row: response of V dc1 to change in I ox .Fourth row: response of V dc2 to change in I ox .Unstable responses are shown in a small window in the corresponding plot.

Fig. 6 .
Fig. 6.HIL results for step changes in weights with converters in mode A1.Vertical lines indicate change in weight values.From top to bottom: power in side 1 (48 Vdc network); voltage in side 1; power in side 2 (375 Vdc network); voltage in side 2; each side weight value.

Fig. 7 .
Fig. 7. HIL results for weighted DCVG with voltage margin control with PEC1 as grid forming.Vertical line indicates change from mode A1 to mode B1.From top to bottom: power in side 1 (48 Vdc network); voltage in side 1; power in side 2 (375 Vdc network); voltage in side 2; each side weight value.

Fig. 8 .
Fig. 8. HIL results for PWDCVG with PEC1 as grid forming.Vertical line indicates change from mode A1 to mode B1.From top to bottom: power in side 1 (48 Vdc network); voltage in side 1; power in side 2 (375 Vdc network); voltage in side 2; each side weight value.
Eigenvalues for PWDCVG and different values for w 1 and w 2 .Left column: case V s1 − I o2 ; right column: case I o1 − V s2 .Top row: the general view with all the eigenvalues; bottom row: a zoom to cover all the dominant eigenvalues.A zoom is presented in the bottom right figure to distinguish overlapping eigenvalues.