A current reference-enhanced strategy endows the GFC in DC microgrids better dynamic response

The constant voltage strategy (CVS) is more suitable for the small-capacity dc microgrid applications to form the dc bus voltage because it can eliminate the steady voltage deviation. However, its dynamic performance is slowed down by the integrator in the voltage loop and negatively influenced by the load steps. Some advanced methods have been conducted in the literature. However, their principle is to increase the system bandwidth, which is limited because the bandwidth is generally designed about one-tenth of the switching frequency and cannot be increased infinitely. This work pays efforts to increase the gain within the system bandwidth to accelerate the transient response and simultaneously eliminate the influence of the integrator. Therefore, the non-integrator disturbance observer-based (NIDOB) strategy is proposed in this work, and it can feed the load current into the current reference to replace the output of the integrator. Compared to the traditional and non-integrator directly-feedforward (NIDF) strategy, it has a better dynamic performance. Compared to the bandwidth-increased strategy, it does not introduce more noise. The theoretical analysis and experimental results prove its advantages.


Load disturbances
The popularity of renewable energy and the increasing global demand for power consumption leads to the emergence of dc microgrids [1][2][3] .The storage converter plays the role of the grid-forming converter (GFC) in some small-capacity applications such as buildings and residences 4,5 , to form the bus voltage and balance the power variation caused by the renewable energy and load-steps 6,7 .In addition, only one GFC can be used in the applications operating in Island mode to form its bus voltage, and it is appropriate to adopt the constant voltage strategy (CVS) 8 .A multi-branch GFC can be utilized to increase the capacity of a single GFC to meet the requirements 9 .However, although the CVS can eliminate the steady voltage deviation, its transient response is limited because of the finite system bandwidth as well as the controller limitations.
The input voltage and load variations generally influence the dynamic performance of a GFC.In particular, the load-steps happen much more frequently than the input voltage variations.When a load-step happens, a voltage sag may occur immediately.Therefore, many strategies are explored to improve the dynamic performance of a GFC, such as the average current mode control 10 , current-programmed mode control 11,12 , and voltage mode control 13 .With these control schemes, the influence of the load-steps on the output voltage can be effectively reduced, but it may not be eliminated.The integrator also influences the dynamic performance in the voltage control loop because its output has to change continuously.There has been no work investigating such influence so far.Some studies adopt the feedforward strategy to accelerate the transient response 14 adds an auxiliary loop to the original control plant, and 15 design the input voltage feed-forward strategy to suppress the input voltage influence.However, the two approaches focus on the specific applications with wide input voltage variation, and the influence of load-steps cannot be eliminated 9 , 16 design a feedforward loop to improve the dynamic response to the voltage sags, but its essence is to increase the bandwidth by adding an additional proportional loop.This approach requires a hysteresis comparator to avoid the negative influence on the steady performance.This method is relatively complicated and limited, since the larger bandwidth brings more noise 17 , 18 propose

Problem formulation
The traditional CVS employed in a GFC can be depicted in Fig. 1, which consists of four units, the voltage controller (VC), current controller (CC), current physical system (CPS), and voltage physical system (VPS).VC and VPS constitute the voltage loop, and CC and CPS constitute the current loop.From Fig. 1, it can be seen that the two proportional-integral (PI) controllers are included, where the inner PI controller (K PIi ) for CC and the outer PI controller (K PIu ) for VC, respectively.The two PI controllers are separately expressed as where k pu and k iu are the proportional and integral parameters in VC, k pi and k ii are the proportional and integral parameters in CC, respectively.
In this traditional CVS, the output can be depicted as where Therefore, the bus voltage (u C ) will be inversely influenced by the load current (i load ).When a voltage variation happens, the K PIu will generate the current reference (i ref ) for the inner current loop to achieve voltage deviation-free.The K PIi is utilized to help the output current (i L ) track i ref , so as to speed the current response.
If practical applications show poor dynamical performance, it is mainly due to an improperly tuned current loop.Therefore 16 , improves the dynamic performance by increasing the proportional gain of the current loop.However, it increases the inner bandwidth, and reduces the ability of anti-disturbance.In terms of parameter design, the bandwidth of the inner loop is generally designed one-tenth of the switching frequency and wider than that of the outer loop.Following the above rule, K PIi can tightly track i ref and does not introduce important noises.The dynamic performance is up to whether the K PIu can immediately generate a desired i ref .However, the transient response of an integrator is relatively slower than that of a pure proportional compensator.Therefore, this work focuses on generating the desired i ref and removing the integrator, instead of increasing the bandwidth of the current loop.The reason for the relatively slow transient response of the traditional CVS is detailly analyzed as follows.

The output of integrator changes continuously
The reason for the integrator in K PIu slowing the dynamic performance is first explained.Since the K PIu can achieve zero-deviation regulation, it can be considered that i L = i ref and u C = U ref in the steady-state.Assuming the system reaches a steady-state at t = t 1 , i ref can be calculated as because the output of the proportional loop is 0. If a load-step happens at this time, the recovery time is needed before the bus voltage goes back to U ref .
Assuming a new steady-state time is t 2 , then It means that the system cannot immediately change from the last steady state to a new one.In other words, since the output of the integrator must change continuously, the instantaneous voltage response will be slowed down.

Current reference limitedly respond the transient components of load disturbance
From Fig. 1, the CPS is coupled with VPS by the u C , which makes it difficult to figure out the relationship between i load and i ref .Therefore, a decoupling feedforward is added after K PIi in CC to conveniently illustrate how the control plant slows the load disturbances transmission.The original current loop can be described as then, the feedforward can be designed as to eliminate the influence of u C , where d denotes the decoupling feedforward.Therefore, the current loop can be described as and the decoupled control system is shown in Fig. 2.
From Fig. 2, the transfer properties from i load to i ref can be illustrated as (1) The current loop bandwidth is limited and only the IF components of i load can be responded, as shown in Fig. 3, which results that the transient |i ref | is smaller than the desired value and the bus voltage recovers slowly.Therefore, the most effective approach is to increase the gain over the IF band to enlarge the i ref as the black curve shows in Fig. 3, without the extra requirement for phase improvement.In this sense, two strategies are proposed and compared in Sect.3.

Proposed Strategies
In Fig. 2, it can be seen that i ref is the output of K PIu and the input of the current loop.But the steady component of i ref is only generated by the integrator of K PIu .Therefore, it is necessary to make its steady output zero to remove the integrator.An effective approach is to find a new variable to replace it.In addition, the gain over the IF band should be increased to accelerate the transient response without introducing more noise.Two strategies are proposed as follows.

The NIDF strategy
Considering i ref = i L = i load in the steady state, i L and i load can possibly function as substitution variables.However, since i L changes passively with i load , and the change is driven by the controller in GFC, it cannot instantly reflect the transient components of i load .Therefore, the approach that directly feedforwards i load into i ref is adopted.( 9) (10)  www.nature.com/scientificreports/Besides, since i load is equal to the steady output of the integrator, the integrator can be directly removed to achieve the non-integrator scheme.In this sense, the PI controller in the voltage loop can be simply designed as a proportional controller, which can be written in, K PIu = k pu .
The NIDF strategy can be depicted in Fig. 4, and the current reference can be expressed as

It can be calculated that
It means that the NIDF strategy can effectively feed the load disturbance into i ref .
Then, the output voltage can be expressed as It can be derived that Therefore, the NIDF strategy can achieve the constant voltage scheme, although the integrator in VC is removed.However, it can be also calculated that which means that the bandwidth of the NIDF strategy is infinite and more noises can be introduced, which is undesired.Besides, another drawback of this strategy is that it needs an extra sensor to measure i load , which increases the volume and investment cost of a GFC.

The NIDOB strategy
The traditional observer-based method has been utilized in many applications [21][22][23] , which can observe the disturbance avoiding using an extra sensor.Since the observer can only be used in the specific structure as shown in Fig. 5, some structure changes of Fig. 2 are made as shown in Fig. 6.Besides, the non-integrator scheme is also achieved through rigid reasoning and design.
The principle of the traditional observer-based method can be explained as follows.In Fig. 5, G n (s) is the control plant and X ref (s) is its reference input, X 0 (s) comes from the controller, Q(s) is a low-pass filter, D(s) is the outer disturbance, and D ob (s) is the observation.The output of the original system without the observer can be expressed as (12)    Equation (19) implies that the transfer property from the controller to the output is not changed, and (21) implies that the influence of disturbance will be eliminated because it can be observed and fed into the reference to reject the physical change.
The observation can be represented as Since Q(s) is a low-pass filter, the transient components over the high-frequency band will be filtered, and the components over the low and the IF band will be observed and fed.It can be derived that Therefore, the observer can be employed in CVS to feed i load into i ref .Before this, it is necessary to figure out G n (s) and X ref (s) first, where G n (s) comes from the physical system, and X ref (s) comes from the controller.
Comparing Fig. 2 with Fig. 5, let then, it is needed to move G cI (s) after i load to constitute the equivalent control plant (G n (s)) together with VPS.Therefore, the equivalent diagram of the NIDOB strategy can be shown in Fig. 6, where the equivalent disturbance and its observation can be expressed as The equivalent control plant is Substituting ( 8) into ( 23), it can be derived that www.nature.com/scientificreports/Therefore, the steady output of the integrator in the voltage loop can be replaced by the observation, and the integrator can be directly removed.
The output of the proposed NIDOB strategy can be expressed as where Because Q(s) is designed as a low-pass filter, (28) can be stated that and Therefore, although the VC is designed as a proportional controller in the NIDOB strategy, the CVS can be achieved and the load disturbance will be effectively eliminated.

Dynamic comparison of the proposed strategies
According to Fig. 6, the transfer property from i load to i ref can be expressed as Substituting ( 23) into (30), it can be reformulated that According to the law of DOB 23 , since the relative degree of G n (s) is 2, the Q(s) can be designed as a two-order filter as and its cut-off frequency can be calculated by no more than the cut-off frequency (ω s ) of the system, where τ f is the time constant of Q(s).In this work, the parameters used in the following theoretical analysis and experiment are gathered in Table 1 in Sect.5, where To illustrate the advantages of the NIDOB strategy, the gain of the current references of the three strategies are compared.Substituting (26) into (34) and then comparing it to (12), it can be found that, the components constituted by U ref of (12) and (34) are the same but a bit larger than that of (9), because the integrator exists in the denominator of the components of (9).Therefore, it is needed to solely compare the transient components constituted by i load in (34), (12), and (9).Hence, let where c tra denotes the component constituted by i load in (9), c DF denotes that in (12), and c DOB denotes that in (34).The gains over the IF band of the three methods are compared in Fig. 7.It can be seen that the NIDOB strategy has the largest gain over the IF band whereas the traditional strategy has the smallest, which means the NIDOB strategy can respond faster to the load disturbance.In addition, the gain of the NIDOB strategy over the highfrequency band decreases rapidly to reject the noises, which solves the problem caused by the NIDF strategy.

Stability comparison of the proposed strategies
According to the parameters listed in Table 1 and the Eqs.( 2), (14), and (28), the stability comparison can be rigidly verified by the locus distribution of the three strategies, as shown in Fig. 8.It can be seen that the traditional strategy has two dominant eigenvalues, but the NIDF strategy and the NIDOB strategy only have one dominant eigenvalue.Furthermore, the NIDOB strategy has the smallest dominant eigenvalue and it is almost offset by a zero, but the #1 dominant eigenvalue of the traditional strategy is close to zero.Therefore, the NIDOB strategy has the largest stability margin, but the traditional one has the smallest.In addition, the locus distribution also indicates that the NIDOB strategy has the fastest transient response, whereas the traditional one has the slowest, which is consistent with the dynamic comparison in Fig. 7.

Comparison of the NIDOB strategy and the Bandwidth-Increased Strategy
Many works improve the dynamic performance by increasing the system bandwidth 19,20 , which may have a faster dynamic response when load-steps happen, but it cannot recover quickly due to the too many introduced noises.As shown in Fig. 9, with the increase of k pu , the bandwidth of the traditional strategy increases, much larger than that of the NIDOB strategy.However, the gain over the IF band of the NIDOB strategy is larger than that of the bandwidth-increased strategy.Hence, the NIDOB strategy has a better dynamic performance than the bandwidth-increased strategy under the condition of the same bandwidth.

Sensitive analysis of the NIDOB strategy
Through analyzing the control structure in Fig. 6, it can be seen the stability of the proposed method is influenced by parameter r, k p , k s , L, C, and τ.
The sensitivity analysis is shown in Fig. 10.The positive number indicates that the parameter will cause the eigenvalues to shift to the right.On the contrary, the negative number indicates that the parameter will cause the eigenvalues to shift to the left.It can be seen that the increasement of r, k s , L, and τ will weaken the stability and the L influence the most.The inductor L will in the equipment are not easily changed, but the control parameter r, k s , and τ is adjustable.Therefore, r, k s , and τ should be set a relatively small value in parameter design.

Experimental results
An experimental islanded dc microgrid setup, shown in Fig. 11, was used to verify the effectiveness of proposed strategy.The experimental system consists of a dc source, three DC/DC converters (GFCs), LC filters, programmed loads, and a dSPACE controller as well as its monitoring platform.The DC source voltage is 200 V, and the output voltage reference of DC/DC converter is set as 100 V.The sampling frequency of dSPACE controller (39)

Case1: effectiveness of the NIDF strategy
The comparison among the NIDF strategy, the integrator-included DF strategy, and the traditional strategy is shown in Fig. 12, where three GFCs adopt different strategies and 500W loads are separately connected and disconnected to three GFCs at t = T 1 and t = T 2 .From Fig. 12(a1), (b1), and (c1), it can be seen that the three strategies have the same steady current reference equal to the load current, hence all of them can realize CVS.However,    Besides, since we directly introduce the load current into the current reference, they have an almost vertical increment at the moment of loads being connected.In addition, as shown in Fig. 12(a3), since the integrator can sense the voltage deviation and output a part of the current reference, the integrator-included DF strategy can respond faster than the NIDF strategy at the beginning of the transient process.However, because the output of the integrator has to change continuously, its current reference decreases more slowly than the NIDF strategy as the voltage deviation decreases.Therefore, the integrator-included DF strategy will cause a voltage overshoot, whereas the NIDF strategy can reach the steady-state more quickly.As shown in Fig. 12(b2) and (c2), the NIDF strategy reaches the steady-state after 0.5 s, whereas the overshoot of the integrator-included DF strategy has not fully disappeared yet.Therefore, the NIDF strategy has a better dynamic performance, but it needs an extra sensor to measure the load current.

Case2: effectiveness of the NIDOB strategy
In Sect. 5 A, the NIDOB strategy has a better dynamic performance.The observer-based strategy with the integrator and the NIDF strategy are compared with it in this case to illustrate the merits of the NIDOB strategy.
The loads are connected and disconnected at t = T 5 and t = T 6 , and it can be seen that the NIDOB strategy and the observer-based strategy can generate larger transient current references as shown in Fig. 13(a1)-(a3), hence they have less recovery time and voltage sag as shown in Fig. 13(b1), (b2) and (c1), (c2).As shown in Fig. 13(a2) and (c2), the steady current reference is equal to the load current, hence it is reasonable to directly remove the integrator, which will not influence the realization of CVS.However, since the observation in the observer-based strategy can provide a large enough current reference to make the bus voltage recover very fast, the integrator functions a little and just generates the minor components of the current reference.Therefore, the transient current references of the NIDOB strategy and the observer-based strategy with the integrator overlap with each other (see Fig. 13(a3)), and they have similar dynamic performance (see Fig. 13(b2) and (c2)).Although there is no obvious improvement after removing the integrator, it can simplify the control structure.

Case3: comparison of the NIDOB strategy and the bandwidth-increased strategy
To illustrate the advantage of the NIDOB strategy, the bandwidth-increased strategy is compared to it in Fig. 13, where k pu = 1 in the NIDOB strategy and k pu = 5 in the bandwidth-increased strategy, and the loads are separately connected and disconnected at t = T 7 and t = T 8 .In Fig. 9, the bandwidth of the bandwidth-increased strategy is larger than that of the NIDOB strategy, but its gain over the IF band is smaller than that of the latter.Therefore, the current reference of the bandwidth-increased strategy has a faster response but has a smaller value than that of the NIDOB strategy, as shown in Fig. 14(a1), (a2).Correspondingly, the bandwidth-increased strategy has a smaller voltage sag, but the NIDOB strategy has much less voltage recovery time, as shown in Fig. 14(b1), (b2) and (c1), (c2).

Case4: Dynamic performance of the NIDOB Strategy under supply voltage disturbance
To evaluate the dynamic performance of the NIDOB strategy under supply voltage disturbance, the validation is conducted in Fig. 15.It can be seen that the supply voltage decreases to 195 V and 190 V separately at t = T 9 and t = T 10 , but the output voltage waveform only produced small fluctuations.Therefore, the proposed strategy has good performance under supply voltage disturbance.

Conclusion
This paper brings forward a NIDOB strategy to improve the dynamic performance of a GFC.This method feeds the load current into the controller to enhance the current reference to accelerate the transient response.In addition, since the feedforwarded value in the steady-state can completely replace the output of the outer integrator, the integrator can be directly removed and the voltage loop can be simply designed as a proportional controller.The proposed method improves the dynamic performance by increasing the gain over the IF band rather than largely increasing the system bandwidth.Compared to the traditional strategy and the NIDF strategy, the NIDOB strategy has better dynamic performance (the recovery time is reduced from 0.5 s to 0.025 s), and it has a simpler control structure than the observer-based strategy with the integrator.Besides, compared to the band-increased strategy, the NIDOB strategy reduces much recovery time without introducing more noises (the recovery time is reduced from 0.35 s to 0.025 s).The theoretical analysis and experimental results can prove its advantages.The NIDOB method is analyzed and proposed for constant voltage control of DC/DC converter, but it is also promising to employ it inverter's control, because the AC component can be transformed into DC component through Park's Transformation.

Figure 1 .
Figure 1.The traditional CVC system.(a) An employed circuit of GFC.(b) The diagram of controller.(c) The physical system.

Figure 2 .
Figure 2. The decoupled control system.(a) The voltage loop.(b) The current loop.

Figure 3 .
Figure 3.The frequency properties of the traditional CVS and the desired gain of an advanced strategy.

Figure 5 .
Figure 5.The typical structure of the observer application.

Figure 7 .
Figure 7.The gain comparison of the three methods over the IF band.

Figure 8 .
Figure 8. Stability comparison of the three strategies.

Figure 9 .Figure 10 .
Figure 9. Dynamic comparison of the NIDOB strategy and the bandwidth-increased strategy.

Figure 12 .
Figure 12.The comparison of the NIDF strategy, the integrator-included DF strategy, and the traditional strategy.(a) The waveforms of current reference.(b) The waveforms of dc bus voltage.(c) The waveforms of load current.

Figure 13 .
Figure 13.The comparison of the NIDOB strategy, the observer-based strategy with the integrator, and the NIDF strategy.(a) The waveforms of current reference.(b) The waveforms of dc bus voltage.(c) The waveforms of load current.

Figure 16 .
Figure 16.Dynamic performance of the NIDOB strategy under desired voltage disturbance.
compared to the NIDOB strategy Traditional CV strategy The NIDOB strategy has faster dynamic performance Integrator-included DF strategy The NIDOB strategy has faster dynamic performance NIDF strategy The NIDOB strategy has faster dynamic performance DOB-based strategy with integrator The two strategies have almost the same dynamic performance, however, the NIDOB strategy has simpler control structure Bandwidth-increased strategy The bandwidth-increased strategy has less voltage sag, however, the NIDOB strategy has much less voltage recovery time

pu k iu k pi k ii
the switching frequency is 2 kHz, and the ω s is designed about 1000 rad/s.The τ f can be calculated by (36) no less than 0.7 ms, but the transient response will slow down as τ f increases, hence τ f is set as 2 ms in this work.

Table 2 .
Comparison results of strategies.