Grid-Coupled Dynamic Response of Battery-Driven Voltage Source Converters
Ciaran Roberts, Jose Daniel Lara, Rodrigo Henriquez-Auba, Bala Kameshwar Poolla, Duncan S. Callaway
aa r X i v : . [ ee ss . S Y ] J u l Grid-Coupled Dynamic Response of Battery-DrivenVoltage Source Converters
Ciaran Roberts ⋆, ‡ , Jos Daniel Lara † , Rodrigo Henriquez-Auba ⋆ ,Bala Kameshwar Poolla † , and Duncan S. Callaway ⋆, † ⋆ Department of Electrical Engineering and Computer Sciences † Energy and Resources GroupUniversity of California, BerkeleyBerkeley, CA 94720 { ciaran r, jdlara, rhenriquez, bpoolla, dcal } @berkeley.edu Abstract —With the increasing interest in converter-fed is-landed microgrids, particularly for resilience, it is becoming morecritical to understand the dynamical behavior of these systems.This paper takes a holistic view of grid-forming converters andconsiders control approaches for both modeling and regulatingthe DC-link voltage when the DC-source is a battery energystorage system. We are specifically interested in understandingthe performance of these controllers, subject to large loadchanges, for decreasing values of the DC-side capacitance. Weconsider a fourth, second, and zero-order model of the battery;and establish that the zero-order model captures the dynamics ofinterest for the timescales considered for disturbances examined.Additionally, we adapt a grid search for optimizing the controllerparameters of the DC/DC controller and show how the inclusionof AC side measurements into the DC/DC controller can improveits dynamic performance. This improvement in performanceoffers the opportunity to reduce the DC-side capacitance givenan admissible DC voltage transient deviation, thereby, potentiallyallowing for more reliable capacitor technology to be deployed.
I. I
NTRODUCTION
As synchronously connected power systems shift towardssystems with high penetration of converter-interfaced genera-tion (CIG), it becomes more critical to understand the dynam-ical and transient behavior of these systems. These converter-dominated power systems are already prevalent in the formof islanded microgrids, motivated by increased resilience tonatural disasters [1], [2]. Recent work has explored the small-signal stability of the DC/AC converter and its interaction withthe grid. A common approach when analyzing the voltagesource converter (VSC) behavior is to model the DC-side ofthe converter as an ideal voltage source [3], [4]. On the otherhand, when studying the dynamics of the DC-side, the gridis often simplified as a resistive load [5], [6]. From a small-signal perspective, an independent analysis of each subsystemseparately may be adequate due to the minimal interaction oftheir control loops. However, this approach gives little insight ‡ Corresponding author. This research was supported in part by the Director,Cybersecurity, Energy Security, and Emergency Response, Cybersecurityfor Energy Delivery Systems program, of the U.S. Department of Energy,under contract DE-AC02-05CH11231. Any opinions, findings, conclusions,or recommendations expressed in this material are those of the authors anddo not necessarily reflect those of the sponsors of this work. This work wassupported by the National Science Foundation under grants CPS-1646612and CyberSEES-1539585. This work was also supported by the LaboratoryDirected Research and Development (LDRD) at the National RenewableEnergy Laboratory (NREL). into the dynamical behavior of these coupled systems duringgrid-scale transient events, particularly, faults or large loadsteps and when the operating conditions differ substantiallyfrom the steady-state operating point used in the linearization.This paper explores the performance of the DC-link capac-itor of a battery energy storage system (BESS) subject to AC-side disturbances, under different DC-side control strategies.The objective of these control loops on the DC-side is totightly regulate the DC voltage across a DC-link capacitor. TheDC-link capacitors act as energy buffers and support a constantvoltage on the DC-side of the CIG. A tight regulation of thisvoltage is critical to the operation of the CIG, as momentarydrops in this voltage restrict the VSC’s power productioncapabilities [6]. Therefore, large electrolytic capacitors areused in order to have a substantial buffer to minimize theDC voltage deviations during disturbances. These capacitorsbeing typically bulky, expensive, unreliable are one of the mostcommon modes of failure in power electronic systems [7]-with system transients and overloading identified as two ofthe primary causes of failure [8].One proposed improvement in converter design is to replacethese electrolytic capacitors with small film capacitors thatare more robust and reliable [9]. As the DC-link capacitanceis reduced, voltage fluctuations during transients increase asthere is a momentary mismatch between the power injectedinto the grid and the power supplied from the DC source e.g., abattery. In order to deploy these small film capacitors, the DC-side control must rapidly correct any difference between thesecurrents to ensure adequate AC-side operation and minimizetransient over-voltages on the capacitor.In this work, we examine different control approaches forminimizing the required DC-link capacitance of a BESS.Specifically, we consider the case of a grid-forming in-verter supporting an islanded microgrid with a BESS as itsDC source. Grid-forming inverters differ from grid-followinginverters–the dominant mode of operation today, in that theformer behave as a controllable voltage source behind acoupling reactance [10]. Consequently, they do not directlycontrol their power injection into the grid but rather controlthe frequency and amplitude of their output voltage [4]. Theirpower injections, therefore, inherently increase or decreaseto balance any changes in load. When choosing a DC-linkapacitor to regulate the DC voltage of a grid-forming inverter,adequate care must be taken that it is appropriately sized toensure satisfactory behavior under the largest expected loadchange and/or fault conditions, thus presenting challenges inthe sizing of the DC-link capacitor for grid-forming converters.This work considers the existing measurements used in thecontrol loop of grid-forming converters as inputs into theDC/DC controller to predict the evolution of DC-link capac-itance dynamics and consequently, improve the regulation ofthe DC bus voltage.For modeling our DC source, we consider a Li-ion batteryas the BESS. In comparison to previous work which modeledthe battery as an ideal voltage behind a resistor [5], [6], weemploy a model of the battery which captures the dynamicsof the electrochemical processes as we increase/decrease thecurrent drawn from the battery. Furthermore, as we reduce theDC-link capacitance and the dynamics on the DC-side becomefaster, it may become more important to model the underlyingbattery dynamics to accurately capture the dynamical responseof the DC source [11].The contributions of this paper are as follows:1) we develop a full-order dynamical model for a battery-driven voltage source converter,2) we examine the impact of battery chemistry dynamicson overall DC-side dynamical response and establish that azero-order model captures the dynamics of interest for thedisturbances considered,3) we improve upon the DC-side controller in [6] by theinclusion of AC-side measured quantities to predict evolutionof DC-side dynamics to compensate for the DC/DC controllerdead-time and DC/DC inductor dynamics,4) we show that, for particular parameterizations of inner-control loops, the behavior of the VSC can help reduce therisk of saturation of the VSC modulation index.II. S
YSTEM M ODELING AND C ONTROL I MPLEMENTATION
A. Grid Forming VSC Control Scheme
Fig. 1: Grid-forming VSC control scheme.
The modeling and simulation of the AC-side, includingthe VSC, is implemented in a Synchronous Reference Frame(SRF), with the mathematical model defined in per unit.The ( dq ) -frame quantities are represented in bold, lower-casecomplex space vectors of the form: x = x d + jx q . The proposed control model depicted in Fig. 1 is based on astate-of-the-art VSC control scheme described in [12]–[14].The power calculation unit computes the active and reactivequantities given by p c + jq c = e g ¯ i g where (¯ . ) denotes thecomplex conjugate. This is followed by an outer control loopthat consists of active and reactive power controllers providingthe output voltage magnitude v c and frequency ω c referencesby adjusting the predefined set points ( x ⋆ ) according to ameasured power imbalance: ω c = ω ⋆c + R pc ( p ⋆c − ˜ p c ) , v c = v ⋆c + R qc ( q ⋆c − ˜ q c ) , (1)where R pc , R qc denote the active, reactive power droop gainsand ˜ p c , ˜ q c represent the low-pass filtered active, reactive powermeasurements of the form: ˙˜ p c = ω z ( p c − ˜ p c ) , ˙˜ q c = ω z ( q c − ˜ q c ) , (2)where ω z is the filtering frequency. The outer-loop voltageset point may be passed through a virtual impedance block ( r v , l v ) , resulting in a cross-coupling between the d - and q -components via a terminal current measurement i g as ¯ v = v c − ( r v + jω c l v ) i g . (3)This new voltage vector set point and the frequency set pointare then fed to the inner control loop consisting of cascadedvoltage and current controllers operating in a SRF ¯ i s = K vp (¯ v − e g ) + K vi ξ + jω c c f e g + K if i g , (4a) ¯ v m = K ip (¯ i s − i s ) + K ii γ + jω c l f i s + K vf e g , (4b)where ˙ ξ = ¯ v − e g and ˙ γ = ¯ i s − i s denote the respectiveintegrator states; ¯ i s and ¯ v m represent the internally computedcurrent and voltage references, e g is the voltage measurementat the converter terminal to the grid, i s is the switching current, K p , K i , and K f are the proportional, integral, and feed-forward gains respectively, and superscripts v and i denotethe voltage and current SRF controllers. The output voltagereference ¯ v m combined with the DC-side voltage v DC generatesthe Pulse-Width Modulation (PWM) signal m .The electrical interface to the microgrid includes an RLC filter ( r f , l f , c f ) and an equivalent impedance ( r g , l g ) modeledin SRF and defined by the angular converter frequency ˙ i s = ω b l f ( v m − e g ) − (cid:18) r f l f ω b + jω b ω c (cid:19) i s , (5a) ˙ i g = ω b l g ( e g − v l ) − (cid:18) r g l g ω b + jω b ω c (cid:19) i g , (5b) ˙ e g = ω b c f ( i s − i g ) − jω c ω b e g , (5c)with v m representing the modulation voltage and v l denotingthe nodal voltage at the load bus. The system base frequencyis represented by ω b and equals the nominal frequency. Thecomplete state-space representation of a single grid-forminginverter, therefore, comprises 13 states of the form ˆ x vsc = (cid:2) e dq g , i dq g , i dq s , ξ dq , γ dq , θ c , ˜ p c , ˜ q c (cid:3) ⊤ . (6)The control input vector u vsc = [ p ⋆c , q ⋆c , v ⋆c , ω ⋆c ] ⊤ providesoperator set points. More details on the overall convertercontrol structure and employed parametrization can be foundin [4], [13], [14]. . DC-side model The modeling of the DC-side consists of a BESS, anidealized DC/DC buck/boost converter with an appropriatelysized inductor, and a DC-link capacitor. This interconnectedsystem is then interfaced to the VSC as shown in Fig. 2.
Fig. 2: DC-side model.
1) DC/DC Controller:
For the DC/DC controller in Fig. 2,we investigate the improved dynamical performance with theinclusion of the measured AC-side quantities into the controllogic. A dual-loop PI DC/DC controller is shown in Fig. 3and modeled as ˙ η = v ⋆ DC − v DC (7a) i ref = K v DC p ( v ⋆ DC − v DC ) + K v DC i η, (7b) ˙ ζ = i ref + i out − i in , (7c) d = K i DC p ( i ref + i out − i in ) + K i DC i ζ + K pred ∆ i out . (7d)The outer-loop (7a)-(7b), maintains a constant DC busvoltage while the inner loop (7c)-(7d), is for current tracking.The inclusion of a feed-forward term i out , in the internal PIcontrol loop is for improving the controller performance by theaddition of information about the disturbance. This disturbancewas primarily a set-point change of the VSC in previousworks [5]. For the case of a grid-forming VSC, however,this disturbance includes unexpected load changes where theadditional required power will be inherently drawn from theDC-link capacitor.The addition of the term K pred ∆ i out in (7d) is motivated by[15], where the authors sought to minimize the required DC-link capacitance for a converter-interfaced three-phase load.In [15] the authors note that the inclusion of a feed-forwardterm alone is inadequate to instantaneously balance the currentflow across the capacitor due to inherent system responsetime delays, mainly due to inductor dynamics. To offset thesedelays, we use a one-step predictor based on the forward Eulermethod to predict the evolution of system dynamics. The feed-forward predicted current, ∆ i out value is approximated by (8) ∆ i out ≈ ∆ P ∆ v DC ≈ T s ( v dm ˙ i ds + v qm ˙ i qs ) v DC , (8)where T s is the switching period of the DC/DC converter, ˙ i ds and ˙ i qs are calculated using (5a). We benchmark the im-provement in dynamical performance for a non-zero K pred against the controller in [5]. The advantage of a one-steppredictor over derivative control in a PID controller is thatwe can predict the evolution of the DC-side dynamics beforethey begin to manifest and minimize noise amplification inestimating the rate of change of the current. The duty-cycle d of the DC/DC converter in this work has a maximum value of . to mimic the behavior of a practical converter [16]. Fig. 3: Structure of the DC-side controller
2) Battery Model:
As previously outlined, prior work onthis topic modeled the electrochemical battery as an idealvoltage source behind a resistor [5], [6]. In the presence ofa large DC-link capacitance and consequently a large energybuffer, this is a reasonable modeling assumption. However,as we reduce the DC-link capacitance, the dynamics of theelectrochemical storage may become more important to model.A common method for parameterizing an equivalent circuitmodel for batteries is electrochemical impedance spectroscopy[11], [17]. This method measures the voltage response toharmonic current input across a frequency range of interest ( kHz to kHz [18]) and an equivalent circuit is adapted to thisdata. These experimental data show that at high frequencies( ≥ − Hz) the battery exhibits inductive behaviorwhile lower frequencies ( ≤ − Hz) have a morecapacitive response [11], [18], [19]. A generalized battery isshown in Fig. 4 where the high frequency behavior is modeledby a series of 2 RL parallel branches and the low frequencybehavior is modeled by a series of 2 RC parallel branches. +_ + _ + _ + _ Fig. 4: A generalized th -order battery model. Within this work we combine the two-time constant RC battery model from [20] with the two-time constant RL model from [18] as shown in Fig. 2. Both of these batteries’chemistries are based on Lithium-ion and offer reasonableinitial parameterization of a dynamic BESS model.
3) DC-side Electrical Model:
In practice, the DC/DC con-verter is a buck/boost converter capable of both charging anddischarging the battery. Here, we focus on the case whenthe converter is operating in the boost mode, i.e., supplyingpower to the grid. A similar analysis holds for the buck modeof operation. The per-unit averaged equations governing theelectrical behavior on the DC-side with the converter operatingin continuous mode, similar to [21], are then given by ˙ i l = ω b l b ( r b ( i b − i l )) , ˙ i l = ω b l b ( r b ( i b − i l )) , (9a) ˙ v c b = ω b c b (cid:18) i b − v c b r b (cid:19) , ˙ v c b = ω b c b (cid:18) i b − v c b r b (cid:19) , (9b) v b = v oc − i b r b − r b ( i b − i l ) − r b ( i b − i l ) (9c) v c b − v c b (9d) ˙ i b = ω b l DC ( v b − (1 − d ) v DC ) , (9e) ˙ v DC = ω b c DC ( i in − i out ) , (9f) v DC i in = v b i b , (9g)where ω b is the AC base frequency, d is the duty-cycle of theDC/DC converter, further discussed in Section II-B1, and i out is the current flowing into the AC grid and given by i out = p inv v DC = v dm i ds + v qm i qs v DC . (10)The full state-space model of the DC-side with a th -orderdynamic BESS model, denoted by ˆ x thDC , is given by ˆ x thDC = [ i l , i l , v c b , v c b , i b , v DC , η, ζ ] ⊤ , (11)with the control input u DC = v ⋆ DC . The nd -order model of theDC-side neglects the inductor dynamics of the battery (i.e.,retains only the 2 RC branches in Fig. 4), while the th -order model further neglects the dynamics of the capacitorand simply represents the battery as a voltage source behinda resistor, as in [5].In the per unit case, the DC-side base power is the samethe AC-side. The DC-side base voltage, however, is two timesthe AC-side peak line-to-neutral base voltage. This is doneto obtain an AC-side voltage of . p.u. from the a DC-side voltage of . p.u. at unity modulation ratio [22]. Thesaturation of the PWM modulation index is implementedsimilar to [23] as v m = min {|| ¯ v m || , v DC }|| ¯ v m || ¯ v m , (12)where ¯ v m is given by (4b) and || ¯ v m || is || ¯ v m || = q ¯ v dm + ¯ v qm . (13)III. M ETHODOLOGY
In this section we outline a methodology for choosing thecontrol gains of the DC/DC converter, in order to understandand improve the dynamical behavior of the DC-side of theCIG. To this end, we use a linearized model of our system, aspresented in Section III-A and identify a set of gains that resultin stable operating points. Subsequently, in Section III-B,we determine the gains from this set which optimize thedynamical performance of the DC/DC controller under largedisturbances. To account for the discrete nature of the DC/DCcontroller we utilize a Pade approximation of the associateddead-time delay. The average output performance for a stepinput of a nd and rd -order approximation is used to modelthe dead-time of the DC/DC controller. A. Small-signal tuning
We express the non-linear differential equations (1)-(9) as ˙ x = f ( x , u , w ) , (14)where x , u , w correspond to the states, inputs, and externaldisturbances (loads), respectively. For the purpose of anal-ysis, we linearize this system around an equilibrium point ( x eq , u eq , w eq ) to obtain a resultant linear system ∆ ˙ x = A ∆ x + B ∆ w, (15) where the matrices A and B are evaluated as A = ∂ f ∂ x (cid:12)(cid:12)(cid:12)(cid:12) ( x eq , u eq , w eq ) , B = ∂ f ∂ w (cid:12)(cid:12)(cid:12)(cid:12) ( x eq , u eq , w eq ) . (16)The task of small-signal tuning involves finding a set ofDC-side control gains K DC = [ K v DC p , K v DC i , K i DC p , K i DC i , K pred ] (17)which satisfy some pre-specified design requirements, e.g., ℜ [ λ i ( A ( K DC ))] ≤ λ crit ∀ i, (18a) ζ i ≥ ζ crit ∀ i, (18b) K DC min ≤ K DC ≤ K DC max , (18c)where λ and ζ correspond to the eigenvalues and the dampingratio of the linearized model respectively, λ crit and ζ crit aredesign requirements, and K DC max and K DC min represent some pre-specified limits on the control gains. We denote this set of allpermissible gains by the set Γ . B. Large-signal tuning
On identifying a set of suitable small-signal gains Γ , anexhaustive search over this set is performed to optimize thedynamical performance of the full non-linear system when it issubject to large disturbances, e.g., large load step changes. Inparticular, we seek to identify the set of gains that minimize theDC voltage deviation from its set point. This can be expressedmathematically as minimizing the ℓ -norm min K DC ∈ Γ || v ∗ DC − v DC ( t ) || subject to (1) − (9) p l ( t ) = p l , p l ( t ) = p l + ∆ p l , (19)where p l is the nominal active power load and ∆ p l representsa disturbance in the form of a step-change increase in the load.We first optimize the DC/DC control gains with K pred = 0 and then benchmark the improved dynamical performance forcases where K pred = 0 . Section IV discusses the design re-quirements and disturbance used in (18) and (19) respectively.IV. R ESULTS
The simulations are performed using the Julia programminglanguage. The ModelingToolkit.jl package is used to constructthe non-linear system and perform the Jacobian evaluations.The power rating of the VSC is kVA and the parametersare taken from [12] while parameters for the DC-side arepresented in Appendix A. The controller design parametersused for both the small-signal and large-signal tuning areshown in Table I. The small-signal parameter search is carriedout by a grid search with step size 0.5. All the analysispresented here is available on Github . TABLE I: Controller tuning parameters
Specification λ crit ζ crit K DC max K DC min ∆ p l Value − .
35 10 0 0 . p.u. https://github.com/Energy-MAC/DCSideBatteryModeling
10 20 30 40 500 . . . . . Time ( − s) V D C ( p . u . ) th − order BESS model nd − order BESS model th − order BESS model Fig. 5: BESS response comparison for non-optimized gains. . . . . . . Time ( − s) v D C ( p . u . ) K pred = 0 K pred = 1 K pred = 5 Fig. 6: Optimized controller performance with one-step predictor.
A. Comparing BESS Models
In Fig. 5, we compare the DC-side voltage of the threeBESS models, i.e., th , nd , and th -orders for non-optimizedcontroller gains under a load step change of ∆ p l = 0 . p.u. Weobserve that all models are in agreement regarding the dynam-ical response (also true for different controller gains). Further,we note that the results here only apply to a Lithium-ion basedBESS for the parameters from [18], [20]. For the case ofcompressed air storage with associated mechanical dynamicsand redox flow batteries, with different underlying chemistry;a higher order model representation may be necessary. B. Impact of one-step predictor
In order to examine the improvement in controller perfor-mance by inclusion of the AC-side measurements, we examinethe response of the system to a load step change of ∆ p l = 0 . p.u. for varying values of K pred . Fig. 6 shows the DC voltagefor three different values of K pred . We observe up to a ∼ %reduction in the maximum DC voltage error after includingthe AC measurements. This reduction, achieved using existingmeasurements readily available in the VSC control loop, offersa means to reduce the severity of transients across the DC-linkcapacitor and reduce overloading in the event of over-voltage,two of the dominant reasons for premature failure [8].Fig. 7 further explores the performance of the optimizedcontroller for varying DC-link capacitor sizing. We see that theinclusion of AC-side measurements does offer some improve-ment, however, due to the saturation behavior of the DC/DC Capacitor Size (mF) K p r e d || v D C ( t ) − v ⋆ D C || ∞ ( % ) Fig. 7: Comparing maximum v DC deviation for varying DC-linkcapacitor sizing. . . Time ( − s) i b ( p . u . ) K pred = 0 K pred = 5 controller delay Fig. 8: Battery current profile with one-step predictor. boost converter this improvement is upper-bounded. Therefore,while the AC-side measurement improves the dynamical per-formance and reduces transient behavior across the capacitor,it only offers a modest reduction in DC-link capacitor sizingfor a pre-specified ℓ norm performance requirement.In order to understand the limiting factor in the response ofthe BESS to regulate the DC voltage, we examine the batterycurrent i b , shown in Fig. 8. We can see that the dead-timeof the DC/DC controller only accounts for a small proportionof the delay in the response. The majority of the delay isdue to the dynamics of the DC/DC inductor, in this case mH. While this is a physical design limitation and there existapproaches to minimize the required inductance to improvedynamic response, e.g., increasing the switching frequency[20] or operating in discontinuous conduction mode [5], thesedesign questions are beyond the scope of this work. C. Examining VSC behavior
One additional benefit of including the AC-side measure-ments, and consequently, better regulation of the DC voltage,is the opportunity to reduce the DC-link capacitor size withoutsaturating the PWM converter.For the simulations considered in this paper with grid-forming inverter control gains from [12], the saturation of thePWM converter was avoided in all cases examined. Fig. 9 . . . Time ( − s) V ( p . u . ) v DC || v m || Fig. 9: DC voltage and AC modulated voltage for K pred = 2 . shows both the DC voltage V DC and magnitude of the mod-ulated AC-side voltage || v m || , for the case of K pred = 2 .The inner control loops of the grid-forming VSC respondon a faster timescale to reduce the magnitude of the ACmodulated voltage and thereby, significantly reduce the risk ofsaturating the modulation index of the VSC. The outer controlloops of the VSC then re-adjust the set points to restore thevoltage to an acceptable operating level. While saturation wasnot an issue in this set up, it may be an issue for differentparameterizations and/or disturbances.V. C ONCLUSION
This work focused on modeling and control of a BESS DCsource grid forming VSC. On the modeling side the DC/DCinductor was observed to be the dominant component dictatingthe dynamical behavior. A 4 th , nd , and th -order model of aBESS was examined and it was found that all three modelswere in agreement for the considered disturbances. For theDC/DC controller, it was found that the inclusion of readilyavailable AC-side measurements into the DC/DC convertercontrol loop could reduce DC voltage deviations by up to ∼ % during large step changes, thereby potentially reducingthe risk of premature failure of the DC-link capacitor. Futurework will focus on the behavior of these controllers underasymmetrical gird faults, additional DC-source technologies aswell as further consideration of how fast inner-control loops ofthe VSC which may help alleviate the potential for saturationof the VSC modulation index.R EFERENCES[1] C. Chen, J. Wang, F. Qiu, and D. Zhao, “Resilient distribution systemby microgrids formation after natural disasters,”
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Table II lists the parameter values used for the DC-sidemodel for simulations [18], [20]
TABLE II: DC-side parameters f s DC/DC c DC l DC r b r b r b . kHz mF mH . m Ω 95 m Ω 0 . m Ω r b r b l b l b c b c b . m Ω 0 . m Ω 35 nH nH . F .7