Constant power loads and their effects in DC distributed power systems: A review

Renewable and Sustainable Energy Reviews 72 (2017) 407–421

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Constant power loads and their effects in DC distributed power systems: A review Suresh Singh, Aditya R. Gautam, Deepak Fulwani

MARK

⁎

Department of Electrical Engineering, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan 342011, India

A R T I C L E I N F O

A BS T RAC T

Keywords: Constant power load Dc/dc converters Ac/dc and dc/ac converters Distributed power systems Dc microgrids Negative impedance instabilities Stability analysis

The penetration of dc distributed power systems is increasing rapidly in electric power grids and other isolated systems to cater demand for cheap, clean, high quality, and uninterrupted power demand of modern society. DC systems are more efficient and suite better to integrate some of the renewable energy sources, storage units, and dc loads. A dc distributed power system usually consists of large number of power electronic converters connected in cascad0ed configuration to satisfy the power quality and voltage magnitude requirements of the sources and loads. Tightly-regulated power converters in the aforementioned settings exhibit negative incremental impedance and behave as constant power loads (CPLs), and tend to destabilize their feeder systems and upstream converters. The presence of CPLs reduces effective damping of the system leading to instability of the whole system and present significant challenge in the system operation and control. In-depth knowledge of the instability effects of constant power loads (CPLs), available stabilizing techniques and stability analysis methods, is imperious to the young researchers, system designers, system integrators, and practicing engineers working in the field of dc power systems and emerging applications of dc power. This paper is intended to fill this gape by documenting present state of the art and research needs in one article. Modeling, behaviour and effects of typical CPL are discussed and a review of stability criteria used to study the stability of dc power systems are reviewed with their merits and limitations. Furthermore, available literature is reviewed to summarize the techniques to compensate the CPL effect. Finally, discussion and recent challenges in the dc distribution systems.

1. Introduction Perennial advancements in the power electronics and control technology have widen the applications of power conversion systems in spacecrafts, aircrafts, ships, telecommunication networks, and electric vehicles to reduce size, weight, cost, and to improve reliability, power quality, efficiency, and flexibility [1]. DC microgrid is one of the recent applications, where power electronic converters are the main power processing units for interfacing renewable sources and to facilitate connection to the conventional power system [2–4]. DC microgrids are gaining increased attention due to number of reasons. The merits and challenges of dc microgrids [2,5–7] are summarized in Table 1. Aforementioned distribution power systems usually consist of large number of power converters in parallel, cascading, stacking, load splitting, and source splitting configurations to ensure the desired design and operational objectives [8]. Such systems are known as mutliconverter power electronic systems or distributed power systems (DPS) [1,8]. Cascading of power electronic converters is a common

⁎

feature of almost every converter dominated power system, helps in ensuring the desired point-of-load regulation. However, a tightlyregulated switched power electronic converter behaves as a constant power load (CPL) and tend to destabilize feeder system and upstream converters [9–12]. Tightly-regulated converters may not behave as ideal CPL in all situations and this does not present worst situation from stability point-of-view [13]. The stability of the non-isolated dc/dc power converters with CPL is analyzed in [14,15] both in continuous conduction mode (CCM) and discontinuous conduction mode (DCM), under voltage mode (VM) and current mode (CM) control. It is shown that all converter with CPL in CCM with both VM and CM control are unstable. The boost converter in DCM is stable under both VM and CM control and buck-boost converter in DCM under both VM and CM control is marginally stable. Buck converter in DCM is stable under VM control and unstable under CM control. In open-loop, basic dc/dc converters with CPL and operating in DCM are stable and in this case control design task becomes similar to that of dc/dc coverters loaded by conventional resistive load [16]. The controllability of six types of non-

Corresponding author. E-mail addresses: [email protected] (S. Singh), [email protected] (A.R. Gautam), [email protected] (D. Fulwani).

http://dx.doi.org/10.1016/j.rser.2017.01.027 Received 26 December 2015; Received in revised form 7 July 2016; Accepted 8 January 2017 1364-0321/ © 2017 Elsevier Ltd. All rights reserved.

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Table 1 Merits and challenges of DC microgrids. DC microgrids merits and challenges Merits • More suitable to integrate renewable energy source like solar PV, fuel cells, and storage batteries, and dc loads. • Relatively simple control strategies. • Better power transfer capacity of lines due to absence of skin effect and reactive current flow. • Better short-circuit protection. • Higher efficiency due to absence of multiple power conversions and more efficient dc loads such as LED lights, BLDC drive based systems etc. • Reduced size and cost of the distribution system. Challenges • Difficult to know demand. • Complicated fault detection and clearance. • Reduced stability margins due to the presence of constant power and pulsed power loads. • Need for dc compatible loads and making existing loads dc compatible, wherever possible. • Lack of well defined standards, government policies and regulatory framework.

Fig. 1. V-I characteristics of a typical CPL.

reference to dc microgrid system are presented in Section 6. Finally, conclusion is given in Section 7. 2. CPLs and negative impedance instabilities As discussed in the previous section, a tightly-regulated power electronic converter behaves as a CPL. A CPL exhibits negative incremental impedance and tends to destabilize its feeder system. The V-I characteristics of a typical CPL is shown in Fig. 1. The simple examples of CPL in a dc system are tightly-regulated dc/dc converters with resistive load and dc/ac inverter drives. A CPL may appear in four configurations as shown in Fig. 2, to have destabilizing effect on the feeder system [22]. It can be seen in Fig. 1, that the current drawn by a CPL increases/ decreases with decrease/increase in its input voltage. Mathematically, a CPL can be modeled as [23].

isolated dc/dc converters with CPL is analyzed in [17] using differential geometry theory and it is shown that all converter with CPL are controllable. The sufficient and necessary conditions for the existence of equilibria in power system with ac or dc sources (linear RLC circuits or switching power converters) with CPL are presented in [18] and upper bounds on CPL power are established. In this paper, the dc/dc converters individually or in a aggregated system are considered to operate in CCM, unless specified. Despite several merits of dc power systems, ensuring system stability is a challenge. This is mainly due to the nonlinearities introduced by the CPL behaviour of the switching power converters and non-linear nature of converters itself. The problem is further aggravated by the interaction among different subsystems and the uncertainties associated with renewable power sources (if present). Therefore, the overall system stability can not be guaranteed, even if individual subsystems are stable. This warrants that the steady-state and dynamic stability of the aggregated system be analyzed under various loading profiles. In [19] small-signal stability of a dc microgrid is analyzed under constant resistance load (CRL), constant current load (CCL), and CPL. It is shown that, higher penetration of CPL makes the system unstable, while the presence of CRL or CCL is not destabilizing in nature. Stability analysis in presence of CPL is also presented in [20,21]. The aim of this paper is to provide comprehensive survey to CPLs in relation to dc distributed power systems found in renewable energy based dc microgrids, telecommunication power system, and transport power systems (land, water, air, and space). Beginning with reasons of occurrence of CPL effect in dc power system, modeling and behaviour of a CPL will be described. Available criteria to study the small-signal and large-signal stability of a dc power system are reviewed, and their merits and limitations are presented. A detailed review of techniques used to compensate the destabilizing effect of CPLs in dc power systems is presented alongwith their merits and associated challenges. In-depth knowledge of the instability effects of constant power loads (CPLs), available stabilizing techniques and stability analysis methods, is imperious for the young researchers, system designers, system integrators, and practicing engineers working in the field of dc power systems. The paper is intended to fill this gape and summarize the required information in a single document. The paper is organized as follow. Sources, modeling, behaviour and effects of CPLs are described in Section 2. The stability criteria to study the stability of a dc power system, in general, are reviewed in Section 3. Merits and limitations of each criterion is also presented in Section 3. The detailed review of CPL stabilization techniques is presented in Section 4. Discussion is presented in Section 5. Recent challenges with

icpl =

Pcpl vcpl

(1)

Where icpl is the current drawn by the CPL, vcpl is the CPL input voltage, and Pcpl is the rated power of the CPL. The rate of change of Pcpl current, for a given operating point (Icpl = V ) using (1) is given by cpl

∂icpl ∂vcpl

=−

Pcpl 2 Vcpl

(2)

At the above operating point V-I curve of the CPL can be approximated by straight line tangent to the curve, given by

icpl = −

Pcpl 2 Vcpl

v+2

Pcpl Vcpl

(3)

Equation (3), gives a small-signal model of a CPL, which can be Pcpl represented as negative resistance (Rcpl = − V 2 ) with a parallel concpl

Pcpl

stant current source (Icpl = 2 V ), as shown in Fig. 3. The constant cpl

current component in CPL small-signal model does not affect the stability, but negative resistance reduces the effective damping of the system and tends to destabilize the system. Such instabilities induced by CPLs are known as negative impedance/resistance instabilities. A summary of CPL's negative resistance characteristics are given in Table 2. In discussion on the stability of a cascaded system (feeder/source and CPL), output impedance of feeder/source subsystem Zo and input impedance of CPL subsystem are frequently referred to. In the next Section, the major stability criteria used to study the open-loop and closed-loop dc power systems will be presented. 3. Stability criteria for DC distribution systems In this section, various stability criteria to study the stability of a dc distributed power system in design and operation phase will be 408

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Fig. 2. CPL and feeder configurations; (a) A tightly-regulated dc/dc voltage regulator with upstream dc/dc converter, (b)A tightly-regulated dc/dc voltage regulator with input LC filter, (c) A tightly-regulated inverter drive with upstream dc/dc converter, and (d) A tightly-regulated inverter drive input LC filter.

G12 =

Table 2 Effects of CPLs in the distributed dc systems. Effect of CPL

1 2 3 4 5

Reduces the equivalent resistance of the system High inrush current as voltage build-up slowly from its initial value Reduced system damping and stability margins Limit cycle oscillation in the system voltage and currents May lead to voltage collapse

(4)

Here GMLG is known as the minor-loop gain (MLG), which is the ratio Z of impedances (GMLG = Z 0 ) at dc link interface. As Middlebrook's in criterion emphasizes on magnitude (gain margin) based analysis without taking phase margin into account, the approach leads to conservative results. In order to apply Middlebrook's criterion Zo and Zin must be known. In [28], authors presented an experimental method to estimate the input and output impedances. The sufficient condition to ensure stability of a cascaded system is given by,

Fig. 3. (a) Large-signal model of a CPL, (b) Small-signal model of a CPL.

Sr. no.

V2out G1 G 2 = V1in 1 + GMLG

∥ Z 0 ∥⪡∥ Z in ∥

(5)

Above relation implies that the resulting loading effect is negligible for all frequency ranges, however this effect only does not necessarily causes system stability problem. In such cases, the system stability can only be ensured by ensuring individual subsystem stability. Furthermore, one may use Nyquist stability criterion which can be applied on minor loop gain of the system, (GMLG). This imply that the system is stable if only if the contour of GMLG does not pass through unit circle with origin at (−1, j 0) and if it passes then net antiencirclement of point (−1, j 0) must be nil. Alternatively, if the contour does not enter the forbidden region as shown in 4(b), the contour can not enclose the point (−1, j 0) also. The unit circle, forbidden region and a circle with radius ∥ GMLG ∥, are shown in 4(b). The radius of circle encompassing the region under stability margin is given by,

discussed.

3.1. Middlebrook's criterion The Middlebrook's stability criterion [24–27] was introduced to ensure the stability and invariance of system dynamics due to the addition of input filter to a feedback controlled system. A cascaded system with two subsystems i.e. upstream subsystem (source converter) and downstream subsystem (load converter) is shown in the Fig. 4(a). Let Z0 and Zin be the output and input impedance of upstream and downstream subsystems respectively as shown in Fig. 4(a), then the overall transfer function can be given as,

∥ GMLG ∥ =

1 <1 GM

(6)

The region (shown in dark) beyond the periphery of this circle shown in Fig. 4(b), is the forbidden region.

Fig. 4. (a) A cascaded system (b) Nyquist plot.

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1

real axis at the point (− GM , 0 ), comes under forbidden region. The forbidden region for minor-loop gain corresponding to individual load can be given by,

⎛ Pin, a ⎞ Re (Z 0 / Z in, a ) ≤ ⎜ − ⎟ ⎝ GMP0 ⎠

where a = 1, 2, …m . Re (Z 0 / Z in, a ) shifts as the ratio of input to output power varies, this causes variation in the forbidden region. The Fig. 6(b) shows Nyquist plots of minor-loop gains for overall system corresponding to (Z in ) and for single load corresponding to (Z in,1) respectively. The forbidden regions (see Fig. 6(b)) left to the lines cutting real axis, for each case is shown using crossed dotted-lines. The condition (9) imposes a limit on the gain margin of Z in, a and if this condition is not satisfied then phase condition should be satisfied [25]. Thus, this criterion has an important requirement of information about phase and gain margin of load subsystems and source for its application.

Fig. 5. Nyquist plot showing GM/PM criterion.

3.2. Gain margin and phase margin criterion (GM/PM) In order to reduce the size of passive filters which are widely used in DC distribution power systems, it is imperative to resolve the conservative nature of Middlebrook's stability criteria for filter design. This challenge motivated the use of a more reliable stability criterion known as gain margin (GM) and phase margin (PM) [29] from classical control theory. This criterion extends the Middlebrook's stability criteria, and states that a system may be stable even if Nyquist plot of (GMLG) encircles the point (−1,0) i.e. ∥ Zo ∥⪢∥ Z in ∥, provided minor loop gain has sufficient stability margins (GM and PM) in some frequency range [25]. Therefore, the two important inequalities to be 1 followed are; (a) ∥ GMLG ∥⪡∥ GM ∥ (b) |∠Zo − ∠Z in | ≤ 180 0 − PM . For a designer, it is good choice to keep PM = 60 0 and GM = 6db . Fig. 5 shows the GMPM criterion graphically, wherein the forbidden region is shown through crossed dotted-lines.

3.4. Energy source analysis consortium (ESAC) criterion This stability criterion is particularly applicable in situations where several subsystems are connected together, with components arranged in different patterns/groupings (i.e. input/output impedance groupings can be different for a same type of subsystems because of their connections). These mismatches may lead to different stability analysis results of the system. The ESAC criterion uses a three-dimensional plot of frequency, phase and magnitude in the admittance space. If upstream subsystem impedance and the forbidden region are known, then load's input admittance can be calculated for a given frequency. Therefore, the stability of system can be ensured if load admittance space does not lie in forbidden region [30]. The ESAC exhibits the smallest forbidden region among all stability criteria discussed above. The root exponential stability criterion (RESC) which is an extension of ESAC, is also available in the literature. For more details on ESAC criterion, readers may refer [26,29,31,30].

3.3. The opposing argument criterion (TOAC) One of the important benefits of this criterion is its applicability to the systems with multiple loads connected to the source, while previously discussed criteria are applicable to single load systems only. The application of opposing argument criterion to analyze the stability of a dc system requires determination of the minor-loop gains for each of load subsystems and then adding them together to get equivalent minor-loop gain. Fig. 6(a) shows a network whose downstream converters are modeled using their individual input impedances. The overall minor-loop gain for the system shown in Fig. 6(a) can be obtained by adding individual minor-loop gains corresponding to each load as follow,

GMLG =

⎛ 1 Z0 1 ⎞ = Z0 ⎜ + ⋯.. + ⎟ Z in Z in, m ⎠ ⎝ Z in1

3.5. Three step impedance criterion (TSIC) As the name suggests, this criterion involves three steps in the analysis of system stability. The application of this method involves the following three steps [25]. 1. Preliminary system stability assessment: In the first step all downstream subsystems are substituted by their corresponding mapped pure impedances using some required transformation [31]. Considering a system with a single regulated upstream subsystem and n-downstream pulse width modulated subsystems, where the load impedance of rth downstream subsystem is given by Zr. Then mapped pure impedance ( 1 ) of rth-downstream subsystem

(7)

So the opposing argument criterion results in a concrete condition for forbidden region [26],

1 Re (Z 0 / Z in ) ≤ − GM

(9)

Gr

can be determined by Z r Dr2− boost and

Zr Dr2− buck

for unregulated boost and

buck type conversion subsystems respectively. Where Dr is the duty cycle of rth load subsystem. In general, the equivalent mapped pure

(8)

This means that the region covered to the left of vertical line which cut

Fig. 6. (a) A Input-output system (b) Nyquist plot showing forbidden region.

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Fig. 7. (a) Multiple I/O network (b) Equivalent two-port network (c) Equivalent single-port network. 1 ) G T (s ) n 1 ∑r =1 G . r

impedance ( 1 G T (s )

can be calculated for n-load subsystems as,

parameters on the system stability.

= 2. Measurement of impedance: In this step, measurement of the output impedance of integrated upstream subsystem and input impedance of each downstream subsystem is carried out. 3. System stability analysis: In this step input to output voltage transfer function of the system is determined, which includes an extended form of minor-loop gain given by.

⎡ 1 1 ⎤ GMLG (extd . ) = Zop (s ) ⎢ − ⎥ ⎣ Z in (s ) G T (s ) ⎦

3.8. Passivity based criterion The passivity based (PB) criterion uses the concept of imposing passivity on the system, in order to determine its stability [25,34–36]. The method provides a sufficient stability condition for a system having n-number of source converters with equivalent output impedance Zs and m-number of load converters with equivalent input impedance Zl, incorporating the power flow direction. This implies that the complete system can be reduced to a two-port network, and further to a single port network (see Fig. 7). This results in the total impedance of the

(10)

Zop (s ),

Zin(s) and GMLG (extd . ) are output impedance of Where upstream subsystem connected to n-downstream subsystems, input impedances of downstream n-subsystems and extended minor-loop gain respectively.

system at the point of interface given by Zbus =

Vb us Ib us

= Zs Zl . In order to

establish stability of the system, the passivity property should hold upon the system. A network is passive if and only if; (a) All the poles of Zbus(s) are in the left-hand side of imaginary axis. (b) Real part of Zbus ( jω) is greater than or equal to zero, or −90 0 ≤ ∠Zbus ( jω) ≤ 90 0 , for all frequency range. System is considered a stable system, if these two conditions hold.

Condition for stability is then obtained using Nyquist plot of GMLG (extd . ) i.e. system is stable if Nyquist plot of extended minorloop gain does not encircle the point (−1,0). 3.6. Small-signal approach (model analysis)

3.9. Mixed-potential function based criterion In this approach, the stability analysis is performed using the linearzed system model. Then using left and right eigenvectors corresponding to eigenvalue λi, the sensitivity of the eigenvalue to the element akj of the linearized system matrix A, and the participation factors (sensitivity of eigenvalue λi to the diagonal element akk of the linearized system matrix A) are determined. In the design phase, the information regarding critical parameter derived through eigenvalue sensitivity analysis and participation factors can be utilized in the control design to ensure desired stability margins and transient performance [32,33].

This criterion is based on nonlinear circuit theory developed by Brayton and Moser. It is a design criterion, used to ensure the largesignal stability of a system and estimate the ROA of an equilibrium point. Application of this criterion results in design constraints on filter parameters e.g. dc-link capacitor to ensure asymptotic large-signal stability of an equilibrium point with a sufficiently large ROA. The application of the criterion involves developing a Lypunov-type Mixed Potential Function using the elements and the topology of the circuit under study, and then imposing one of the five Brayton and Moser's theorems, under certain conditions [37–39]. The criterion of Mixed Potential Function is used to study the large-signal stability of dc source feeding CPL(s) with input LC filter in [37], and large-signal stability and to estimate ROA of cascaded dc power system in [38]. The detailed information large-signal stability tools such as multimodel approach, block diagonalized quadratic Lyapunov functions (BDQLF), and reverse trajectory tracking can be found in [39].

3.7. μ-sensitivity criterion μ-sensitivity approach is basically motivated from robust stability analysis and robust controller design from robust control theory [33]. The concepts of Linear Fractional Transformation (LFT), structured singular value μ, skewed-structured singular value ν, and μ-sensitivity are some of the important terms in the stability analysis using this approach. The μ-sensitivity based stability analysis involves following steps; (a) Symbolically linearzed model of the system is obtained at the equilibrium points. If there are some nonlinearities in the system matrix, they must be replaced with their polynomial approximation, (b) Obtaining LFT based Model, and (c) Compute μ-sensitivities. The value of μ-sensitivity with respect to a particular system parameter, is an indication of how critical is that parameter from stability point of view. The method of μ-sensitivity is applied to a PI controlled dc/dc buck converter system with input LC filter in [33], wherein μ-sensitivities for equal small perturbation in all parameter are determined. Based on obtained μ-sensitivities values, it is shown that the filter parameters (Rf , L f and Cf) are most critical for system stability. Compared to other stability criteria such as Middlebrook's and Model analysis, this approach gives broader and direct insight into the influence of system

3.10. Phase-plane analysis In phase-plane analysis, the system differential equations are graphically solved (plotted), which gives an insight about how the system dynamics evolve with time [40–42]. This technique can be used to study global close-loop behaviour of the converters loaded with CPLs. The method does not give a particular solution to the differential equations describing the dynamics of a system. The phase-portrait of a buck converter loaded with a CPL is shown in Fig. 8. In [40], the phaseplane analysis tool is used to analyze the large-signal closed loop behaviour of a PD controlled dc/dc buck converter loaded with CPL, and to determine region of attraction (ROA) of an equilibrium point. It has been shown that ROA depends on the controller and converter parameters, and it does not include origin. Same approach is applied to 411

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In the following subsections, different CPL compensation techniques are reviewed.

SEPARATRIX

4.1. Passive damping

Inductor current

LIMIT CYCLE

In this method, in order to compensate the negative incremental impedance effect of the CPLs, the system damping is increased by adding passive components (resistances, resistance-capacitance, and resistance-inductance) to the system under consideration. This approach results in an increased size, cost, and weight of the system. Furthermore, passive components lead to high power dissipation, particularly when resistance is used in parallel with filter capacitor, which is detrimental to the system efficiency. The application of the loss free resistance (LFR) [45] is used to reduce the power dissipation. The challenge with LFR is that, it increases the system size, complexity, and cost. In [46], the interaction of CPL's small-signal negative input resistance with input LC filter is analyzed and a passive damper consisting of a series RC branch in parallel with filter capacitor is proposed to stabilize the system. Cespedes et al. [47], have proposed three different passive dampers to stabilize the input filter of a CPL and presented an analytical theory to determine the required value of damper parameters is also presented. The design of the dc bus capacitor, to ensure the desired stability margins using impedance criteria under three different droop control schemes, is presented in [48]. The considered test system consists of a dc aircraft power system having parallel sources driving a CPL. The influence of the converter paracitics (switch ‘ON’ resistance, inductor resistance, and diode resistance) in the presence of CPLs is analyzed in detail in [49], under both CCM and DCM operation. Furthermore, design recommendations are presented to avoid CPL induced instabilities in a dc DPS feeding pure CPL and combination of CPL with conventional resistive loads.

Fig. 8. Phase-portrait of an ideal buck converter loaded with a CPL (Vin = 380 V , D=0.5, CPL Power P=4 kW, L=2 mH, C=1 mF).

study the large-signal behaviour of a state feedback controlled dc/dc boost converter loaded with a CPL in [41], and ROA of equilibrium points is determined. It is shown that the boost converter has two equilibrium points in first quadrant. Furthermore, authors show that the boundary of ROA to ensure stability is determined by stable manifold of the unstable equilibrium point. Authors in [42] presented analysis of start-up and transient response of an average current controlled dc/dc buck converter feeding a CPL using phase-plane approach. Furthermore, design criteria to ensure the stability during start-up, transients and current limiting value to avoid converter clamping, are also proposed. 3.11. Bifurcation analysis

4.2. Active damping

In bifurcation analysis, the region of stable operation is determined through the search of hopf bifurcation points (where eigenenvales of system Jacobian matrix consists of one pair of pure imaginary and others with non-zero real part) [43]. This gives an insight into how the variations in the system parameters influence region of stable operation. This knowledge can be effectively used by the system designers to ensure the stability of the actual system. Authors in [43], presented Hopf bifurcation search using indirect (eigenvalue analysis) and direct (numerical computation of Hopf points) methods to determine the boundary of the stable region, for an equivalent dc microgrid loaded with CPL. The Boundary of the stable operation is determined by selecting filter parameters (R, L, and C) and load power P as bifurcation parameters. In [44], the stability analysis of two configurations of the DC systems feeding CPLs is presented using numerical continuation method. A limit on CPL power and corresponding ROA is determined using Hopf bifurcation search. Furthermore, state feedback controllers are proposed to enlarge the ROA. Several criterion for stability analysis are summarized in Table 3 also.

The underlying concept of active damping is to create the damping effect of series/parallel resistances and dc bus capacitance through the modifications in the control structure of the feeder or load subsystem. In addition to this, an auxiliary circuit can also be connected at the load terminals to inject a compensating current or to emulate variable impedance, so as to mitigate the CPL induced instabilities [50,24]. Next, the active damping techniques realized at feeder side, load side and through auxiliary system will be discussed separately. 4.2.1. Feeder side active damping In this section, active damping methods implemented through the modification in control loops of the feeder subsystem are reviewed [50–61,33]. The compensation of the CPL effect at the upstream feeder level is applicable only when, the upstream feeder subsystem is a switched dc/dc or ac/dc converter. When the feeder subsystem of the CPL is an input LC (inductance-capacitance) filter or uncontrolled rectifier, CPL compensation at the feeder side is not possible. In active damping at the feeder side, the additional compensation loops modifies the output impedance Z0 of the feeder converter, so as to satisfy the impedance stability criterion. The major advantage of this approach is that the system can be stabilized without compromising the load performance [24,15]. An active damping technique which emulates a resistance in series with the converter inductor to stabilize the basic dc/ dc converters loaded with a CPL is presented in [50]. The measured inductor current is passed through a feedback coefficient RLA and is subtracted from control voltage to emulate a resistance in series with the inductor, which increases the system damping (see Fig. 10). Furthermore, the technique is extended to the isolated dc/dc converters loaded by a CPL. Through active damping, only a limited amount of the CPL can be compensated. In [51], the global behaviour of a dc/dc buck converter is analyzed using phase-plane analysis, wherein the system is controlled using current feedback loop with

4. CPL stabilization techniques The basic concept of CPL compensation involves increasing the effective system damping through some modifications at feeder/source level, load level or the use of some additional circuits [24]. These modifications can be done in system hardware or in their control loops. The techniques based on hardware modifications are known as passive damping techniques and those based on modifications in the control structures are known as active damping techniques. The techniques based on some specialized control approaches, discussed separately here, are also usually considered under active damping. A broad classification of the CPL compensation techniques is shown in Fig. 9. 412

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Table 3 Summary of criteria for stability analysis. Criterion

Merits

Challenges

Middlebrook's Criterion

Most basic and easiest to apply; Ensures both, stability and performance. Considers both, magnitude and phase of MLG; Smaller forbidden region. Applicable for single/multiple load-source systems; Less conservative; In case of multiple loads, considers individual MLGs. Less sensitive to component grouping; Smallest forbidden region. More Generalized; No need to check individual subsystem stability. Very useful in design phase; Simple procedure. Broader and direct insight into influence of system parameters on stability. Large-signal stability; Reduces a multi-port system to a single port network. Large-signal stability; Applicable for multiple loads system also.

Small-signal stability; Zout of source and Zin of load should be known; Does not use impedances phase information Small-signal stability; Focuses on individual subsystems; Requires knowledge of GM and PM of each subsystem Small-signal stability; Results are valid in limited frequency range; Require knowledge of PM and GM of source and each load subsystems

Gives global behaviour of the closed-loop system; Very useful in design phase. Gives boundary of stable operation; Very useful in design phase

Does not give a particular solution to system differential equations

Gain Margin and Phase Margin Criterion (GMPM) The Opposing Argument Criterion

ESAC Criterion Three Step Impedance Criterion Model Analysis μ-sensitivity Criterion Passivity Based Criterion Mixed-potential Function based Criterion Phase-plane Analysis Bifurcation Analysis

Small-signal stability; Intensive calculations Small-signal stability Small-signal stability; Not reliable in system stability Applicable to LTI systems only; Small-signal stability Does not give stability margins The system under consideration should be ‘topologically complete’

Small-signal stability;Considers open-loop system

converters in a CPL dominated dc microgrid, two active compensators are proposed in [55] using two different approaches based on linear theory. In addition to stabilize the feeder converters, the active damping loops also improve the dynamic performance of the microgrid. Active damping control to emulate virtual resistance in a source dc/dc buck converter supplying power to paralleled CPLs with their input filter is presented in [56] (Fig. 12). The major disadvantages of the proposed method are; in order to stabilize the system the closed loop bandwidth of source converter should be greater than the resonant frequencies of the input LC filter and resonant frequencies of the filters must be different. Shafiee et al. [57], have proposed a concept of dc active power filter (APF) to stabilize a dc microgrid under load changes and while connecting it to other dc microgrids. A small-signal model of the system is derived to analyze the effect of CPLs and tie line impedance on the stability. To implement the active damping current loop, the information of load current and current disturbance in tie line are used and compensation gain is selected through root locus analysis. For more feeder side active damping methods, see [58–61,33], the references therein.

Fig. 9. Broad classification of CPL compensation techniques: (a) Feeder side compensation, (b) Load side compensation, (c) Compensation using auxiliary circuits.

4.2.2. CPL Side active damping Under the situations when the feeder subsystem of a CPL is an input LC filter or an uncontrolled ac/dc rectifier (behaves as LC filter), CPL compensation from feeder side is not possible, due to the absence of control loops associated with feeder subsystem [62–72]. In such cases, there are two alternatives available for CPL compensation: CPL side compensation and the use of an active shunt damper between feeder and load subsystems. In this section, the active damping methods based on CPL side compensation will be reviewed. In these methods, a compensating current/power is injected into the CPL control loops to modify the input impedance Zin of the CPL subsystem, such that Middlebrook's stability criteria is satisfied. The main drawback of this approach is that the compensation loop dynamics may interfere with the main control loop, and may deteriorate the load performance. On the other hand the approach is advantageous as CPL itself is utilized to mitigate the negative impedance instabilities. In [62,63], a nonlinear system stabilizing controller (NSSC) is presented to mitigate the negative impedance instabilities as shown in Fig. 13(a), where n is a real number. The controller is tested on a tightly-regulated induction motor drive and a dc/dc converter with input LC filter, controlled through a nonlinear PI controller. It is shown that, the

Fig. 10. Active damping of dc/dc converters.

hysteresis and PI voltage controller. In [52], it is shown that peak and valley current mode control can be used to stabilize a dc/dc boost converter loaded with a CPL. The stability of the system is analyzed in a small-signal sense. Furthermore, concept of load current feed forward is used to improve the transient response. Authors in [53], proposed active damping of a bidirectional voltage source converter (VSC) interfacing a dc microgrid, by injecting a damping signal in its outer, intermediate, and inner control loop (concept is shown in Fig. 11). The stability of the compensation is analyzed in small-signal sense and sensitivity analysis of the compensation and voltage control dynamics is also presented. It is shown that the intermediate loop dynamics provides best performance in terms of damping capabilities and its influence on the voltage control loop. In [54], active compensators are proposed to reshape input admittance of the tightly-regulated VSCs in a hybrid ac/dc distribution system to stabilize the system in the presence of interaction dynamics and negative impedance effect of the CPLs. In order to stabilize the feeder 413

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Fig. 11. Active damping of grid-connecting VSC in a DC microgrid.

Fig. 12. Active damping of a dc/dc buck converter loaded with CPL.

controller stabilizes the system, however, significantly deteriorates the load performance. A negative input-resistance compensator (NIRC) is proposed in [64] as shown in Fig. 13(b), to stabilize a brush-less dc motor drive exhibiting a CPL behaviour. The compensator design using small-signal analysis and sensitivity analysis with motor performance is also presented. In order to further reduce the effect of the compensator on motor performance and to improve immunity to input voltage disturbances, an improved version of NIRC, known as state feed-forward stabilizing controller (SFSC) (Fig. 13(b)) is proposed in [65]. The controller takes input filter inductor current and input voltage as its inputs. In [66], the local stability of a permanent-magnet synchronous motor (PMSM) inverter drive with an input LC filter, and tightly controlled using linear controllers, is analyzed using Nyquist and bode plots. An additional compensation block consisting of a band-pass filter and a proportional controller is proposed to compensate for the input voltage oscillations and to reduce dc bus capacitor size. Compensation block parameters can be tuned to get an optimum motor performance and suppression of oscillations. Authors in [67], have proposed a reference voltage based active compensator (RVC) and its improved version to mitigate the negative impedance instabilities in a PMSM drive. It has been shown that, low-pass filter (RVC-1) and band-pass filter (improved version, RVC-2) active compesnators stabilize the system without compromising the motor torque and speed performance. Second configuration RVC-2 is found to be more effective, resulting in reduced interaction dynamics between compensator and motor-drive main control (see Fig. 14). Magne et al. [68], have presented a small-signal stability analysis of a system consisting of an inverter motor-drive, dc/dc converter with resistive load, and a bidirectional dc/dc converter (BDC) interfacing a supercapacitor. A

Fig. 14. Reference voltage based active compensators (RVC-1 and RVC-2).

central stabilizing controller is proposed to ensure the system's global stability and to reduce the size of input filter components. The main drawbacks of the proposed scheme are requirement of large number of sensors and high control bandwidth. To reduce the number of sensors required, the authors proposed an observer in [69] to estimate the load voltages. In [72], the active stabilization of a CPL supplied through a LC input filter is formulated as linear H∞ optimization problem with an objective to minimize the degradation of load performance while ensuring desired stability and robustness. It has been shown that main CPL control bandwidth is limited by LC filter's resonant frequency. Details about more CPL side active damping methods can be found in references [70,71,73] and the references therein. 4.2.3. Active damping using auxiliary circuits In this method, to mitigate the destabilizing effect of CPL, an additional circuit is connected between feeder and load subsystems, leaving the feeder and load systems intact. This additional circuit is usually a dc/dc converter which is controlled to inject a desired compensating current in the entire operating range of the main system. The method, although eliminates the challenges of the above two approaches, results in increased cost and increases overall complexity of the system. In [74], a BDC interfaced with a storage capacitor is connected between CPL and its input LC filter to eliminate the

Fig. 13. (a) Nonlinear system stabilizing controller, (b) Negative input-resistance compensator (NIRC) and state feedforward stabilizing controller (SFSC).

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oscillations in the input voltage. The controller uses voltage and current variables of the filter and BDC, to place the poles of the overall dynamic system at the desired location. Furthermore, a second order observer is also proposed to reduce the number of sensors required. Authors in [75], have proposed the placement of a suitably sized capacitor and a PI controlled BDC with storage, at the terminals of a tightly controlled inverter drive with an input LC filter, to stabilize the dc bus voltage. The concept of an auxiliary smart active damper is presented in [76] to stabilize a dc telecommunication power system and data center dc microgrids. The active damper which emulates the RC damper characteristics, is realized through a non-isolated BDC without any additional storage, and communicates with source and load subsystems in real time to determine the desired damping current required to stabilize the system under various input and load conditions. The inner loop of the damper is controlled in peak-current mode at a fixed frequency, while outer loop eliminates the deviation in the peak and average current of the inductor.

Fig. 15. Block diagram of the Pulse-adjustment control technique.

pole placement at the desired location. A PD controller is proposed and the sensitivity analysis of system parameter mismatch on the performance of the linearizing function is also presented.

4.3. Feedback linearization 4.4. Pulse adjustment Linearizing a nonlinear plant about an operating point ensures stability only in small-signal sense. Feedback linearization is a nonlinear control approach used to compensate CPL effect in dc DPSs, wherein a nonlinear feedback is chosen to cancel the nonlinearities introduced in the system due to the presence of CPL [77]. Basically, this involves a nonlinear coordinate transformation which allows access to the system nonlinearities through input channel, such that the resultant system is linear [78]. Consequently, control system can be designed using conventional linear control theory. In contrast to the active damping technique, feedback linearization can compensate any amount of CPL and stabilizes the system in large-signal sense. The major drawbacks of this approach are its noise sensitivity due to the presence of differentiator and slower transient response compared to techniques which handle CPL nonlinearity directly, such as sliding mode control and synergetic control [79,80]. Authors in [78], used feedback linearization through nonlinear coordinator transformation to stabilize a dc/dc buck converter feeding CPL. It is shown through Lyapunov analysis that the transformation results in an extension of local asymptotic stability. Stabilization of a dc/dc buck converter driving a combination of resistive load and CPL is presented in [77] and the large-signal stability of the system is proved using Lyapunov approach. Rahimi et al. [81], have proposed loop cancellation technique to stabilize all basic dc/dc converters feeding a resistive and CPL using suitable nonlinear feedback, which cancels nonlinearity introduced due to the presence of CPL. It is shown that the value of feedback gain to cancel CPL nonlinearity depends on input, load and the converter parameters. To overcome this problem, the value of feedback gain is chosen such that, under all the operating conditions, the sign of resultant nonlinear term remains positive. This implies that the resultant nonlinear term can be represented by a positive equivalent resistance, which helps to increase the system damping. In [82], a nonlinear coordinate transformation is applied to a dc/dc buck converter loaded with a pure CPL to obtain its linear model. To obtain near exact linearization, the converter parameters (L and C) to be entered in the controller are assumed to be equal to their actual values. Furthermore, a reduced order observer is proposed to estimate the CPL power and its derivative, to ensure the accuracy of linearization in entire operating range, i.e., to improve the transient performance. A full-order feedback controller is proposed for the linearized converter model. The sensitivity analysis of parameter mismatch on the performance of observer and closed loop system is also presented. A technique based on linearization via state feedback (LSF) is presented in [83] to stabilize a medium voltage shipboard dc power system in the presence of CPLs. The method involves defining two functions, one to linearize the nonlinear system and other to realize the

The pulse-adjustment control [84,85], is a digital control technique in which the task of converter output voltage regulation is achieved by supplying high and low-power pulses to the converter. Depending on measured actual output voltage and the reference voltage, the controller chooses either high or low-power pulse. If vout < Vref , the controller generates switching pulse of duty ratio DH (high duty ratio) until desired voltage level is reached, otherwise switching duty ratio DL (low duty ratio) is selected to regulate the output voltage to its reference value. The ratio DH presents a trade-off between output DL voltage ripple and the voltage regulation, and can be chosen to satisfy a particular application requirement. The selected value of the high pulse duty cycle DH is such that the converter operates in DCM. The output voltage sampler and switch driver being synchronized, the technique ensures constant frequency switching of the converter. A block diagram of the pulse-adjustment technique is shown in Fig. 15. In [84], the pulse-adjustment technique is applied to stabilize a dc/ dc buck-boost converter loaded with a CPL. A model of the converter loaded with CPL and operating in DCM is derived, which is then used to analyze system stability and to determine the output voltage variations during high and low-power pulses. Furthermore, a detailed sensitivity analysis of the output voltage variations and stable CPL power range with respect to switching frequency, input voltage, reference voltage, and converter parameter (L and C) variations is presented. It is shown that the output voltage contains undesirable disturbances under the input voltage variations, if not filtered properly. The authors proposed a modified pulse-adjustment technique in [85] with variable DH and applied to a buck-boost converter to minimize the effect of input voltage variations on the output voltage. The technique of pulse-adjustment is inexpensive and simple to implement using digital tools, gives fast response, and does not require a detailed small or large-signal model of the converters. The main limitation is that it can stabilize the system in the limited range of CPL power.

4.5. Digital charge control Digital charge control is yet another digital control technique used to compensate the effect of CPL in DC systems. The block diagram of the digital charge control technique is shown in Fig. 16. In [86] the digital charge control method is applied to a dc/dc boost converter feeding a CPL, and small-signal analysis is also presented. An improved version of the technique known as digital forecast charge control is also presented to eliminate the undesirable phenomenon of duty cycle jumping. The salient features of this technique include simple implementation and fast response. 415

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Fig. 16. Block diagram of digital charge control technique.

4.6. Sliding mode control

Fig. 17. Equivalent and Discontinuous SMC, shown in Red and blue respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Sliding mode control (SMC) is a robust nonlinear control technique which falls under variable structure system control (VSSC) [87]. In SMC, depending on the switching conditions a switching system ( f (x, u , t )) can be decomposed into a set of subsystems, wherein each subsystem exhibits a fixed characteristics in a specified region of state space. An important task for a designer is the selection of a stable switching function (s ) and a switching control law (u), such that the control forces the representative point (RP) from an arbitrary point to a predefined switching surface (s=0), within a finite time. The existence of sliding mode and stability of switching surface must be proved for the chosen switching function. The existence condition is proved by satisfying η-reachability condition,

sT . s˙ < − η|s|

boost cascade topology interfaced with battery has been proposed in [90]. The authors have shown the minimum switching action of the controller for stabilization of CPLs with different bandwidth. A SMC using a washout filter for a bidirectional converter feeding a mixed load is proposed in [91]. Authors in [92], have proposed a PWM based sliding mode control using a nonlinear switching function, to mitigate negative impedance instabilities in a dc/dc buck converter supplying mixed load (CPL and CVL). Mitigation of CPL induced instabilities in a dc/dc boost converter with a pure CPL has been presented in [93,79] using a nonlinear switching function based sliding mode controller.

(11)

The existence of sliding mode results in reduced order dynamics and during sliding mode the system dynamics is insensitive to the matched uncertainties and system parameters. The system dynamics during sliding mode is completely governed by the parameters of the chosen switching function. The stability of system dynamics during sliding mode (s=0) can be proved using linearization or nonlinear Lyapunov approach. SMC approach can be used to design conventional discontinuous controller or fixed frequency PWM controller. The main challenge of discontinuous SMC is practical bang bang operation i.e. an infinite frequency discontinuous operation which causes chattering. In order to overcome the chattering phenomenon a fixed frequency based equivalent SMC can be used, however, it degrades the robustness. The discontinuous and equivalent control laws can be defined as follows,

udisc = K . sign (s ) ueqv =

(SB )−1Sf

(x , t )

4.7. Synergetic control Synergetic control [94,95] is a non-linear technique which encompasses dissipative structure algorithms. This control technique shares similarity with SMC and ensures constant frequency switching. The control design follows an analytical procedure using state space approach. The steps involved in control design through synergetic control are as follows, 4.7.1. Plant modeling In this step, a mathematical model of the dynamic system is described using differential equations of the following form, (14)

x˙ = f (x, u , t )

(12)

where x is the state vector of dimension n, and u is the control vector of dimension m. Then, a macro variable ψ (x ) and control law are designed, such that the control law forces the system trajectory from an arbitrary initial condition, towards the predefined invariant manifold, ψ (x ) = 0 and constrain it to manifold then on. The macro-variable can be any function of the state variables. The number of macro variables should be less than the number of control channels.

(13)

where S has dimensions of mxn (m-inputs, n-order), whose elements are the derivatives of s with respect to the state variables, B is input matrix and K is a tuning parameter. The two different methods of SMC are shown in Fig. 17 in different colors. A SMC can be designed with continuous equivalent control law, discontinuous control law, or a combination of two. SMC has a wide range of applications due to its robustness and simple implementation. It has been widely used to control dc/dc and ac/dc power converters in general and mitigation of CPL induced instabilities in particular. Emadi et al. [88], have presented a simple SMC for a dc/dc buck converter which ensures supply of constant power to the load. One of the limitation of proposed SMC is that it does not ensure the regulation of converter output voltage. Authors in [89], have proposed a sliding mode duty cycle ratio controller for buck converter feeding a CPL, to stabilize the dc bus voltage, in an application of medium voltage dc shipboard power system. The designed control law, in addition to equivalent control term, contains a switching term which provides robustness to line and load disturbances during reaching phase. A geometric control based on a circular switching surface for constant power load stabilization in buck and

4.7.2. Control law synthesis To synthesize a control law, a dynamics governing the evolution of the macro-variable towards the manifold is defined. The required dynamic evolution of the macro-variable is given by

Tψ˙ + ψ = 0;

T>0

(15)

where T is a parameter of the above dynamics which controls the speed of convergence of trajectory toward the manifold. The control is obtained by solving (15) with (14) for u. The order of the system on the manifold is reduced to (n-m). In [96], authors have proposed synergetic controllers for dc voltage stabilization and dynamic current sharing between two buck converters with constant power load and operating in CCM, and for voltage regulation of a single buck converter with CPL, considering DCM operation of the converter. The authors extended this work and 416

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of CPLs, the system differential equations are re-formulated in terms of rate of energy or instantaneous power. This basically results in a linear system, thus eliminates the nonlinearity introduced by the CPL. The power shaping control strategy is relatively easy in design and implementation, and results in desired regulation of dc bus, while maintaining large-signal stability. The authors in [104], have proposed stabilization of a dc distribution system supplying constant power loads using power stabilization control strategy.

proposed a generalized synergetic control strategy in [97], for the dc voltage regulation and dynamic current sharing among m-number of paralleled buck converters feeding constant power load. 4.8. Passivity based control Passivity based control (PBC) is a non-linear control approach to design static/dynamic controller for a physical system described by the Euler-Lagrange (EL) [98–100]. The central idea of passivity based control design is to passivize the system by applying following two steps

4.10. Coupling based techniques An amplitude death solution or coupling based technique, is basically coupling induced stabilization of the equilibrium points of an unstable dynamic system [105–107]. The sufficient strength of coupling and different natural frequencies of the systems being coupled, are the two main requirements for amplitude death. The technique originally belongs to nonlinear dynamical systems and has recently been applied for open-loop stabilization of the dc-dc converters in a dc microgrid in the presence of CPLs. Authors in [105], have proposed a heterogeneous and time-delay coupling to stabilize a dc/dc buck converter supplying a CPL. Konishi et al. [106], have presented a bifurcation analysis of instability phenomenon of dc bus voltage in the presence of CPLs and proposed a delayed feedback control to stabilize the system. The concept of delayed feedback control has been further extended in [107], to a networked system having multiple dc bus systems, connected through resistive links. The delayed-feedback control is applied to each unit, in a decentralized manner to stabilize the system. Moreover, it has been shown that stabilization is independent of the number of dc buses and the network topology. The block diagram of the techniques discussed in this section is shown in Fig. 19.

(1) Energy shaping by assigning a closed loop storage function to compensate the energy difference between the energy of the system and energy injected by the controller. This results in modification in the potential energy function (PEF) only, in order to get the strict local minimum of PEF at required equilibrium point. Basically, PBC works on the principle of energy conservation, i.e.

Esupplied = Estored + Edissipated

(16)

(2) Modification of the dissipation energy function by damping injection in order to make equilibrium point a globally asymptotic stable point. This is achieved by adding a virtual impedance matrix. Some researchers have used Port Controlled Hamiltonian (PCH) model, instead of Euler-Lagrange equations, to a nonlinear electrical dc power system and to implement interconnection and damping assignment (IDA)-PBC. A PBC combined with IDA technique used for stability analysis and to design a linear PD (proportional-derivative) controller for a buck converter, and a nonlinear inverse quadratic PD controller for a boost, and buck-boost converters in a dc microgrid application, have been proposed in [101]. However, the PD controller poses noise susceptibility issue, therefore an appropriate filter is needed. Furthermore, IDA technique has been designed with fixed parameters (i.e. for specific operating point of CPL), which is not always the case in practical systems. To mitigate this problem, a complementary PI (proportional-integral) controller along with adaptive IDA-PBC technique for dc/dc boost converter is proposed in [102]. A PBC with Immersion and Invariant controller has been proposed for dc bidirectional converter interfaced with a battery in [103]. This combined control ensures improved transient performance of the converter feeding a mixed load (CPL and resistive load). Two different PBC design approaches using PD and IDA controllers for dc bus regulator are shown in Fig. 18.

4.11. Pole placement In [108], the pole placement control has been used to relocate righthalf s-plane pole of a buck converter loaded with a CPL to stabilize the system. By placing the unstable poles at the desired location, the effect of CPL is compensated. It is shown that, the technique results in the reduced size of the filter capacitor while maintaining system stability. In order to use pole placement technique all the states are required to

4.9. Power shaping stabilization In power shaping control strategy to mitigate the destabilizing effect

Fig. 19. Block diagram of coupling based techniques: (a) Heterogeneous and delayed coupling; (b) Delayed feedback control.

Fig. 18. PBC control techniques.

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Table 4 Comparison of CPL compensation techniques for DC DPS. Stabilization Technique

Salient features

Challenges

Damping improvement in CPL dominated system

Passive Damping

Simple to implement; No change in source or load hardware.

High power dissipation; Modifies source or load hardware

Active Damping

No change in source or load hardware; Higher efficiency and reliability.

Feedback linearization

Can compensate any amount of CPL; Can achieve stability in large-signal sense; Uses conventional linear control design techniques.

Pulse Adjustment

Fast dynamic response; Insensitive to system parameters; Inexpensive implementation; Reduced switching losses and EMI noise, due to DCM operation Insensitive to matched uncertainties; Largesignal stability; Fast response Fixed frequency switching; Large-signal stability; Suitable for digital implementation.

Can compensate limited amount of CPL; May interfere with other control objectives; Switching frequency affects the effectiveness. Sensitive to noise in the output channel; Dynamic response is not comparable to that offered by nonlinear controls such a SMC and synergetic control. Sensitivity to input variations; Stable in limited range of CPL.

CPL adds negative impedance to the system. This results in decrease of system overall damping. Adding passive components (R,L,C) to improve the system damping System damping can be improved using active damping methods such as adding virtual impedance, bandwidth control, improving system stability margins etc Feedback gain adds a positive impedance term, equivalent to CPL negative impedance to the control, such that resultant load impedance appears a positive impedance to front end converter Region of stability of a system feeding CPL can be increased by increasing ON time of duty cycle and number of pulses or decreasing the number of OFF time pulses per period Sliding mode makes system immune to system parameters variations like CPL negative impedance Similar to sliding mode control, in this technique also, RP on system trajectories is forced to track a predefined stable manifold and keep system immune to parameter variations Low damping caused by presence of CPL is improved by inserting an impedance matrix in control and reshaping of energy function Energy stored in output filter capacitor of converter feeding a CPL is a function of Vc2, i.e. square of capacitor voltage. Control of capacitor power controls output voltage indirectly. Instability induced by CPL can be eliminated by properly reshaping the power balance

Sliding mode control Synergetic control

Variable frequency switching and chattering issue; Higher sensor requirement Sensitive to high frequency noise; Higher sensor requirement.

Passivity based control

Simple Implementation; Robust; Energy based modeling.

Sluggish transient response

Power Shaping stabilization

Power based modeling; Large-signal stability; Reduced control bandwidth requirement.

Increased computation needs.

Coupling based techniques

Low implementation cost.

Digital charge control

Simple to implement; Fast response.

Limited to open-loop stabilization; Implementation issue, when sources are at different locations Higher computational needs.

State space pole placement

Unstable poles can be placed at the desired location; Can compensate desired CPL amount.

All system states need to be sensed, if there is no observer; Required feedback gains may vary with load.

New converter topologies

Duty ratio can be kept low; Higher efficiency; Improved dc-bus power quality.

Can compensate limited amount of CPL; More number of switching devices, i.e. complex control.

equation i.e. 0.5CVc2 = Pinput − CPL Exchange of energy between coupled systems mitigates destabilizing effect of CPLs This method can forecast voltage jumping caused by low damping of system such as CPL fed dc-dc converter. For forecasting, charge value proportional to reference voltage is considered Suitably designed feedback gain vector relocate unstable poles of control to voltage open loop transfer function of CPL fed converter, in order to make closed loop system stable around a given operating point New converter topologies with improved output impedance characteristics are used to compensate the CPL effect

a dc power system both, in small-signal and large-signal sense. Middlebrook's, GMPM, the opposing argument and ESAC stability criteria use Nyquist stability approach to determine the system stability in a small-signal sense. Methods such as phase-plane and bifurcation analysis are very useful to determine the boundary of stable operation with respect to variations in the system and controller parameters, and to design a stable system. Passivity based criterion and Mixed-potential function based criteria ensure the system stability in large-signal sense. Methods ensuring large-signal stability of the system are obvious choice due the highly nonlinear nature of the dc power systems in the face of uncertainties associated with renewable sources and the presence of CPLs. The compensation of CPLs can be applied at the feeder subsystem level, CPL level itself or through some auxiliary dc/dc converter. Feeder level CPL compensation is applicable to the cases when feeder of CPL is a switched converter, on the other hand, CPL compensation at the load system level has the major disadvantage that, the compensation loop may interfere with main control loop, and may degrade the load performance. The third alternative is based on the introduction of an auxiliary dc/dc converter between the feeder and load systems. In this method, no modifications are applied to the feeder and load subsystem. However, it increases the overall size, weight, cost, and complexity of the system. From system stability point-of-view, some of the techni-

implement the state feedback control. 4.12. New converter topologies Hou et al. [109], have presented impedance analysis of the interleaved boost converter with coupled inductor (DPCI) loaded with CPL and compared the same with that of conventional boost converter. It is shown that, the DPCI exhibits better output impedance characteristics compared to conventional bidirectional dc/dc converter, such that the negative impedance effect of the CPLs is reduced. The stability analysis is presented in small-signal sense based on Bode and Nyquist plots. Comparison of several techniques for CPL compensation are summarized in Table 4 also. 5. Discussion Tightly regulated power electronic converters behave like CPL. Its negative incremental impedance reduces overall damping of power system and its interaction with other neighborhood converters induces instability in the overall system. Therefore, ensuring stability in the presence of CPL and the sub-system interaction-dynamics is of prime importance. Several criteria have been set forth to study the stability of 418

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ment schemes is very important to improve the overall system efficiency and smooth mode transitions. Selection of current sharing control scheme, switching control of source side converter between MPPT and voltage control mode, inclusion of battery state-of-charge (SoC) and its equalization, in the current sharing controller of storage units, are some of the important decisions which largely affects the efficiency of a power management scheme.

ques utilize the linearized system model and ensure the stability in the small-signal sense. While, the techniques which handle the system nonlinearities directly ensure system stability in large-signal sense. Given the overall system objectives, operating conditions, and other considerations, a suitable compensation technique can be chosen. To evaluate the performance of a particular CPL compensation technique, the amount of CPL it can compensate, robustness, the speed of response, noise immunity and its ability to ensure large-signal stability, are some of the most important parameters. The active damping technique, although widely applicable, is operating point dependent, and can compensate only a limited amount of CPL. Feedback linearization ensures compensation of any amount of CPL and large-signal stability, however, performs poorly in terms of robustness, speed of response and noise immunity. Power shaping stabilization, in which a linear model of the system is obtained by reformulating it in term of instantaneous power, ensures better robustness and noise immunity, compared to that with feedback linearization, however, it requires current loop bandwidth to be sufficiently higher than that of voltage-squared loop. Pulse adjustment technique ensures robustness and fast response, however, it has limitations of poor line rejection, operates in DCM and ensures stability with a limited range of CPL. Passivity based control has poor noise immunity due to the presence of differentiator and is sluggish in response. Synergetic control, similar to sliding mode control in many respects becomes problematic in DCM and is sensitive to high frequency noise. Furthermore, the dynamics of the macro-variable does not ensure finite time converge, thus reaching phase response is slow, because response becomes extremely slow in the vicinity of switching function. Sliding mode control technique ensures invariance to matched uncertainties and variations in the system parameters, can compensate any amount of CPL, and ensures system stability in a large-signal sense. Furthermore, sliding mode control of dc/dc converter provides better steady-state and dynamic response, less EMI, and results in an inherent order reduction, compared to linear controllers. Depending on system specific requirements a suitable CPL compensation technique may be chosen.

7. Conclusion The stability of the dc distributed power systems has been a major challenge due to the inherent nature of uncertainty in power generation of renewable energy source, nonlinearities induced by indispensable power converters and interaction among the dynamics of different subsystems. The pervasiveness of tightly regulated power converters i.e. constant power loads in modern dc microgrids make them more prone to instability. Nonlinear nature of CPLs adds complexity to the control and its negative incremental impedance characteristics not only induces instability at the point of load, but also reduces the effective damping of the system. Low damping and instability caused by negative incremental impedance of CPLs in a dc microgrid, are the real challenges. This paper has presented an overview of modeling, behaviour, and effects of CPLs in dc power systems. Various stability criteria to study the stability of a dc power system in the presence of CPLs have been reviewed with the merits and associated challenges of each criterion. These criteria are discussed for both single-input singleoutput (SISO) and multiple-inputs multiple-outputs (MIMO) system stability point of view. This helps reader/researcher in the selection of suitable stability criterion accordingly for their ease in the analysis of complex nonlinear systems. Furthermore, several techniques to compensate the destabilizing effect of CPL are classified and an extensive review has been presented for each technique with its salient features. Linear and nonlinear control techniques are discussed for the mitigation of the CPL nonlinearities in the context of small-signal and largesignal disturbance around operating points. Finally, a discussion and brief overview of recent challenges in dc microgrids have been presented.

6. Recent challenges in DC distribution systems References Renewable sources based dc microgrids, a class of dc distributed power systems are gaining increased attention to integrate local renewable sources and storage units to supply high quality and reliable power to the local loads. A dc microgrid may be connected to the utility grid for bidirectional power exchange, and can operate in island or grid-connected mode. The microgrids present a complex dynamics due to a number of sources in parallel, tightly-regulated converter behaving as CPL, uncertainties associated with renewable power sources, and interaction-dynamics in different operating modes. To ensure the stability of dc microgrids in different operating modes, it is necessary that detailed model of each subsystem be developed, to obtain an aggregated model of the system. The aggregated model can be used to analyze the effect of different controller parameters (voltage, current and droop control), system parameter including resistance of connecting cables and CPL, on the system stability. In an ac power system, supply frequency being universally constant around the network is used to estimate the load demand. However, in dc power systems, there is no such parameter to estimate the load demand. Therefore, load demand estimation in dc power systems is really a challenge, particularly when load consists of CPL, CVL and constant current components. In dc microgrid serving large number of electric vehicles, the aggregated charging load profile is of pulse power load (very high load for some time and low or no load for other time duration, having a low frequency periodicity). Such high magnitude pulse power loading may be detrimental to the system stability. Therefore, extensive analysis of dc microgrids with pulse power loads and its mitigation could be an important topic of research. Development of efficient power manage-

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