Robust Controller for Limit Cycle Elimination of Two Input DC-DC
Converter with Constant Power Load in Dc µGrid Applications
CH. NAYAK BHUKYA, B. AMARENDRA REDDY, ALLAM VENKATESH, T. R. JYOTHSNA
Department of Electrical Engineering,
Andhra University,
Vishakhapatnam, Andhra Pradesh,
INDIA
Abstract: - Direct Current (DC) µgrid is gaining more attention than Alternating Current (AC) µgrid, because
of their simplicity in structure, which raises the utilization of dc power sources. However, for proper operation
of dc µgrids, dc-dc power converters are required. MIMO converter systems are more commonly used in µgrid
applications because of their vast benefits over SISO converter systems. The power converter operates under
the previous state of tight control regulation which acts as constant power loads (CPL). Most loads in dc µgrid
are CPLs, hence, one of the prime challenges in dc µgrid with CPLs is, an exhibition of negative incremental
impedance (NII) which causes limit cycle behavior in the system. which tends to undesirable operations of the
upstream converter and dc µgrid. The proposed double integral sliding mode controller (DISMC) control
technique is designed to control and eliminate the dc µgrid bus voltage variations due to uncertainties of load
and CPLs limit cycles. This control technique guarantees fast trainset response over system uncertainties and
CPLs limit cycles. To validate the robustness of the proposed DISM controller, it is designed and simulated in a
MAT lab environment.
Key-Words: - Dc µgrids, dc-dc two input power converter, CPLs, DISMC, INI, Limit cycle of CPLs, nonlinear
loads.
Received: January 28, 2024. Revised: May 12, 2024. Accepted: June 3, 2024. Published: July 29, 2024.
1 Introduction
As you can see from the title of the paper you must
in the recent past, the adoption of dc µgrids is
increasing in off-grid applications such as electric
vehicles, data communication centers, electric
aircraft, and shipboard, among others as depicted in
Figure 1 over the alternating current (ac) µgrids.
The Direct current (dc) µgrids are gaining more
popularity. Its main benefits include resilience, ease
of control, simplicity in integrating renewable
energy sources, and the absence of reactive power
management requirements (RES), etc., [1], [2]. The
main problem with a dc grid is integrating
renewable energy sources (RES), which can be
made easier by using dc-dc power converters, [3].
The dc-dc power converters are either Single Input
converters (SIC) or Multi Input converters (MIC).
As compared to SICs, MICs have remarkable
benefits like flexibility, reliability, and efficiency in
operation. Owing to this, MICs are very commonly
used in the integration operation of dc µgrid, [4].
Most loads linked to multi-input dc-dc power
converters in dc grid applications are tightly
regulated, forcing the loads to act as constant power
loads (CPLs). The loads have negative increment
impedance (NII) characteristics because of CPL's
behavior. This will cause the output to exhibit limit
cycle behavior or oscillatory response. As a result,
the switching components will be under extreme
stress, which will raise the temperature and cause
problems with voltage instability in the converter
system, [5]. As a result, many methods for dc µgrids
with CPLs have been discussed to overcome the
instability difficulties; among these methods, sliding
mode controller (SMC) is widely employed in the
dc µgrid application, [6].
SMC is also known as a variable structure
control system since it is a nonlinear controller
whose control law alternates between two distinct
continuous structures (VSS). In comparison with
other types of nonlinear controllers, SMC gained
more attraction due to its robust capabilities and
ensures the stability towards variations in the
parameters and load uncertainties. Moreover, the
design choice of SMC is easy and more flexible, [6],
[7]. Because of its viable nature, SMC is the most
adoptable base line controller in the field of
integrated power converter dc µgrid with CPLs, [8].
But frequently, power converters' inconsistent
operating frequency, chattering around the
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.22
Ch. Nayak Bhukya, B. Amarendra Reddy,
Allam Venkatesh, T. R. Jyothsna
E-ISSN: 2224-2678
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switching surface, and steady state error prevent
them from being used for real-time SMC adoption,
[9]. Several adaptive strategies using SMCs help to
alleviate the variable frequency issue. Higher-order
sliding mode control solutions are implemented to
mitigate the chattering tendency. Many scientists
have attempted to reduce SSE by including an extra
integral term in the SMC state variable; this method
is known as an integral sliding mode controller
(ISMC), [10], [11], [12].
Controller
DC Boost
Converter
Load
AC
Constant Power Loads
(CPLs)
Constant Voltage Load
(CVL)
Tightly regulated
dc-dc converter
Telecom System
Electric Vehicle
(EV)
SMPS
DC-DC Converter
DC-DC Converter
DC-DC Converter
DC-DC Converter
DC BUS
DC-DC Converter
DC-DC Converter
DC-DC Converter
DC-DC Converter
AC-DC Converter
Solar Energy
Wind Energy
Fuel Cell Energy
Battery Energy
Fig. 1: Structure of dc µgrid with CPLs and CVL
The main drawback of ISMC is that, will not
mitigate the SSE of the converter completely, [13].
The mitigation of SSE can be achieved by the
increment in order of the ISM controller, i.e.by
adding an additional second order integral term,
which is called double integral sliding mode
controller (DISMC), [14], [15], [16].
Hence, the main objective of this article is to
present, the modeling and robustness analysis of a
double integral sliding mode controller (DISMC) for
two input dc-dc power converter with CPLs in dc
µgrid applications.
The existing research is mainly focused on the
regulation of SISO dc-dc converters with constant
voltage loads. Here, the voltage across the load is
maintained constant and the current variation is
observed depending on changes in load resistance.
Most of the literature discusses SISO converter
regulation using conventional SMC. The main
drawback of SMC is the chattering effect due to
high frequency phenomena weakens the system
stability. To overcome this drawback an integral
sliding mode controller is implemented. The main
limitation of an integral sliding mode controller is it
will not eliminate the steady-state error. To
overcome this double-integral sliding mode
controller (DISM) is used. The novelty of this paper
is the regulation of a two-input integrated converter
system under constant power load. Constant power
load requires voltage and current product constant at
load, and it is a non-linear characteristic
(hyperbola). To maintain constant power at the load
terminals of a multi-input and multi-output dc-dc
converter double integral sliding mode controller
(DISMC) is considered which can be used for dc
µgrid applications.
A detailed modeling, design, and analysis of
two input dc-dc power converters with CPLs in dc
µgrid applications with DISM controller are
enumerated as follows: (i) section 2 problem
formulation discussed (ii) section 3 problem to
solution discussed (iv) section 4 brief description,
modes of operation of two input dc-dc converter
with CPL and its state space equations are discussed
(v) section 5 Constant power load and its limit
cycles impacts are discussed (vi) section 6
mathematical modeling of DISM controller are
discussed (vii) section 7 Simulation results under
different conditions are discussed.
The contribution of this article is as follows.
1. Modeling of two input dc-dc with CPL.
2. Performed impact analysis of CPL on the
proposed system.
3. Elimination of Limit cycles with the help of
DISM controller.
4. Validating the robustness of the proposed DISM
controller under uncertainty situations.
2 Problem Formulation
When CPLs are under the umbrella of a dc µgrid
through a two-input dc-dc converter, it is prone to
characteristics of negative increment impedance
(NII), owing to this there will be a limited cycle
behavioral operation of the system. It will lead to an
unstable situation, and switching devices will
undergo stress and be prone to failure of the system,
it is the major challenge in dc µgrid with CPL's
operation.
3 Problem Solution
To mitigate the challenges associated with CPL's
operation in dc µgrid applications, a robust
controller is desirable. This work deals with a robust
sliding mode controller. The main limitation of a
single integral sliding mode controller is it will not
eliminate the steady state error caused by the CPLs
completely. To overcome this, an addition of
another integral component to the single integral
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DOI: 10.37394/23202.2024.23.22
Ch. Nayak Bhukya, B. Amarendra Reddy,
Allam Venkatesh, T. R. Jyothsna
E-ISSN: 2224-2678
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Volume 23, 2024
sliding mode controller, which is known as a
double-integral sliding mode controller (DISM). In
this article, DISMC is used to alleviate the SSE and
eliminate the limit cycles due to the CPL's
operation. The proposed DISMC gives accurate
results for the operation of two input dc-dc power
converters with CPLs in the application of dc µgrid.
The DISMC's design and guiding principles are
covered in detail in the section that follows.
4 Two Input DC-DC Converter with
CPL and CVL
The two input SEPIC converters with CPL and CVL
are depicted in Figure 2. It consists of two switches,
these switches are regulated independently, and
each may undergo either of two positions (ON and
OFF), and collectively behave as a control function
of the converter circuit. The two switches are
represented by S1 and S2. The switch positions are
represented by “u1 and u2”, which can take the
discrete values of {0,1}, these switch positions are
controlled by the duty cycle ratios (). The
duty cycle of the converter is controlled by pulse
width modulation, and it is extremely important for
power processing. The two switches (S1, and S2) can
be regulated in three different ways based on
switching periods of duty-cycles. These are: (i) d1=
d2 (ii) d1 < d2 (iii) d1 > d2.
ic2
E1
S1
S2
CVL
u1=1
i1
i2
ic1
io
L2
u1=0
u2=1 u2=0
C1
BUCK SEPIC Converter
C
P
L
C2
Parallel Connected Loads
(CPL+CVL)
Source Bus Load Bus
L1
E2
Fig. 2: Schematic diagram of two-input SEPIC
converter with CPL and CVL
To accomplish the anticipated output of the two-
input converter, it will undergo three modes of
operation.The state-variable modeling of this
converter circuit is obtained by applying Kirchhoff’s
current and voltage laws. Here, L&C elements
without parasitic resistance are considered in writing
the equations. The three modes of operation of the
converter are given in the form of differential
equations, state-space averaging has been proven to
be an effective analysis method. which are written
as follows:
Mode-1: (S1=u1=1, D1=u1=0; S2=u2=1, D1=u1=0)
(1)
(4)
Mode-2: (S1=u1=1, D1=u1=0; S2=u2=0, D1=u1=1)
Mode-3: (S1=u1=0, D1=u1=1; S2=u2=0, D1=u1=1)
It is observed that the mathematical state space
equations of the three modes of operation of the
converter have inter relation to each other, which
influences the output of the converter.
5 Constant Power Loads and its Limit
Cycles
Constant power loads are electrical devices or
systems that consume a constant amount of power
regardless of variations in voltage or current. The
basic CPL model depicted in Figure 3(a), according
to Ohm's law like resistive loads, CPLs have a fixed
resistance and consume constant power (P=V2/R),
[17], [18], [19].
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C
P
L
Vcpl
Icpl
Vcpl Icpl=P/Vcpl
(a)
Icpl
Vcpl
ΔIcpl
ΔVcpl
Pcpl =Vcpl*Icpl = Constant
ΔVcpl/ΔIcpl <0
(b)
Fig. 3: (a) CPL model. (b) Constant power load V-I
Characteristics
In Figure 3(a) Icpl is current through the CPL
terminal which is expressed in (13)
(13)
For a given operating point
, the CPL
small signal model can be approximated by a
straight-line tangent to the curve as illustrated in
Figure 3(b), which is computed as:
(14)
Equation (14) can be represented with a
negative resistance
and constant
current source “I” given in (15)
(15)
In general, Source-side converters and CPLs are
often close to each other in power electronic
converter systems, like EVs, electric shipboard, and
the telecom sector. A tightly controlled Point of
Load (POL) converter behaves like as CPL depicted
in Figure 4(a) and is susceptible to instability, due to
negative incremental impedance (NII)
characteristics. This can cause limit cycle
oscillations in the system, reduce system damping,
voltage collapse, high stress on switching devices,
and even system shutting down. The limited cycle
oscillations which are the biggest challenge in dc
µgrid with CPL's operation, [20], [21], [22],
depicted in Figure 4(b).
Two Input
Source
Converter
DC bus
Tightly
Regulated
Load Converter
L
O
A
D
Controller
Constant Power Load (CPL)
Source-1
Source-2
(a)
(b)
Fig. 4: (a)Point of Load converter acts as CPL (b)
limit cycle oscillations of CPL
6 Double-Integral Sliding Surface
Application to PWM-Based
Indirect SM Controller
This section explores the application of DISM
configuration to the PWM-based SM controller for
buck SEPIC converter with CPLs of voltage and the
current control, [23], [24], [25]. Furthermore there
are several kinds of nonlinear controllers; SM
Controllers are the most often used due to their
enormous advantages when it comes to handling
uncertainty and elimination of limit cycles caused
by CPLs.
A practically single integral sliding mode
controller (ISM) can eliminate steady state and
transient error to some extent only i.e. it will not
alleviate completely. To overcome these limitations
an additional integral term is added to ISM, and it is
called double integral sliding mode controller
(DSIM) which is represented in equation (16).
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The use of a double-integral term in a controller
to improve steady-state accuracy while addressing
stability concerns. The additional double-integral
term i.e. for i=1, 2…n-1, is introduced
to correct errors in the indirect integral computation
of Integral Sliding Mode (ISM) controllers. By
incorporating an integral closed-loop, the steady-
state errors of the controlled state variables are
indirectly reduced. This approach is referred to as
the double-integral (indirect) sliding mode (DISM)
controller, [21].
6.1 Control Law of DISMC for dc-dc
Converter
Figure 5 describes the realization scheme of the
PWM-based DISM controller for two input dc-dc
converters with CPL. The form of control law used
in the design of the DISM controller for a two-input
dc-dc converter is given in (17) and the sliding
surface is defined using equation (18).
(17)
(18)
where ‘µ’ represents the control logic for regulating
the power switch, and α1, α2, α3, and α4. represent the
desired sliding coefficients.
The procedure to derive double integral control
law for voltage control and current control loops of
integrated dc-dc converter is enumerated as follows.
(i) Define the switching surface equations for
voltage and current loops of the integrated dc-dc
converter. (19)
(20)
In a two-input dc-dc converter, the state
variables xi, i = 1, 2, 8, are utilized to define the
switching surfaces of the voltage and current loops.
Current error and voltage error are described by the
variables, x1. And x5. The variables, 2. And 6.
describe the dynamics of the voltage error and the
current error, respectively. The variables x2 and x6
describe the voltage and current error integrals,
respectively. The double integrals of the errors of
the voltage and current are described by x4 and x8,
respectively. Equations (21) and (22) are used
respectively, to provide a mathematical description
of these.
(21)
(22)
where Vref, signifies reference voltage and
current, Vo, and are instantaneous voltage and
current outputs of two-input integrated converter
respectively. βv, denotes the feedback network
ratios of voltage and current loops respectively.
(iii) By considering the first order
differentiation of the equations (21) and (22) the
dynamics of the stated state variables are derived
(22). The dynamic model of this integrated dc-dc
converter system is described using equations (23)
and (24).
(23)
(24)
(iv) Determine the linear part of sliding mode
control law and it is denoted by “Ueq” i.e.,
equivalent control. The equivalent control signal of
voltage and current loops of the integrated converter
of DISM are denoted by and . To obtain the
equivalent control laws for voltage and current
loops , consider the derivative of switching surface
equations defined using (19) and (20), and equate
these to zero. As the derivative of switching surface
equations are functions of the equivalent control
signals and equated to zero results and
These are given in equations (25) and (26).
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(25)
(26)
Where
(v) (v) PWM is generated by comparing ramp signal
with VC. Use an indirect SM control approach to
develop a set of equations for the control signal (VC)
of voltage loop and a ramp signal (Vramp) with peak
magnitude, [26], [27]. VC is defined as in (27).
Similarly, iC is defined as in equation (28).
(27)
(28)
(vi) Compare the equation (27),(28) with equations
(25),(26) and coefficients are defined as follows.
(29)
(30)
where ‘’ and ‘’ are continuous and bounded
by 0 and 1.
,
.
E1
Vo
E2i2
i1
Vc1ic1
ic2
Vc2
ic2
1
2
i1
i2
ic1
io
2
1=0
2=1 2=0
1
Two Input DC-DC Converter
2
Parallel
Connected Loads
(CPL
+CVL)
Source Bus Load Bus
1
2
X
u
D1 D2
Double Integral SMC
Vc1
Vc2
ic1
ic2
i2
i1
E2
E1
PWM
PWM
Iramp Ic
Vramp Vc
=
111
X
u
Fig. 5: realization scheme of the PWM-based DISM
controller for two input dc-dc converter with CPL
7 Simulation Results and Discussion
The theoretical considerations are verified to
confirm the performance of the proposed Double
Integral Sliding Mode Controller with CVL and
CPL, it is executed with the structure of Figure 5 in
MATLAB environment. Two-input dc-dc converter
with CVL and CPL simulations are executed under
different situations. The system parameters are
chosen as illustrated in Table 1.
Table. 1. System and Control Parameters
Description
Value
Input voltages (E1, E2)
Output voltage (Vo)
Switched frequency (fBW)
Inductances (L1, L2)
Capacitances (C1, C2)
feedback network ratio of voltage
(βv)
feedback network ratio of current
()
Voltage Coefficients (K1v, K2v,
K3v)
Current Coefficients (K1i, K2i,
K3i)
60V,36V
48V
25kHz
650µH,600µH
20µf,220µf
0.1250
0.4
-2.255,11.231,600
-3.04,13.24,854
Figure 6, Figure 7, Figure 8, Figure 9, Figure
10, Figure 11 and Figure 12 show the unstable,
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stable operation of the converter under the impact of
CPL, and uncertainties in source, load is modeled.
The converter operates with duty cycles of d1>d2 as
depicted in Figure 6(a) and Figure 6(b).
Performance results were analyzed in three different
situations. (i) Consider the converter without
DISMC and with CPL, the converter exhibits limit
cycle behavior. The converter state-variables Vc2 ,
IL1 are depicted in Figure 7, and Vc1, IL2 are depicted
in Figure 8. (ii) Consider the converter with DISMC
and with CPL, and the closed-loop converter does
not exhibit the limit cycles the converter state-
variables Vc2 , IL1 are depicted in Figure 10, and Vc1,
IL2 are depicted in Figure 11. Here, the DISM
controller eliminated limit cycles caused by the
CPL. (iii) Consider the converter with DISMC and
with CPL under source and load perturbations
depicted in Figure 12. The closed-loop converter
rapidly reaches its reference value without any limit
cycles. The results of state variables are depicted in
Figure 13 and Figure 14. Under these situations the
robustness of the DISM controller reflects provides
constant power to CPL without limited cycles and
disturbances.
(a)
(b)
Fig. 6: Simulated Duty Cycles of the proposed
system: (a) Control input of Switch, S1; (b) Control
input of Switch, S2
(a)
(b)
Fig. 7: Simulated response of proposed system
without DISMC: (a) Capacitor Voltage, Vc2; (b)
Inductor Current, IL1
(a)
(b)
Fig. 8: Simulated response of proposed system
without DISMC: (a) Capacitor Voltage, Vc1; (b)
Inductor Current, IL2
(a)
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(b)
Fig. 9: Simulated response of the proposed system
with DISMC: (a) Input Source Voltage, E1; (b) Input
Source Voltage, E2
(a)
(b)
Fig. 10: Simulated response of the proposed system
with DISMC: (a) Capacitor Voltage, Vc1; (b)
Inductor Current, IL2
(a)
(b)
Fig. 11: Simulated response of the proposed system
with DISMC: (a) Capacitor Voltage, Vc2; (b)
Inductor Current, IL1
(a)
(b)
Fig. 12: Simulated uncertainties in source and load
response of the proposed system with DISMC:(a)
uncertainty in Input Source Voltage, E1;(b)
uncertainty in Input Source Voltage, E2
(a)
(b)
Fig. 13: Simulated uncertainties in source and load
response of the proposed system with DISMC:(a)
Capacitor Voltage, Vc1; (b) Inductor Current, IL2
(a)
(b)
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Fig. 14: Simulated uncertainties in source and load
response of the proposed system with DISMC:(a)
Capacitor Voltage, Vc2; (b) Inductor Current, IL1
8 Conclusion
This work presents the design aspects of the DISM
controller for a two-input dc-dc converter with CVL
and CPL. DISM controller eliminates limit cycle
behaviour caused by the constant power loads in dc
µgrid applications specifically in electric shipboard,
electric vehicles etc., The simulation results prove
that the use of a double integral sliding mode
controller offers robustness, fast response, reduced
steady-state error, and disturbance rejection, and
handles the negative increment impedance (NII)
effect sensibly well. It is an effective control
solution for applications where precise regulation of
output voltage is required despite source and load
uncertainties. Results gave a close treaty between
theoretical study and simulation. Future research
will examine the potentiality of DISM controllers
for fault tolerant converters introduced in dc
microgrid to achieve extreme reliable operation and
quick steady state response.
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Allam Venkatesh, T. R. Jyothsna
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Ch Nayak Bhkuya, B.Amarendra Reddy, and
A.Venkatesh have carried out the mathematical
modelling and prescribed analysis of this article.
- Ch Nayak Bhukya has written this article and has
performed simulations and taken the results for
this article.
- B.Amarendra Reddy, and T.R.Jyothsna have
supervised this article.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2024.23.22
Ch. Nayak Bhukya, B. Amarendra Reddy,
Allam Venkatesh, T. R. Jyothsna
E-ISSN: 2224-2678
205
Volume 23, 2024
