Intelligent and Fast Model-Free Sliding Mode Control for Shipboard DC Microgrids
Mahdi Mosayebi , Seyed Mohammad Sadeghzadeh, Meysam Gheisarnejad , and Mohammad Hassan Khooban ,Senior Member, IEEE
Abstract— This article proposes an intelligent and fast con- troller for the parallel dc–dc converters in the islanded dc microgrids (DCMGs) in the shipboard power system (SPS) appli- cations. The existence of constant power loads and propulsion loads in the SPS makes the voltage regulation an important issue in the islanded DCMG. In this article, the goal is to provide a fast response and proper voltage regulation and optimal current sharing in the stand-alone shipboard DCMG. To achieve this, a nonlinear I−V droop control is first designed in the primary control level to ensure the basic performance. Then, a new model- independent technique, entitled intelligent single-input interval type-2 fuzzy logic controller (iSIT2-FLC) combined with sliding mode control (SMC), is suggested to further improve the voltage regulation and current-sharing accuracy. In this control scheme, an extended state observer (ESO) is employed to estimate the unknown DCMG dynamics, whereas the SMC is adopted to eliminate the ESO estimation error. Moreover, the dynamic consensus algorithm (DCA) is implemented at the secondary control level to achieve coordinated control between DGs in the SPS. The effectiveness and applicability of the new suggested control approach are validated with hardware-in-the-loop (HiL) experimental results.
Index Terms— DC–DC converter, hardware-in-the-loop (HiL), intelligent control, shipboard dc microgrid (DCMG).
I. INTRODUCTION
I
N RECENT years, dc microgrids (DCMGs) due to the advantages, such as simple control, high efficiency, no reac- tive power control, no harmonic issues, no synchronization of power sources, and the easy interface between power sources and electronic loads, tend to be significantly preferred over ac microgrids [1], [2]. Moreover, increasing concerns with regards to pollution and greenhouse gas emissions have led to the increasing use of renewable energy sources (RESs) and energy storage systems (ESSs) in marine vessel applications [3], [4]. In addition, the utilization of fuel cells (FCs), which is known as a low-emission power source with other RESs andManuscript received July 17, 2020; revised October 7, 2020 and November 21, 2020; accepted December 17, 2020. Date of publication December 31, 2020; date of current version August 24, 2021.(Corresponding author: Seyed Mohammad Sadeghzadeh.)
Mahdi Mosayebi and Seyed Mohammad Sadeghzadeh are with the Fac- ulty of Engineering, Shahed University, Tehran 3319118651, Iran (e-mail:
[email protected]; [email protected]).
Meysam Gheisarnejad is with the Department of Electrical Engineer- ing, Najafabad Branch, Islamic Azad University, Isfahan 14778-93855, Iran (e-mail: [email protected]).
Mohammad Hassan Khooban is with the Department of Electrical and Computer Engineering, Aarhus University, 8200 Aarhus, Denmark (e-mail:
Digital Object Identifier 10.1109/TTE.2020.3048552
ESSs, is introduced to improve the reliability and the efficiency of the shipboard power systems (SPSs), thus becoming the trend of the future all-electric ships (AESs) [5], [6].
In the configuration of the SPS, there exist several types of loads, such as ship’s lightening, pulsed power loads, propulsion loads, and constant power loads (CPLs), which are connected to the dc bus through paralleled power electronic converters [7]. With paralleling the converters in the stand- alone DCMG in the SPS, voltage regulation and current- sharing issues among converters have appeared. Inappropriate current sharing among paralleled converters in the SPS may lead to overloading of converters. Droop control is one of the most useful methods, which is used in the paralleled converters in the DCMGs. The droop control method often uses a virtual resistance (VR) in order to improve the current sharing among paralleled converters [8]. Generally, the droop control method is applied in the primary control level inside the hierarchical control structure to guarantee the current sharing. Furthermore, the current sharing degrades when the line impedance of each load or converter is not the same, which is inevitable in practical applications [9], [10].
In previous works, various control strategies have been proposed in order to improve the current sharing among paralleled converters and enhance the voltage regulation in the islanded DCMGs. In [11], an adaptive droop control method is proposed for balancing the state of charge (SoC) for battery management systems. In [12], secondary and tertiary control levels are suggested to eliminate the voltage deviation and improve the current sharing and energy management between DCMGs in a cluster. Nonlinear droop control is illustrated in [13] to achieve voltage restoration in paralleled droop- controlled converters. In addition, to achieve fast dynamic response performance in the DCMG, an I−V droop control is proposed in [14].
However, through all loading conditions from light to heavy load conditions, the aforementioned methods suffer from poor current sharing. In addition, in the SPS with pulsed power loads, the response of the controller in the primary and secondary control levels plays an important role in the stability of the stand-alone DCMG. In addition, the slow response of the controller can prolong the recovery time of the voltage and current, which has an adverse effect on the stability of the SPS. Moreover, most of the current-sharing methods are based on the high-bandwidth communication network. In [15], a centralized configuration of the communication network is
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used to adjust the VR of each converter based on the rated power and average voltage of the DCMG. Meanwhile, the centralized method suffers from reliability problems since it exposes a single point of failure (SPOF). Therefore, distributed control schemes have drawn a lot of attention due to the advantages, such as reliability, flexibility, modularity, and the absence of SPOF [16].
The dynamic consensus algorithm (DCA) is one of the most promising distributed control approaches, which is imple- mented at the secondary control level [17]. In the DCA, the paralleled converters exchange information with neighbor- ing units on a sparse communication network, which decreased the communication and computation costs. In [18], a DCA in the secondary control level is proposed to regulate the voltage in the DCMG. However, the proposed conventional controller is not fast enough to compensate for the voltage fluctuations during a fast transient caused by pulsed power loads.
Robustness to the uncertainties and nonlinearities makes fuzzy logic control (FLC) a powerful tool for a wide range of control problems. This is due to some inherent advantages of FLCs, such as ameliorating the performance and flexibility of the control systems by expert knowledge in the control design stage. Another control scheme is to design the sliding mode control (SMC) based on a fuzzy model [19] for compensating the disturbances to offer precise power sharing. However, the efficiency of the suggested controller depends on a large number of rule-base and membership functions (MFs) for handling the load changes. In [20], a fuzzy sliding-mode control (FSMC) approach is adopted for reliable power sharing in a hybrid distributed energy plant, including FC, BESS, and photovoltaic in a microgrid configuration. However, the syn- thesis of a complete fuzzy rule set in the FSMC is a complex task when the number of parameters of the fuzzy is more than two. Based the sliding surface, the rule base of the Mamdani fuzzy is designed in [21] to ameliorate the power quality in the power distribution network during power system perturbations.
However, obtaining the control law of the suggested scheme needs an accurate model of the power system to achieve a desirable control performance from the power engineering point of view.
To overcome the aforementioned drawbacks, this article applies a novel nonlinear droop control approach in the primary control level to achieve fast and proper power sharing among RESs and ESSs, as well as desired current sharing from light- to heavy-load conditions. Recently, the advanced forms of FLC, such as interval type-2 FLC (IT2-FLC), have caught considerable attention, particularly for control of the unstruc- tured and nonlinear systems [22], [23]. Despite the majority of the research work accomplished was concentrated on using IT2-FLCs, it was been demonstrated in [24] that a single- input IT2-FLC (SIT2-FLC) gives better functional features and easier realization. Motivated by this idea, this work proposes a model-independent-based intelligent single-input IT2 fuzzy PI (iSIT2-FPI) and SMC in the primary layer to regulate the voltage of the SPS. By using the DCA in the secondary control level, the average value of the voltage is calculated, and the voltage restoration can be achieved to realize the accurate current sharing and voltage restoration. The main advantages
DC-DC converter DC Bus
D1
DC-DC converter
DC-AC converter DC Loads
Pulsed Power Loads
AC Loads DC-DC converter DC-DC converter
DC-DC converter
Fig. 1. Islanded DCMG in the SPS.
of the novel proposed controller can be stated as follows: 1) robustness over all types of disturbances and uncertainties; 2) easy implementation on hardware applications; 3) compared with the model-based control schemes [e.g., model predictive control (MPC) and linear matrix inequality (LMI)], any accu- rate dynamics and structural data of the power plant’s model for the controller design does not require to be identified; and 4) unlike the conventional FLCs (Type-1 FLC and IT2-FLC) that suffer the complexity of designing fuzzy sets, the iSIT2- FPI controller is designed by a single parameter, called the footprint of uncertainty (FOU).
II. SPS DC MICROGRIDMODELING
As illustrated in Fig. 1, in the islanded shipboard DCMG, several parallel DGs, including the PV systems, FCs, and ESSs, are connected to the dc bus, which is controlled with droop control to achieve power sharing among DGs. Further- more, CPLs, pulsed power load, and ac loads are connected to the dc bus via related dc–dc converters or dc–ac converters.
Moreover, to implement the distributed coordination con- trol among all units in a stand-alone DCMG in the SPS and to restore the voltage deviation caused by the droop control, bidirectional communication links are required. The communication network is used to share information between the neighboring DGs. Thus, a DCA distributed coordination control is applied at the secondary control level to ensure voltage restoration. In the following, each of the DGs and ESSs in the SPS is further investigated.
A. Photovoltaic Systems
The electrical characteristic of the photovoltaic systems (PVs), which converts the solar energy to electrical energy, can be represented as [25]
IPV=NpIph−NpIrs
exp
q VPV
kTTAIFNsNc
−1
(1) where VPV and IPV are the voltage and current of the PV system, respectively; Iph is the string’s short circuit current;k is Boltzman’s constant;TT is the p-n junction temperature;q is the unit electric charge; Irs is the diode reverse saturation current; Nc is the number of the series cells per module; Ns
and Np are the numbers of the series-connected modules per string and parallel string, respectively; and AIF is the ideality factor. Moreover, the PV temperature characteristic can be
considered as a function of the solar irradiation and ambient temperature as follows:
d TT
dt = 1 CPV
kin,PVφ−VPVIPV
APV −kloss(Tc−Ta)
. (2) The PVs are connected to the dc bus via boost or buck converters based on the dc bus voltage and output voltage of the PVs.
B. Battery Energy Storage System
The reliability and efficiency of the islanded DCMG in the SPS can be improved with the utilization of the ESSs, such as lithium-ion batteries. The ESSs can be modeled with a controllable voltage source and internal resistance as follows:
E=E0− KQ Q−
i dt +Aexp
−B
i dt
. (3)
Moreover, the SoC of the battery can be expressed as SOC=100
1− 1
Q
i dt
(4) where Q is the capacity of the battery. More details of the battery modeling and definition parameters can be found in [3].
C. Fuel Cell Systems
An FC generates electricity from chemical energy through an electrolysis process with low to zero emissions. Generally, FCs have four basic components known as FC stack, which is generated current; fuel processor to convert fuel into a usable form; current converters; and heat recovery systems.
The dynamic model of the FC can be expressed as qH2
pH2 = Kan
√MH2
(5) wherePH2is the partial pressure of hydrogen and the dynamic of the partial pressure of hydrogen can be defined as follows:
d PH2 dt = RT
Van
qinH2−qH2out−qrH2
(6) qrH2 = NoNsIFC
2F =2KrIFC. (7)
PH2 = 1/KH2
1+(Van/KH2RT)S
qH2in −2KrIFC
. (8) The output voltage of the FC system can be computed by
V =E−Bln(C IFC)−RintIFC (9) where
C=No
Eo+ RT 2F log
PH2
√PO2
PH2O
. (10)
The details and definition of the FC system can be found in [26].
As mentioned earlier, the paralleled DGs in the stand- alone DCMG are controlled with dc–dc converters. However, in order to have proper voltage regulation in a transient condition and current sharing among DGs in the SPS, the par- alleled dc–dc converter should be controlled fast and robust.
In Section III, to achieve this goal, the new proposed controller is presented.
Fig. 2. Equivalent circuit of two parallel DGs in the SPS.
III. PROPOSEDCONTROLSCHEME
A. Nonlinear I–V Droop Control
Different lengths of the cable of each DGs in the DCMG in the SPS lead to unequal resistance, which degrades the current sharing among the DGs in the SPS. A typical scheme of two DGs that are connected to the dc bus is shown in Fig. 2, whereV1andV2are the output voltages of the DG1 and DG2, respectively. RL is the equivalent resistance of the common load. R1 and R2 are the line resistances between each DG to the load terminal, respectively. Here, the droop resistance of DG1 and DG2 are Rd1 and Rd2, respectively. The droop resistance can be defined as
Rdi = V Imax
(11) whereV is the desired voltage deviation from the reference voltage. Moreover,Imaxis the maximum output current of each DG. The total resistance from the DGs to the load can be calculated as follows:
Rd1 = Rd1+R1 (12) Rd2 = Rd2+R2. (13) Applying Kirchhoff’s laws to the equivalent circuit in Fig. 2, the difference between the current of the two paralleled DGs is given by
I1−I2= 2(V1−V2) Rd1
+ Rd2
+(Rd1)·(Rd2 )
RL
+ V1· Rd2
−V2· Rd1 Rd1
· Rd2
+ Rd1
·RL+ Rd2
·RL
. (14) As discussed in [9], using droop resistance can improve the current sharing among the paralleled converters in the SPS. Although selecting high droop gains yield better current sharing, it affects the voltage deviation [27]. Therefore, in the droop control strategy, it is a tradeoff between current sharing and voltage regulation. Furthermore, the unequal line resis- tance, which is nonnegligible in practical application, affects the current sharing.
In order to improve current sharing and achieve good voltage regulation in the droop control method, the droop resistance trajectory should be optimized. Several droop curves can be considered for a specified voltage deviation, from no- load to full-load operations of the dc–dc converters in the
Fig. 3. (a) Droop gains based on the output current for different values of n. (b) Different droop gain based on the proposed method forn=1, 3, 5, 7.
stand-alone DCMG. The droop control method can be denoted by the following equation:
Vi =V∗−Rdi· Iin
,n ∈ R+ (15) where Vi is the output voltage of the dc–dc converter, Rdi is the VR, and V∗ is the reference voltage. Here, Rdi can be expressed as follows:
Rdi= V Ii,max
n. (16) Fig. 3(a) shows the droop gains based on the output cur- rent of the converters for different values of n. As shown in Fig. 3(a), it is clear that the linear droop method has a small droop gain in the heavy-load condition, which degrades the current sharing. Also, the nonlinear droop methods suffer from poor current sharing in the light-load conditions. To overcome the limitation of the conventional linear and nonlinear methods and to satisfy good performance in the current sharing and voltage restoration, a novel nonlinear I − V droop-based control is proposed.
The I–V droop control can be achieved by increasing the current reference when the bus voltage decreases as follows:
Iref,i = V∗−Vi
Rdi . (17)
By expanding (15), with respect to the linear droop gain (i.e., n =1), the general equation of the droop curve can be achieved as follows:
Vi =V∗−Rdi,1·Ii−Rdi,n·Iin. (18) The selection of n (i.e., n = 3,5,7, . . .) is depending on the amount of voltage deviation and the maximum current value in the ith converter in the DCMG [13]. It should be noted that it is preferred to select the droop curves with odd
values to have a symmetrical performance for the bidirectional converter. Therefore, (18) can be rewritten as
Rdi,1·Ii =V∗−Vi−Rdi,n·Iin. (19) Finally, (19) can be express as
Iref,i = V∗−Vi
Rdi,1 − Rdi,n
Rdi,1 ·Iin
(20) where Rdi,1 is the linear droop gain and Rdi is defined in (11). As illustrated in Fig. 3(b), it can be concluded that the proposed droop method has both good features of linear and nonlinear droop methods. To put it another way, throughout all loading conditions from no-load to full-load for the converters in the SPS, the proposed method has maximum available droop gain, which satisfies the good current sharing among DGs.
B. Dynamic Consensus Algorithm in the Secondary Control Level
The secondary control level in the DCMGs can be imple- mented in centralized and distributed configurations. The centralized methods suffer from SPOF, which degrades the reliability of the DCMG [28]. Moreover, the implementation cost of this configuration is higher than the distributed method because the communication links between all DGs and the central controller are required. To overcome these drawbacks and to increase the flexibility of the system, a DCA is applied at the secondary control level in the DCMG in the SPS.
In this method, the information is exchanged through the communication network between neighboring DGs to reach an agreement on a specific quantity. Here, each DG in the DCMG only communicates the voltage of the related converter to updates the voltage reference. Therefore, all DGs can converge to the desired voltage.
The DGs in the DCA method are defined as nodes, and the communication links between DGs are considered as edges in a connected graph. For N-paralleled DGs in the DCMG in the SPS, the connected graph can be expressed as GDCMG = (N,E). Therefore, the DCA can be formulated as follows [29]:
xi(k+1)=xi(0)+ε·
j∈Ni
δi j(k+1) (21) δi j(k+1)=δi j(k)+ai j.
xj(k)−xi(k)
(22) wherexi andxj are the states in nodesi and j, respectively.
Let i,j = 1,2, . . . ,N,{i,j} ∈ E, and ai j indicates the connection status between node i and node j. Therefore, if there is a communication link between node i and node j, then ai j =1. Ni denotes the number of the DGs that are connected to nodei. Moreover, in (22),δi j(k)represents the accumulative difference between two units, and δi j(k) = 0.
In the DCA, the final consensus is xave = (1/N)N i=1xi(0) irrespective of any changes in xi(0), and the algorithm will converge to a proper average value [29]. Moreover, a constant edge weight that minimizes the convergence time for a given communication network can be expressed as follows:
ε= 2
λ1(L)+λn−1(L) (23)
Fig. 4. Proposed control scheme in the SPS.
where λ1(L) and λn−1(L) are the first and second largest eigenvalues of the Laplacian matrix. The Laplacian matrix of the network can be written as
L=
⎡
⎣
j∈N1a1j · · · −a1NT
·· ·
· ·
−a1NT · · · ·
j∈NTaNTj
⎤
⎦. (24)
Since, in the considered DCMG, only the voltage of the DGs in the DCMG is exchanged, (21) and (22) can be rewritten as
Vref,i(k+1)=Vi(0)+ε·
j∈Ni
δi j(k+1) (25) δi j(k+1)=δi j(k)+ai j.
Vj(k)−Vi(k)
(26) where Vref,i is the reference voltage of the DCMG.
However, in the case of pulse loading conditions in the DCMGs, the conventional controllers in the secondary control level are not fast enough to compensate for the voltage.
In Section IV, a new fast and robust controller with high bandwidth is proposed for the DCA-based secondary control level in the SPS.
IV. CONTROLLERDESIGN
In this section, a novel model-free control scheme that con- sists of three subcomponents is introduced for the stand-alone DCMG in the SPS. First, by the knowledge of the control input and output parameters, an extended state observer (ESO) is used to estimate the process uncertain data. Second, a model- free-based iSIT2-FPI is proposed to suppress the high-order derivative output, enter required control specifications, and reduce the complexity of the existing controller. Finally, an SMC is implemented to eliminate the ESO estimation error.
The overall control diagram of the proposed control strategy is shown in Fig. 4. As shown in Fig. 4, each dc source is connected to the dc bus via a related dc–dc converter. In the secondary control level, the voltage reference of the DCMG in the SPS is calculated by the DCA, and the compensating voltage term (V)is injected into the primary control level.
In the primary control level, the proposed nonlinear droop control generates the current reference based on the measured capacitor voltage (Vc) of the dc–dc converter, the voltage reference, and V. Finally, the proposed model-free iSIT2- FPI adjusts the duty cycle of the dc–dc converter. In Section IV-A, a novel model-free iSIT2-FPI controller is presented.
Fig. 5. Schematic of (a) SIT2-FPI structure and (b) triangular MFs.
A. PI Type SIT2-FLC
1) Basic Structure of the Interface Mapping IT2-FPI:
Fig. 5(a) shows the general form of one to one inference map- ping SIT2-FPI controller. The error signal is normalized by the scaling factor (SF)Ge.λis selected as an input coefficient of the SIT2-FPI structure. Therefore, the error signal (e) is converted into λ. A simple normalization ensures thatλ will be within the universe of discourse [−1, +1]. As depicted in Fig. 5(a), the output of the designed iSIT2-FPI controller is transformed into the control signal (u)as follows:
u(t)=kpψ+ki
ψ (27)
where
kp=kp0Gu; ki =ki0Gu (28) where kp0 and ki0 represent the baseline PI controller coef- ficients and Gu denotes the output SF, which is defined as Gu =G−e1.
2) SIT2-FLC Design Scheme: The rule structure of each SIT2-FLC is defined as [30]
ri :IFe is ˜Ai THEN eIT2 is Bi, i =1,2,3. (29) As depicted in Fig. 5(b), the antecedent MFs are made by the triangular type of IT2 fuzzy sets (IT2FSs) ˜Ai. The values of the crisp consequents Bi are generated as B1= −1, B2 =0, andB3=1. Also, based on the lower MF (μ)and upper MF (μ), the IT2FSs can be defined. In Fig. 5(b),li’s (i =1, 2, and 3) are the height of the lower MFs. Here, the symmetrical MFs are adopted with the following assumptions:l1=l3=1−δIT2
and l2 = δIT2, where δIT2 is the main IT2-FPI coefficient, which should be adopted according to the required system performance.
The output of the IT2-FLC can be defuzzified as ϕo = (ϕor+ϕol)/2 by using the center of set type reduction scheme [24]. ϕro and ϕol are the end points of the type reduced set,
TABLE I
DEFINITIONS OFSIT2-FLC UNDERDIFFERENTCONSTRAINTS OF
Fig. 6. Illustration of the control surfaces with different values ofδIT2.
which is achieved by [31]
ϕol = L
n=1μ¯A¯n(σ)·Bn+N n=L+1μA¯
n(σ)·Bn
L
n=1μ¯A¯n(σ)+N n=L+1μA¯
n(σ) (30)
ϕor = R
n=1μA¯
n(σ)·Bn+N
n=R+1μ¯A¯n(σ)·Bn
R n=1μA¯
n(σ)+N
n=R+1μ¯A¯n(σ) (31) where L and R denote the switching points.
Moreover, the fuzzy mappings (FMs) of the SIT2-FLC can be expressed as [32]
ψ(σ)=σ·k(|σ|) (32) wherek(σ)is a nonlinear coefficient derived from SIT2-FLC, which is described as
k(e)=1 2
1
δIT2+e−αλ +δIT2−1 δλ−1
. (33)
As described in Table I, by definingεo(σ)=ψ(λ)−λ, for various constraints ofδIT2, the three mapping IT2-FLC modes based on control curve (CC) can be obtained. In Fig. 6, for δIT2 = 0.18, δIT2 = 0.375, δIT2 = 0.52, δIT2 = 0.631, and δIT2 = 0.795, the examples of A-CCIT2, S-CCIT2, and M- CCIT2 are presented. More details about the CCs generations of the SIT2-FLC can be found in [24].
Remark 1: In comparison with the conventional IT2-FLC [33], the SIT2-FLC does not need to adjust MFs and rule base. In fact, by adjusting a single coefficient (δIT2), the desired control specification (aggressive, smooth, and moderate) will be provided. This scheme can reduce the complexity’s design
Fig. 7. Schematic of the iSIT2-FPISMC.
and computational burden, which is a very valuable form a practical perspective.
B. Intelligent PI Type SIT2-FLC-Based Sliding Mode Control 1) Intelligent PI Type SIT2-FLC: For a general single- input/single-output (SISO) nonlinear system, based on an ultralocal model, the model-free method is characterized as follows [34]:
y(v)(t)=+u(t) (34) wherey(v)(t)is the derivative of the orderv∈ N of the output y(t), which is usually chosen as 1 or 2.∈ Ris a nonphysical design constant, which is selected so that u(t) and y(v)(t) have the same order of magnitude.represents an unknown term, which is identified by the control signalu(t)and output y(t).
Let and be well-known items, and an iSIT2-FPI can be described as
u(t)= 1
−˜ + ˙y∗+kpψu +ki
ψu
(35) wherey˙∗is the output desired reference and ˜is the estimated value of(t). The error equatione(t)=y(v)∗(t)−y(v)(t)by combining (34) and (35) can be obtained as
e(t)+˜ +kpψu+ki
ψu =0. (36) In this work, the estimated disturbance ˜will be calculated by using an ESO. The ESO is designed by a third observer as
˙
q1 =q2−μo1 fal(F, ω1, ρ)
˙
q2 =q3−μo2 fal(F, ω2, ρ)
˙
q3 = −μo3 fal(F, ω3, ρ) (37) whereμo1,μo2, andμo3 represent observer constants,q1,q2, andq3represent the intermediate states, andFis the estimated error of observer. Since, in the ESO scheme, q3 denotes the estimation generated by the observer,q3=.˜
2) Design of Model-Free Sliding Mode Control: Based on (35), the performance of the iSIT2-FPI controller depends on the baseline PI gains of the SIT2-FLC, FOU design coefficient, and estimation of . The performance of the model-free technique degrades if the estimation is inaccurate with large error. To eliminate the estimation error, an extra controller
SMC is proposed. The overall structure of the hybrid iSIT2- FPI and SMC is highlighted in Fig. 7. The additional SMC input control is denoted by usmc and yields
u(t)= 1
−(˜ t)+ ˙y∗+kpψu+ki
ψu
+usmc. (38) Let the estimated error of the observer be defined as ˜= ˆ −. Based on that, one has ˜ < m, where m is an upper bound. Substituting (38) into (34), the error equation is generated as
˙
e(t)+kpψu+ki
ψu+usmc+˜ =0. (39) According to the SMC design scheme, a switching function is introduced as
σ =γe+ ˙e (40)
whereγ >0 is a design coefficient. The derivative of (40) is calculated as
σ(˙ t)=γe˙+e¨=γe˙−kpψu−ki
ψu−usmc−(˜ t). (41) The extra inputuSMC that consists of equivalent controlueq and switching controluswis defined as uSMC=ueq+usw. In order to reach the desired sliding mode condition in (41),ueq
is obtained by replacing ˜withm as ueq= 1
γe˙−kpψu−ki
ψu−m
. (42)
In order to eliminate the chattering effects, usw is defined as
usw= 1
(η1[sat(σ, ε)]+η2σ) (43) where η1 and η2 denote the switching gains, ε denotes the boundary thickness, η1 >0, η2>0,ε >0, and
sat(σ, ε)=
⎧⎪
⎨
⎪⎩
1 ifσ > ε σ/ε ifσ> ε
−1 ifσ <−ε.
(44) Substituting the extra inputuSMCinto (34), the total control signal can be rewritten as
u(t)= 1
−(˜ t)+ ˙y∗+η1[sat(σ(t), ε)]+η2σ(t) . (45) 3) Stability Analysis: Consider a Lyapunov function as
V = 1
2σ2. (46)
The derivative of (46) can be written as V˙ =σσ˙ = −σ
η1[sat(σ, ε)]+˜ −m
−η2σ2. (47) Therefore, the boundedness of ˜is sufficient to satisfy that V˙ <0. For|σ|< ε and|σ|> ε, the Lyapunov stability can be guaranteed under two situations [35].
Case1: If|σ|> ε, the existing condition will be converted as
V˙ = −σ
η1+˜ −m
−η2σ2. (48)
Fig. 8. Experimental setup.
According to the boundedness of ˜, it is to meet thatV˙ <0 if one hasη1>2m.
Case2: If |σ| < ε, the existing condition will be changed as
V˙ = −σ
η1σ/ε+˜ −m
−η2σ2. (49) Defining the condition of (25), the term negative right-hand side is met
−σ˜ −m
−(η1/ε+η2)σ2<0
|˜ −m|< (η1/ε+η2)σ. (50) Considering the boundedness of ˜and|σ|< ε, one can get (η1+εη2) >2m. In Section V, to validate the effectiveness of the proposed method, the experimental results of the case study islanded DCMG are provided.
V. EXPERIMENTALRESULTS
In this section, to verify the effectiveness of the presented control scheme, a shipboard DCMG consists of three DGs is considered, which are connected to the dc bus through dc–dc converters. The nominal voltage of the system is considered to be 400 V. The maximum output power of each dc is assumed to be 1.2 kW. The experimental setup is depicted in Fig. 8.
The hardware-in-the-loop (HIL) experimental tests are con- ducted using the dSPACE platform, which is provided real- time data acquisition and control operations. The readers can refer to [36]–[38] to achieve more information regarding HIL implementation tests. The gains of the suggested controller are positive coefficients that are designed by the trail-and- error method and the designer experience. The electrical and control parameters are listed in Table II. In the following, the effectiveness of the proposed method is validated for various scenarios.
The experimental results for the heavy loading condition are presented in Fig. 9. A common load of 3 kW is considered in the dc bus in this case study. As depicted in Fig. 9, it can be observed that the current sharing among DGs is real- ized accurately with the suggested control scheme. Moreover, in comparison to the linear droop method, in heavy loading
TABLE II
EXPERIMENTALPARAMETERS
Fig. 9. Performance of the DGs in the SPS during the heavy loading condition. (a) Conventional droop method. (b) Proposed control method.
conditions, the proposed method has the desired performance and good voltage restoration.
Fig. 10 shows the experimental results of the proposed control method under light loading conditions in the SPS.
To realize a light load condition in the stand-alone DCMG in the SPS, a 1-kW CPL is applied in the dc bus. As illustrated in Fig. 10, it can be seen that, with the proposed control scheme, the current sharing among DGs is improved in com- parison to the conventional droop methods.
From Figs. 9 and 10, it can be concluded that, by applying the proposed controller for the DCMG system in the SPS, during all loading conditions, the desired currents sharing among DGs can be achieved.
To illustrate the performance of the proposed controller under dynamic loading conditions, a step change load from 2 to 3 kW can be considered in the dc bus. From Fig. 11(a) and (b), one concludes that with the novel proposed control structure, the power of the pulsed load is shared fast and accurate among DGs in comparison to the conventional con- troller, and the power sharing among DGs is prolonged in several seconds [29]. Fig. 12(a) depicts the comparison of the proposed controller, conventional controller with SMC, and conventional controller with fuzzy SMC for a step change load
Fig. 10. Performance of the DGs in the SPS during the light loading condition. (a) Conventional droop method. (b) Proposed control method.
Fig. 11. Performance of the DGs in the SPS during the step change load condition. (a) Conventional droop method. (b) Proposed control method.
in the dc bus. From Fig. 12(a), it can be seen that, with the proposed controller in the secondary control level, the restora- tion of the dc bus voltage is realized fast during the load changes, and the voltage is retained in the acceptable range (i.e., 5% of the reference value [9]). Furthermore, to show the advantage of the proposed controller in the current reference tracking in the SPS, Fig. 12(b) depicts the comparison of the proposed controller and the conventional controller for a step
Fig. 12. (a) Comparison of the proposed controller, conventional controller with SMC, and conventional controller with fuzzy SMC for the restoration of the dc bus during step change load in the SPS. (b) Current reference tracking during step change load in the SPS.
Fig. 13. Current of DGs for plug-and-play experimental results with the proposed controller and the proposed nonlinear droop control method.
change load in the dc bus. As shown in Fig. 12(b), it can be concluded that the suggested controller offers a superior dynamic performance than the designed control methodologies during a step change load and follows the current reference fast and accurately.
In order to show the plug-and-play capability of the pro- posed control scheme in the DCMG in the SPS, Fig. 13 illus- trates the current of the DGs during this condition. As depicted in Fig. 13, beforet =2.8 s, DG1 and DG2 are injected power to the load. Att =2.8 s, DG3 is connected to the DCMG and starts the power injection to the loads. From Fig. 13, it can be concluded that the proposed nonlinear droop control method can guarantee accurate power sharing among DGs during this situation and provides the plug-and-play capability for DGs in
the DCMG. In addition, the proposed iSIT2-FPI controller has a fast and proper transient response during the plug-and-play condition.
Remark 2:Compared with the prevalent control methodolo- gies, the following advantages are assigned to the suggested model-independent scheme.
1) In comparison with the model-based schemes, such as MPC and nonfragile fuzzy control [39], [40], the sug- gested model-independent approach does not need to complete knowledge of the controlled plant model.
Thus, the mathematical complexity of control design is reduced to cope with the unknown dynamic phenomena and high perturbations.
2) In comparison with the iPI scheme [41], the iSIT2-FLC controller provides higher resilience against the system uncertainties.
3) In comparison with the prevalent ultralocal model-based schemes (e.g., model-free nonsingular terminal SMC) [42], a supplementary SMC controller is adopted to com- pensate for the observer estimation error and ameliorate the system performance.
VI. CONCLUSION
In this article, an intelligent and fast controller was proposed for the voltage regulation and current sharing in the stand- alone DCMG in the SPSs. The suggested controller employed a nonlinear I −V droop control in the primary level to eliminate the voltage oscillation in the dynamic situation and improve the current sharing among paralleled converters. Also, in order to achieve fast and accurate voltage regulation and current sharing among DGs in the SPS, the SMC scheme was introduced into the model-free iSIT2-FPI in the primary control level. The SMC part was added in the structured model-free controller to eliminate the ESO estimation error.
The DCA was provided the coordinated control for the DGs to achieve voltage regulation in the secondary control level.
Finally, HIL experimental results verified the effectiveness and applicability of the proposed method during various loading conditions in the islanded DCMG in the SPS. The future work can be devoted to preserve a correlation between various terms of marine/ship systems, such as design constraints, operational modes, and energy management for an actual ship.
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