IET Power Electronics
Research Article
Explicit discrete modelling of bidirectional
dual active bridge dc–dc converter using
multi-time scale mixed system model
ISSN 1755-4535 Received on 13th March 2020 Revised 24th August 2020 Accepted on 6th November 2020 E-First on 17th February 2021 doi: 10.1049/iet-pel.2020.0293 www.ietdl.org
Mohammad Tauquir Iqbal
1, Ali I. Maswood
1, Hossein Dehghani Tafti
2, Mohd Tariq
3, Zhong Bingchen
11School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Ave, Singapore 639798, Singapore 2School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia
3Department of Electrical Engineering, ZHCET, Aligarh Muslim University, Aligarh, Uttar Pradesh 202002, India
E-mail: [email protected]
Abstract: Dual active bridge (DAB) is one of the most popular high power isolated bidirectional dc–dc converter. It offers high
power density, a low number of passive components, high efficiency. It is essential to model the converter so that it can be used in applications such as solid-state transformer, micro-grids, and dc distribution systems. Moreover, modelling a system is used to design inductor, select the switches, and controller design. The advantage of the discrete-time models is that they explicitly describe state trajectories in all sub-intervals of converter operation, meaning they are capable of providing exact solutions for ac state variables. This study proposes time scaling mixed system model of the DAB. The proposed method provides explicit solutions of the large and small signal, which can be used to study the steady-state and transient nature of the converter. The experimental results validate the proposed model with high accuracy in the open and closed-loop operation.
1 Introduction
Dual active bridge (DAB) converter is one of the most suitable converters for high power applications [1]. It has a minimum number of external elements (inductor or capacitor). Morever, it offers high power density, high efficiency, soft switching, step-step, or step-down operation [2]. The advantage of a low number of external components increases converter reliability [3]. The DAB converter consists of two H-bridge separated by an external leakage inductor and a high-frequency transformer. It is of uttermost importance to model the converter for the design of inductor, selection of the switches, controller design [4].
The conventional approach for averaging a dc–dc converter assumes the current to have a small ripple [4]. However, the inductor current in the case of a DAB is purely ac, and the average value of the inductor current is zero [1, 2]. Therefore, the conventional averaging technique cannot be applied in the DAB. The following methods are used in the literature to overcome this challenge.
Generalised average modelling (GAM) was first proposed in [5] to model pulse width modulation (PWM) and resonant converters. Instead of only dc terms, GAM also includes the additional number of state variables like a fundamental component of the inductor current. Since the power flow in the DAB is the sum of the harmonic power of the fundamental and higher-order frequency [6], the accuracy of the GAM depends on the number of high-frequency state variables added in the final state-space model. In [7], the only fundamental frequency of inductor current has been considered in the modelling. Therefrom, the gain response of the small signal has a significant error at low frequency [8]. The low-frequency error in the GAM is mitigated using a correction factor γ [8]. The enhanced GAM has been proposed in [9] for the DAB converter. A loss correction (taking effect of the system's resistance) has been implemented to improve the converter's large-and small-signal model. The third harmonic component of the inductor current is added in the state model [10]. The addition of the third harmonic improves the overall accuracy of the GAM. However, adding more terms in the state variable increases the complexity of the model.
The average value modelling (AVM) of any converter is developed by replacing the semiconductor switches with their average current or voltages. The AVM model, considering parasitic
elements like core loss, copper loss, and switch resistance for the phase-shifted DAB converter, is reported in [11]. The AVM lowers system complexity and gives converter system-level insightful information. However, the AVM becomes complicated for advanced modulation techniques, where the number of sub-intervals is relatively high. The reduced AVM model for phase shift plus duty cycle control (triple-phase-shift control) [12] has been proposed in [13]. However, the non-ideality of the system is not considered. The absence of the non-ideality in the final model reduces the accuracy of the model.
The modelling of the DAB converter can also be obtained using the Fourier-series model [14–16]. The inductor current and output voltage can be represented as a sum of Fourier series. An optimum number of harmonic number is selected to minimise the overall error in the model [14–16]. The fundamental and up to the seventh harmonic component is used to model the system. An adaptive controller is designed based on the proposed Fourier series based model [14]. The feed-forward-based controller is proposed to enhance the DAB converters' step load response [15]. Moreover, dead-time compensation is also employed to improve the accuracy of the light load condition [15]. A similar concept is used to model the three-phase DAB converter for the solid-state transformer application [16].
The AVM and Fourier-series based models are reduced models that neglect the effect of the inductor current dynamics in the final model. Therefore, system stability cannot be studied by AVM or Fourier series model [11–16]. The problem mentioned above of ignoring inductor current dynamics can be mitigated by discrete modelling [17–22].
The advantage of the discrete-time models is that they explicitly describe state trajectories in all sub-intervals of converter operation, meaning they are capable of providing exact solutions for ac state variables. In the case of DAB converters, discrete models are able to predict transformer currents accurately and, the current zero-crossings critical to zero voltage switching (ZVS). The sample data discrete modelling of the dc–dc converter goes back to the early 80s [17]. Later, it has been extended to the resonant converter [18]. The model developed in [18] is then extended to DAB in [19, 20]. The discrete model can be directly employed for the control and stability analysis of the DAB. Therefore, the discrete transfer function obtained from the DAB sample data model can be used for the controller design and tuning the control
parameters. The delay due to A/D converter can also directly be incorporated in the digital system with ease. However, the discrete-time implementation of the AVM and GAM models with the digital processors needs to model delay by Pade approximation, which results in adding the additional pole and zero to the final model [13].
The solution of the discrete model in the literature [17–22] contains matrix exponential, which can be calculated using first- or second-order approximation. The first approximation gives a significant error in the state variable [20–22]. The second-order exponential expansion leads to higher complexity [20]. The discrete model in [18–22] includes the matrix exponential which increases the complexity of the system. The problem in the discrete modelling leads to the motivation of finding a modelling method that should be simple and must have an explicit final model equation.
This paper proposes a two time-scale modelling for the DAB converter with reduced complexity and high accuracy. The proposed time scale model reduces the system's complexity by removing exponential matrix terms from the final model. Since the inductor current has zero dc value and large ripple while output voltage has quite a low ripple, the inductor current can be labelled as a fast variable while output voltage as a slow variable. The dynamics of the DAB converter behave as a mixed system consisting of fast and slow variables. Hence, it can be modelled using the method of averaging for mixed systems with two-time scales (TTS). The comparison between the proposed TTS with the other modelling methods proposed in the literature is listed in Table 1.
2 Averaging theory of TTS mixed system
The scale of fast and slow variable timescale is normally denoted by ε. The fast variable can be represented by Y, while the slow variable is denoted by X. ε can be put in the right-hand side of the state variable equation to specify the corresponding variable as slow. While a mixed system equation has been described below where Xϵ Rk, Y ϵ Rl, each functions is in the format of f:
R+× Rk× Rl and g: R+× Rk× Rl are discontinuous functions vector and, as in equation (1), ε > 0 is a small parameter. Therefore, the slow varying system can be described by
X˙= ε f (t, X), XϵRk where X(t) = X
0 at t = t0 (1)
While the mixed system can be described by
X˙ = ε f (t, X, Y), XϵRk1 where X(t) = X
0 at t = t0 (2)
Y˙ = g(t, X, Y), YϵRk2where Y(t) = Y
0at t = t0 (3) For ε ≪ 1, the state equation can be segregated into TTS. On the fast time-scale, X is assumed to a fixed value while the state equations of Y are solved. On the slow timescale, Y can be treated as algebraic, rather than dynamic, while the state equations of X are solved.
DABs contain a passive component, reactive elements, and switches. Due to switch on and off operation, the circuit dynamics have a non-linear discontinuous differential equation. The governing converter function (2) switches from one to another
X˙ = fj(t, X, Y) and Y˙= gj(t, X, Y) (4) The above equation can be reduced to (4), (5) if (3) is a linear and the switch action is instantaneous
X˙= ϕsx(j)X+ ϕsy(j)Y+ ψs(j)U (5) Y˙= ϕ(f xj)X+ ϕf y(j)Y+ ψ(f j)U (6) where ϕsx(j), ϕsy(j), ψs(j), ϕf x (j), ϕ f y (j), and ψ f
(j) are similar to conventional state-space approach j = 1, 2, 3…, n given by (17).
In (4) and (5), ε does not occur explicitly in the converter model. However, parameters can be easily classified as slow and fast variables. The actual simple procedure of obtaining average model of two timescale discontinuous system is first taking slow variable X in the fast variable (5). Thereafter, finding the operating point of the fast variable Y¯ (5). Once the steady-state solution of Y
is obtained, its value Y¯ is substituted in (4). Finally, averaging the
left side equation (4) over one switching cycle to get the slow variable operating point X¯.
3 Converter operation
The DAB has two full-bridge separated by a high-frequency transformer and an external inductor shown in Fig. 1. The full-bridge can be made from MOSFET or IGBT, depending on the application. The emergence of new semiconductor types like gallium nitride and silicon carbide helps in increasing the power density of the converter. The high-frequency transformer helps in step up and step down the input voltage to the desired voltage level. An external inductor is used to limit the maximum power flow in the circuit. The input side of the DAB is normally high voltage in some applications like solid-state transformer while the output is low voltage dc-link, which can be configured as an RC circuit. The duty cycle of semiconductor devices in each bridge is 50%.
The power flow direction depends on the phase shift between the primary and secondary bridge. When the primary bridge leads the secondary side, power flow in the forward direction, and vice versa. The steady-state waveform of the DAB in the forward mode is shown in Fig. 2. The primary bridge is leading secondary bridge by time dTs/2. In the phase-shift dc–dc converter, there are four intervals possible. The circuit representation of each sub-interval is shown in Fig. 3.
Sub-interval I [tn, tn′]: The primary side switches S1 and S4 are turned on at tn. The voltage across the primary bridge in this sub-interval VPR is Vi. The secondary side bridge voltage VSC is −Vo because switches S6 and S7 are conducting. Since the voltage across the inductor is positive, the inductor current increases from
I1 to I2. The negative inductor current at turn-on depicts the switch S1, and S4 is turned on at ZVS.
Sub-interval II [tn′, tn+ 1]: The secondary side switches S5 and S8 are turned on at tn′. The voltage across the primary bridge in this
sub-Table 1 Comparison of different DAB modelling
Type of model Orderof model Study of open-loop response Steady-state accuracy Explicit
model complexityModel
optimal harmonic based model [13] reduced ✓ varies on harmonic number ✗ low full harmonic model [14] reduced ✗ high ✓ low discrete time model [15, 16] full ✗ not discussed ✗ high average switch model with losses [17]
reduced ✓ high ✓ high
fundamental harmonic based model [18]
full ✗ low ✓ medium
bilinear modelling [19] full ✗ not discussed ✗ high proposed
model full ✓ high ✓ low
interval remains the same as Vi. The secondary side bridge voltage
VSC is Vo because switches S5 and S8 have been turned-on. The voltage across the inductor can be positive or negative depending on the Vi> Voor Vi< Vo, the inductor current increase from I2 to
I3. The positive inductor current at turn-on depicts the switch S5, and S8 is turned on at ZVS.
Sub-interval [tn+ 1, tn′+ 1]: The primary side switches S2 and S3 are turned on at tn+ 1. Therefore, the voltage across the primary bridge in this sub-interval changes its polarity as −Vi. The secondary side bridge voltage VSC is Vo like sub-interval II. Since the voltage across the inductor is negative, the inductor current decreases from
I3 to I4. The negative inductor current at turn-on depicts the switch S1, and S4 is turned on at ZVS.
Sub-interval IV [tn′+ 1, tn+ 2]: The secondary side switches S6 and S7 are turned on at tn′+ 1. The voltage across the primary bridge in this sub-interval remains the same as Vi. The secondary side bridge voltage VSC is Vo because switches S5 and S8 have been turned on. The voltage across the inductor can be positive or negative similar to sub-interval II, the inductor current increases from I4 to I1. The positive inductor current at turn-on depicts the switch S5, and S8 is turned on at ZVS.
4 Conventional discrete model of DAB
The state variables in the DAB are inductor current and the output capacitor voltage. The state equation of the four sub-interval of the DAB can be written as
x˙α1(t) = A1xα1(t) + B1Vi (7)
x˙α2(t) = A2xα2(t) + B2Vi (8)
x˙α3(t) = A3xα3(t) + B3Vi (9)
x˙α4(t) = A4xα4(t) + B4Vi (10) where xα(t) = [iLvc]T is a state variable, A1− A4 and B1− B4 are the circuit parameter matrices which depend on the operating condition, Vi is the source input voltage. The state matrix A1− A4 and B1− B4 in each sub-interval can be written as
A1= A4= 0 L1 −1 C − 1 RC and A2= A3= 0 −L1 1 C − 1 RC B1= B2= 1 L 0 and B3= B4= −L1 0 (11) The solution of state equation (7)–(10) in each sub-interval can be solved by conventional linear algebra methodology. The solution of these equation at (k + 1)th switching instant depends on the (k)th state of the circuit, i.e. [iL(k) vc(k)]T, state exponential matrix (eAjtk j) and input voltage Vi. The expression for these solution can be written as xk+ 1= fnk(xk, Vi) = eAjtk jxk+ ΓjVi where, Γj=
∫
0 tj eAjtn jB jdt = Ai−1eAjtn j− I Bj (12) where j = 1, 2, 3, 4 defines the state vector x in the time period tj. The phase shift of the DAB remains the same for each switching period. The phase-shift in each sub-interval is treated as constant. The input voltage varies very slowly with time due to the presence of a big capacitor at the input voltage source or battery. Therefore, the input voltage can be treated as a slow varying time function. The final solution of (7)–(10) can be determined with the help of (12) as written below:xk1= eA1tk1xk+ Γ1Vi (13)
xk2= eA2tk2xk1+ Γ2Vi (14)
xk3= eA3tk3xk2+ Γ3Vi (15)
xk+ 1= eA4tk4xk3+ Γ4Vi (16) where xk+ 1, xk3, xk2, and xk1 represent the state variable at the end of the sub-interval IV-I, respectively. xk and xk+ 1 represent the state variable at the kth and (k + 1)th. The four sub-interval of the DAB repeats over a switching period. At steady-state operation, net change in the state variable over a switching period is zero, i.e. the state variable at kth sample will be equal to the state of the variable at (k + 1)th sample. The state variable [iL vo] at (k + 1)th switching instant can be achieved by putting the expression of xk3, xk2, and xk1 in (16). The final expression of xk+ 1 can be written as
xk+ 1= ξ(Φk)xk+ ζ(Φk)Vi where ξ(Φk) = eA4tk4eA3tk3eA2tk2eA1tk1 ζ(Φk) = eA4tk4eA3tk3eA2tk2Γ1+ eA4tk4eA3tk3Γ2 +eA4tk4Γ 3+ Γ4 (17)
The state variable xk+ 1 in (17) involving matrix exponential is far too complex and difficult to conclude any insightful information about the system characteristics. Therefore, the model cannot be used for controller design. The model needs to be simplified using a matrix exponential evaluation method either by matrix exponential expansion. The simplest matrix exponential expansion Fig. 1 Circuit diagram of the DAB dc–dc converter
Fig. 2 Steady-state waveforms of the DAB dc–dc converter
is using the first-order term of the series or second-order and neglecting higher-order terms, i.e.
eAjtn j= I + A
jtn jor I + Ajtn j+
Aj2tn j2
2 (18)
Although the first approximation using (18) gives an accurate large-signal value of the inductor at an operating point, the error in a second state variable vo is significant as discussed in [16]. Therefore, the second-order approximation (18) is used in the model equation (17). However, the second-order approximation gives a complex model that contains a very long equation. The problem of matrix exponential can be easily solved by using a TTS model which is discussed in the next section.
5 Proposed TTS modelling of DAB
The procedure discussed in Section 2 will be used to model DAB since the inductor current is changing rapidly while output and input voltage change very slowly. First, the inductor current dynamics will be solved to calculate its average value. Later, the slow variable is calculated. All components are assumed ideal, and the transformer turn ratio is ‘a’. The steady-state waveform is shown in Fig. 2. The equation for first [tn, tn′] and second [tn′, tn+ 1] sub-interval can be described by (19) and (20), respectively
iL(t) = iL(tk) + avi(tk) − v0(tk) t′/L for tk< t < tk′or 0 < t′< d′Ts/2
(19)
iL(t) = iL(tk′) − avi(tk) + v0(tk) t″/L
for tk′< t < tk+ 1or 0 < t″< dTs/2 (20) where t′= t − tk and t″= t − tk. iL(tk) and iL(tk′) are the inductor current at time interval tk and tk′, respectively. vi(tk) and vo(tk) are input and output voltage respectively. Since vi(tk) and vo(tk) are considered as slow variable, they are considered as constant for a switching period [tk− tk+ 1]. The expression for the inductor current after half-switching period Ts/2 can be achieved from (19) and (20) as iL(tk+12) = iL(tk) − vo(tk) Ts 2L+ avi(tk) (1 − 2D)Ts 2L (21)
Once, the steady-state equation of fast variable (21) is obtained. the slow variable, i.e output voltage can be achieved by averaging over one switching cycle
vo(tk+12) = vo(tk) +C1 Qk− vo(tk) R (tk+ 1− tk) where Qk=
∫
tk tk+ 1 iLdc(τ) dτ (22)The second bridge rectifies the inductor current iL to iLdc. In the sub-intervals I and II, the rectified inductor current is equal to the inductor current. Therefore, the output voltage after half-switching period can be solved by putting the inductor current expression (19) and (20) in (22). The final expression can be defined as
vo(tk+12) = iL(tk) Ts 2L+ vo(tk) 1 − Ts 2CR− Ts2 8LC +avi(tk) 1 − 2d2 Ts2 8LC (23) The state variable of DAB for half-cycle is represented by (21) and (23). These equations are combined to form standard discrete difference equation given by (24). Similarly, the discrete equation for the sub-intervals III-IV can be derived
Xk+12= ϕnXk+ ψkU where ϕk= 1 −2LTs Ts 2C 1 − Ts 2CR− Ts2 8LC ψk= (1 − 2d)Ts 2L (1 − 2d2) Ts2 8LC (24)
Steady-state inductor current and voltage can be found by using (24). The inductor current at the kth event is equal to negative of the inductor current at the (k + 1)th event, i.e. iL(k) = − iL(k +12). However, the output voltage at the kth event is equal to output voltage at the (k + 1)th. Therefore, the steady state of the inductor current and output voltage can be written as
Xss= (W − ϕ)−1ψ Viwhere W = −1 0 0 1 (25) iL= Ts 4L aVi(2D − 1) + Vo and Vo= aViTsR 2L (D − D 2) (26)
Since DAB has two patterns per cycle, i.e. odd half-cycle symmetry inductor current waveform while even half-cycle symmetry output voltage. The state variable in the next half cycle is governed by the same functions ϕ and ψ with transformed state vector as
Fig. 3 Equivalent circuit diagram at each operational subinterval of the
DAB dc–dc converter
Xk+ 1= Wϕk W Xk+12 + WψkU= ϕk fXk+ ψk fU where ϕk f= Wϕk 2Xk and ψk f= Wϕk+ I Wψk
(27) The above equation represented the final dynamical equation of DAB over a switching period Ts. The above equation can also be used to find the steady-state value of the inductor current and the output voltage by assuming that the net change in these state variable over a switching period is zero.
6 Linearisation around the operating point
The non-linear state equation given by (27) can be rewritten separately for two state variable as
x1(tk+ 1) = ∇1(x1(tk), x2(tk), vi(tk), dk) (28)
x2(tk+ 1) = ∇2(x1(tk), x2(tk), vi(tk), dk) (29) The digital control of power electronics converter means that the phase-shift of the DAB is fixed over a switching period Ts. It is paramount to have the knowledge of a converter dynamic properties about its operating point ‘d=D’ to a small changes of control variable 'Δd'. Moreover, the linearising also helps in modifying non-linear equations (28) and (29) to a linear equation (30) so that a simple linear control can be applied
x^k+ 1= ϕ˙n fXssd ^ + ϕk fx^n+ ψ˙k fVid ^ + ψn fv^ik where ϕ˙k f= δϕk f/δd and ψ˙k f= δψk f/δd (30) The exact expression of ϕk f, ψk f, ϕ˙k f, and ψ˙k f can be written as using (24) and (27) ϕk f= 1 + Ts2 4LC − Ts 2L Ts 2RC+ Ts2 8LC −2CTs 2RCTs + Ts 2 8LC 1 − Ts 2RC− Ts2 8LC 2+ Ts2 4LC ψk f= Ts3 16L2C(1 − 2D 2) Ts2 4LC(2D − 1) + Ts2(2D2− 1) 8LC Ts2 8LC+ Ts 2RC− 2 ψ˙k f= − DTs3 4L2 C Ts2 2LC+ DTs2 2LC Ts2 8LC+ Ts 2RC− 2 , ϕ˙k f= 0 (31)
The standard output equation is given in (32) where dk and xk are the control and state variable at the kth cycle, respectively. The small signal variable is shown as subscript ‘s’ for clarity, ϕs= ϕk f,
ψs= ϕ˙k fXss+ ψ˙k fVi and Ψs= ψk f. x^k+ 1= ϕsx^k+ ψsd ^ k+ Ψsv^ik where ϕs= ϕ11 ϕ12 ϕ21 ϕ22 and ψs= ψ1 ψ2 (32) The open-loop and small-signal transfer function can be obtained by applying z-transform on (27) and (32), respectively, as given by (33) with E = [01] G(z) = E(zI − ϕk f)−1ψk f Gvd(z) = E(zI − ϕs)−1ψs Gvi(z) = E(zI − ϕs)−1Ψs (33) where (zI − ϕs)−1= zI− ϕ1 s z− ϕ22 ϕ12 ϕ21 z− ϕ11
Since the diagonal term of ϕk f or ϕs is dominant, i.e.
ϕ11ϕ22≫ ϕ12 ϕ21. The inverse of the matrix (zI − ϕs). The open loop output voltage and small signal model of the DAB using (33) can be written as G(z) = aVi Ts2 4LC(2d − 1) + Ts2(2d2− 1) 8LC Ts2 8LC+ Ts 2RC− 2 z− 1 − Ts 2RC− Ts2 8LC 2− Ts2 4LC (34) Gvd(z) = aVi Ts2 2LC+ dTs2 2LC Ts2 8LC+ Ts 2RC− 2 z− 1 − Ts 2RC− Ts2 8LC 2− Ts2 4LC (35) Gvi(z) = Ts2 4LC(2d − 1) + Ts2(2d2− 1) 8LC Ts2 8LC+ Ts 2RC− 2 z− 1 − Ts 2RC− Ts2 8LC 2− Ts2 4LC (36)
7 Validation of DAB discrete model
7.1 Open-loop validation
The validation of the proposed TTS model is carried out in MATLAB and by an experimental prototype whose parameter is given in Table 2. Moreover, PLECS software is used for the simulation and small-signal model. The average value of the output voltage given by (26) is compared with the experimental value, simulation model, and second-order approximation of the matrix exponential (18) in Fig. 4. The experimental results, TTS model, second-order matrix approximation model, and simulation model at
d=0.1 are 55, 51.5, 52.5, and 54 V, respectively. The maximum
error between the experimental and model is <2% at a higher value of d, as shown in Fig. 4. The model accuracy can be increased by adding the effect of the resistance of the system. However, the inclusion of the resistance increases the model complexity. The TTS model accuracy is similar to the second-order matrix approximation model over a wide range of the phase-shift (d).
The open-loop model can be verified by a step change in load resistance, phase shift, and input voltage. The output voltage transient with a step-change in load resistance is shown in Fig. 5. The set up was initially operated at d=0.15 and load resistance of 100 Ω with input voltage 250 V. At t=1.32 s, a step increase in phase shift from d = 0.15 to d = 0.175 is carried out. It can be seen that the waveform obtained from the model matches very well with the experimental results. Similarly, the step decrease is carried out from d = 0.175 to 0.15 at t = 0.96 s. Similar to the previous case, the model matches accurately with the experiment (Figs. 6–8).
The model verification with step change in load resistance is carried out next. The set up was initially working on same operating point like the previous case with phase shift of d = 0.15. The load resistance is increased from 80 to 100 Ω at 0.625 s, The resultant model output voltage follows the experimental results with high accuracy. Similarly, the load resistance is changed back
Table 2 Experimental parameter
Parameter Value
input voltage supply 250 V switching frequency 50 kHz dc blocking capacitor 18.8 μF external leakage inductor 425 μH transformer turn ratio 0.97 output filter capacitor 68.8 μF load resistance 100 Ω semiconductor device IXFL210N30P3
from 100 to 80 Ω at t=0.675 s. Again there is very much similarity between the model and the experimental results.
Finally, the model verification is done for a step change in input voltage. Since the variable power supply has big capacitor installed in it, the variation in input voltage is not step wise. As described in [14] that the step down takes a significant time to change. Therefore, the model verification cannot be achieved for step decrease in input voltage. The open loop model verification is done
with linear increase of the input voltage from 235 to 255 V at
t= 0.4 s. The input voltage increases linearly from 235 V to 255 V in 0.15 s. The linear increment is modelled in the MATLAB as shown in Fig. 9. The model input voltages and the output voltages corresponding to the changes in input is well matched.
7.2 Small-signal validation
The set up to obtain the small signal is shown in Fig. 10. The steady-state phase shift PWM is supplied through DSP. The perturbation in the phase shift is passed through the ADC pin of the DSP. The perturbation is connected to Channel 1 of the analogue discovery 2, while the other Channel 2 is connected to the output of the voltage sensor, as shown in Fig. 10. The ADC range of the DSP is 0–3 V. Therefore, an LEM voltage sensor LV-25p with voltage multiplying factor of 0.02 is used to reduce the noise and step down the output voltage. Since the ADC is not compatible with a Fig. 4 Comparison of the steady-state output voltage of different models
with respect to phase-shift
Fig. 5 Output voltage transients with a step increase phase shift change
Fig. 6 Output voltage transients with a step decrease phase shift change
Fig. 7 Output voltage transients with a step increase of load resistance
Fig. 8 Output voltage transients with a step decrease of load resistance
Fig. 9 Output voltage transients with a linear increase of input voltage
Fig. 10 Experimental set up to measure loop gain
negative voltage, the voltage of amplitude 1.5 V with an offset of 1.5 V is fed to the ADC of the DSP. The magnitude of the perturbation is controlled in the MATLAB as full-range voltage has less noise to signal ratio. Moreover, the offset in phase shift is removed in the MATLAB environment. The change in phase shift can only be possible at the switching instant, which adds a time delay in the system. In the s-domain, the delay is expressed by the Pade approximation, as explained in [14].
The magnitude of the perturbation should be carefully chosen. If the magnitude of the perturbation is too small, the output voltage change will be very little. The noise will dominate the change of output voltage due to perturbation. Similarly, if the magnitude of the injected signal is significant, the loop response will be non-linear. The non-linearity in the response will again affect the accuracy of the small signal. Therefore, at the low frequency, the perturbation of 0.01 is used up to 500 Hz. The injected signal magnitude then increases to 0.02 when the injected frequency is more than 500 Hz. The injected signal magnitude is increased from 0.01 to 0.02 to improve the SNR. The advantage of using a discrete system is that the time delay does not need an approximation. It can be modelled by adding the inverse of z in (35). The voltage sensor is a low-pass filter with a cut-off frequency of around 50 kHz. The voltage sensor starts contributing significantly to the kHz range. The two film capacitors of 25 μF have been used at the output filter, and four 4.7 μF film capacitors are installed near the full bridge to absorb high-frequency transient. Since the dissipation factor of the film capacitor is <0.1%, the equivalent series resistance of the thin film is in the range of few mΩ.
The magnetic elements inductor and transformer have been designed in the Mn-Zn ETD ferrite core. The transformer and inductor have been made by litz wire of 30 strands of 0.2 mm diameter. The number of turn for transformer and inductor is 21 and 30, respectively. The flux density is kept around 150 mT to prevent the core from overheating due to core loss. The dc blocking capacitor is also installed before the primary to stop the transformer going to the saturation region. The capacitor value needs to be large enough so that the loop gain does not get affected. The control to output voltage bode plot given by (35) with a time delay is compared with the experimental and simulation model in Fig. 11. PLECS multi-tone analysis is used to obtain the small-signal model [23]. The gain response of the Gvd(z) obtained from the TTS model at phase shift d = 0.15 agrees with the experimental value and simulation model, as shown in Fig. 11. However, the phase response loses some accuracy at the higher frequency. Similar results have been observed at the phase shift
d=0.15 gain response of all three models ( TTS, experimental and
simulation model) is similar. There is some difference in the phase response of these three models. The accuracy of the model can be increased by including the resistance in the model. However, it increases the complexity of the model [11].
7.3 Closed-loop validation
The closed loop is verified with a PI controller whose proportional and integral gain is chosen as 0.001. The model has been changed into their per-unit system so that floating-point mathematics can be utilised. The integer quotient mathematics ensures that the mathematical operation is fast and accurate [24]. The DSP can handle 32-bit data effectively. Therefore, per unit methodology gives a resolution of around 2−32. Moreover, the floating-point helps change the integer point bit according to the resolution needed by the operation. The internal clock frequency of the TMS320F28335 DSP is 150 MHz. Therefore, the clock cycle of the PWM for the 50 kHz switching frequency will be 3000. The resultant resolution of the phase-shift with TMS320F28335 will be 3000 steps. The closed-loop validation is first done for a step-change in reference from 75 to 90 V. The experimental and model results both show the system to be little over-damped with <5% overshoot. Similar results can be seen when a reference voltage is changed from 90 to 75 V. The load resistance then decreased from 100 to 80 Ω and again reverted back to 100 Ω (Fig. 12). The model and the experimental data are almost the same for the transient as well as the steady state, as shown in Fig. 13. The closed-loop frequency response of the DAB converter for a reference voltage of 75 V is shown in Fig. 13. The steady-state phase shift for the converter to operate at is around 0.15. Therefore, the closed-loop transfer function Gcl(z) with a controller transfer function H(z) can be represented as Gcl(z) = Gvd(z)H(z) 1 + Gvd(z)H(z) (37) where H(z) = kp+ ki z z− 1 (38)
The cross-over frequency is kept around 6.8 kHz. The closed loop has an infinite phase margin and a 17.5 dB gain margin as shown in Fig. 14. The sufficient gain margin helps in proper controlling of the output voltage to the desired level. The controller is tuned for the specification shown in Table 1. However, when the load Fig. 11 Control to output transfer function Gvd(z)
(a) d = 0.15, (b) d = 0.35
demand or reference voltage is changed, the optimum value of the controller proportional and integral gain needs to be adjusted. Therefore, advanced control, like non-linear control, model predictive control, or adaptive controller, can be used. The explicit state-space representation proposed in this paper helps in the proper design of these advance controllers.
8 Conclusion
A novel modelling, based on the multi-time scale based mixed system model, for DAB converters have been proposed in this paper. The state variable has been segregated into a fast and slow variable. The proposed method thus removes the exponential matrix terms from the state model. There is no approximation involved in this method, which improves the accuracy of the model. The resultant model accurately predicts the important state
variable information. The large-signal model has been verified by the validating step response of the state model with the switching model of MATLAB. The small signal is later validated by an experimental setup to achieve the Bode plot of the system transfer function. The model achieved by the TTS model gives an accurate small-signal model. The model has been later verified for different operating points by changing the load resistance.
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