Modular converters are becoming increasingly attractive due to increased demand for compact and highly efficient isolated DC-DC power converters for different applications, such as fast chargers, uninterruptible power supplies (UPSs), photovoltaic systems and solid state transformers. These structures can be formed by cascading multiple modules of DABs in different configurations, such as input-parallel output-series (IPOS) [1]- [2] and ISOP [3]- [8] or both IPOS and ISOP combined into a single converter [9]. When considering the ISOP configuration, different control strategies have been proposed for balancing the input and regulating the output voltages. The control method for a single module of the converter can either use single phase-shift (SPS), dual phase-shift (DPS), extended phase-shift (EPS) or triple phase-shift (TPS). In [3], a decoupling control method was applied in order to both obtain voltage sharing between the modules and to control the output voltage. A decoupled input and output voltage controller was proposed in [4]. In [5], another decoupled control method was applied to balance the input and regulate the output voltages based on lag compensators. A surge voltage suppression technique was presented in [6] to achieve high power and efficiency for wide load range. However, this method used SPS for power flow between the inputs and the output. Therefore, this method is unable to achieve the maximum efficiency of the system. An input voltage sharing strategy was presented in [7]- [8] to enable proper operation of the converter. All the above mentioned articles, except [6], used the small signal analysis of each module of the ISOP to implement their control method. Advanced control techniques, such as DPS, EPS and TPS, were proposed in [10]- [17] to improve efficiency and to decrease the RMS current, the reactive power loss and the peak current stress.
To further improve upon the control techniques presented in [10]- [17], this paper presents a control scheme based on optimal TPS control for both power sharing control and RMS current minimization. The goal is to achieve balanced power transmission, even under mismatched leakage inductance or shutdown of a DAB module of the ISOP. This work has analyzed all the operation modes and zones of the converter in order to obtain the best operation region for achieving high efficiency.
A standard ISOP DAB converter is shown in Fig. 1. The ISOP is composed of multiple cells configured through series inputs and parallel outputs. Fig. 2 shows the AC side voltage of the ¿rst and second bridge voltages, V1 and V2, respectively, and the inductor current, IL, for one cell of the ISOP. D1 and D2 are the duty cycles of the first and second converters, respectively, and ij is the phase-shift between V1 and V2. All three parameters are de¿ned in per-unit. D1 = D2 = 1 is the case of the square wave and ij = 1 is 90ƕ phase-shift. The simplest method of operation of DAB is considering D1 = D2 = 1 and adjusting ij to regulate the power. This method covers the entire power capability range of the DAB, and the RMS current is at the minimum for the same voltage level V1 = V2. However, when V1 differs from V2, unnecessary RMS current and the circulating power generate losses and heat the converters and the transformer. The most general and Àexible method of operating DAB is to vary all three parameters, D1, D2, and ij. Depending on the values of these three variables, V1 and V2 waveforms are different, and hence, the calculation is different. To start the calculation of the current and power, all of the possible modes or operation regions should be determined.
In Fig. 2, the rising and falling times of V1 and V2 are de¿ned as x1 and x2. y1 and y2 are the rising and falling times of V2 and y3 is the rising up from negative of V2.
When considering the half of the period which contains x1 and x2, there are three areas in which y1, y2, and y3 can lay. However, it is impossible to have y3 > x2, as it is a logical contradiction. Knowing y3 < x2 eliminates a set of possibilities which are not feasible.
y3 and y1 cannot exist at the same time in half a period. The fact that only y1 and y2 or y3 and y1 can happen at the same time removes several possible combinations.
Unlike the previous impossible combinations which describe a set of possibilities, this combination is a speci¿c case which is a logical contradiction.
This condition describes another speci¿c impossible case:
All of the remaining feasible possible combinations are considered for the derivation of the equations. The names of each region come from the three dimensional regions in which each condition is satis¿ed. Fig. 3 shows one example waveform of each region. Figs.
The boundaries of all the regions are listed in Table I.
The ¿rst step in calculating the RMS current and average power is to formulate the instantaneous expressions. The magnetizing current of the transformer is neglected in comparison to the leakage current. The instantaneous current is calculated piecewise for each region of operation based on (7). ܫ ሺݐሻ ൌ ܫ
In ( 7), zA and zB can be 0, x1, x2, y1, y2, y3, or 2, depending on the region of operation and the piece of the current. The output current may not be vertically symmetrical. However, after the transient, the current is symmetric around the x axis. Hence, the average of the current needs to be calculated and subtracted from the instantaneous current. The average calculation has been done in a piecewise manner in (8).
The average power is calculated piecewise in each region from (9), where V1 and I are variables, depending on the piece in each region. The RMS current is obtained from (10).
The average power equation for each region is obtained in (11). PaveFront, PaveRight and, PaveLeft are independent to ij, D1, and D2, respectively, as they could be predicted by examining Figs. 3(a)-3(c).
The expression of the average powers after the common term of భ మ ଼ is the per-unit power. The following conclusions can be made from the maximum achievable per-unit power as in (12):
There is a common term between all of the RMS current equations which is written in (13). The RMS currents for all of the regions are obtained in (14). Although some average powers are independent to some variables, all of the variables affect RMS current in all of the regions.
IV. OPTIMIZED TRAJECTORIES
The goal of the optimization is to minimize RMS current for a given average power. The variables are D1, D2, and ij. The objective function, f, is the RMS current which needs to be minimized. The constraint, g, is the average power which the converter should be able to provide, regardless of the current. The Lagrange Multiplier method is used for the constraint optimization problem of the DAB. Based on Lagrange optimization methodology, a Lagrange function is the combination of the objective function and the constraint multiplied by Ȝ, as formulated in (15).
The optimal solution of the constraint optimization is described by a set of equations of the gradient of the Lagrange Multiplier, as written in (16) for the DAB. Although ȁ/Ȝ = 0 is part of ȁ = 0, it does not reveal any new information, as it is simply the constraint function, g, itself.
There are ¿ve regions and each has its own f, g, and ȁ. As shown in (16), there are three equations with four unknowns, which means there is a relation between the variables rather than an exact value. To ¿nd the optimal trajectory in each region, Ȝ needs to be found from one of the three equations of ( 16) and substituted in the others. The expression ij should be calculated from one of the remaining equations and substituted in the others. The last equation reveals the relation between D1 and D2 and this relation should be substituted back into the ij expression. If the relation describes a trajectory outside of the boundary of a region, the boundary is the best feasible solution.
The derivations of the optimal trajectories for all of the regions have been done, but for each set of conditions the optimal RMS current of one of the regions is minimum. Only the derivation of the best region is presented here. Since the only region that can cover powers above 0.5pu is the Rear region, the comparison of regions has been done only up to 0.5pu. For power higher than 0.5pu, only the optimal trajectory in the Rear region needs to be derived. Fig. 4(b) shows the minimum RMS current that each region can provide for the range of per-unit power of 0-0.5. In this ¿gure, V1 = V2 and the best result is the Rear (red) region. The corresponding trajectory in the Rear region is shown in Fig. 4(r), where D1 = D2 = 1, which is the square wave operation. The optimal trajectories in the other regions for this condition are shown in Figs. 4(f), 4(l), 4(m), and 4(g). Fig. 4(s) shows the optimal trajectory regardless of region, which is identical to the optimal trajectory of the Rear region in this case. This means if the DAB is being used only for isolation purpose with V1 = V2, then the square wave operation is the optimal operation. Fig. 4(c) shows the minimum RMS current when V2 > V1. The Right and Middle regions have the minimum current and exactly the same value. In Figs. 4(n) and 4(i), the optimal trajectory lies on the boundary of the Right and Middle regions, which explains why they have the same current. The Left region has the highest current. After 0.5pu, the only optimal trajectory is the Rear region of Fig. 4(t). Fig. 4(u) shows the overall trend that the optimal trajectory should be on the boundary of the Left and Right regions: íD1 +D2 +ij = 0 plane. For higher power, when D1 is saturated to 1, it is on the D1 = 1 plane. Eventually, when both D1 and D2 are saturated, it is on D1 = D2 = 1 line. A dual situation is shown in Fig. 4(a) when V2 < V1. The optimal trajectory for P < 0.5 ʌ is the boundary of the Left and Middle regions or D1 í D2 + ij = 0 plane. For higher power, D2 would be saturated to 1 and the trajectory is on D2 = 1 plane up to the point where D1 becomes saturated as well. The last part of the optimal trajectory is on D1 = D2 = 1 line. It can be observed that the optimal trajectory of any condition is never inside of a region, but always on the surface or boundary of the regions.
There are three zones on the optimal trajectory, as shown in
The ¿rst zone is low power when D1 & D2 1. For V2 > V1, the optimal trajectory is on íD1 + D2 + ij = 0. Based on the Lagrange Optimization, V1D1 = V2D2.
The trajectory lies on the boundary of the Right and Middle regions, so the power equation of either of them can be used. The per-unit power is shown in (18).
This zone ends when
Substituting this ij into (18) yields P12, the power at the boundary of zones 1 and 2.
The case of
The second zone is medium power when D1 is reached to the limit of 1 for V2 > V1. The optimal trajectory is D1 = 1 plane. By solving the Lagrange Multiplier in the Rear region and eliminating two of the equations via substitution, (23) can be derived. This is the solution of a quadratic equation which relates D2 to ij.
Substituting D1 = 1 and D2 from (23) into the Rear region power equation gives the power in zone 2. This zone ends when D2 = 1:
Similarly, for the dual case of ܸ ଶ ൏ ܸ ଵ the following equations can be derived.
The third and last zone is the high power zone when both D1 and D2 are saturated to 1. Varying ij is the only way to increase the power. Regardless of the relation between V1 and V2, the per-unit power in this zone has the same equation of the square wave. This can be found by substituting D1 = D2 = 1 into the power equation of the Rear region, as is written in (27).
The optimal closed loop control block diagram is shown in Fig. 6. PI controllers generate the duty cycles and phaseshift variation that compensate the power variation, which can occur in cases of mismatched transformer leakage inductance or when one converter is shut down. These variations are incremented to the optimal duty cycles in order to produce the final optimal D1, D2 and ij.
Fig. 789show the simulation results for the ISOP. Four scenarios are considered based on Fig. 5(a). Table II presents a summary of the simulation results. Fig. 7 (a)-(c) show the independent and total powers, the primary and secondary voltages and the leakage inductor currents of the transformer, and the output voltage with the three independent RMS currents of the three DAB cells of the ISOP. This was done with the system balanced under SPS control method, which is the ideal AC square wave voltage case. Fig. 7 (d)-(f) present the same powers, voltages and current waveforms when the TPS optimization control method is applied. It can be seen that the proposed control method effectively decreases the RMS current of the system. Fig. 7 (g)-(i) show the key voltage and leakage current for each operating zone of Fig. 4. From 0.4 to 0.6 seconds, the converter of Cell-1 is shut down. To compensate for this power deficit, the other two operating converters increase their power level to meet the load requirement. To further validate the power sharing algorithm, 25% mismatch leakage inductance is introduced in the converter of Cell-1. In this scenario, the algorithm changes the phase-shift and the duty cycles of the unbalance cell in order to keep the ISOP balanced. This can be seen in Fig. 8 (a) and (c). However, Fig. 9 shows the performance of the ISOP without the power sharing method. It can be seen that Cell-1 power is less than the other two DAB cells, meaning the system is unable to meet the load requirement. This paper proposes a power sharing control method added to the RMS current minimization technique based on Lagrange optimization for an ISOP of three modules. The TPS control method has been applied to regulate the ISOP. Detailed analyses of the RMS current and average power transferred in the DAB for each mode and the power sharing controller for all operation zones are presented in the paper. Four scenarios of the simulation results are summarized in Table II. It can be seen that the TPS method has significantly minimized the RMS current of the DAB, especially at low load condition, and improved the efficiency for the overall power range. The power sharing method has been applied to maintain equal power between the converters under various conditions, such as converter shutdown and mismatch in leakage inductance of the transformers. Simulation results have proven the effectiveness of all the control techniques employed.
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