Multidimensional Ripple Correlation Technique for Optimal Operation of Triple-Active-Bridge Converters

This article presents a multidimensional ripple correlation search technique of optimal operating points of triple-active-bridge (TAB) converters. Such converters present multiple modulation parameters that should be exploited to achieve high operation efficiency. On the other hand, the several degrees of freedom available make the identification of optimal parameters a challenging task, not easily tackled analytically or in closed form. A model-free online search method based on the ripple correlation technique is then proposed in this article. The proposed method finds the optimum modulation parameters of TAB converters utilizing a three-dimensional ripple correlation control. The key property of the proposed solution is the adoption of orthogonal perturbation signals, where the orthogonality is simply obtained using different injection frequencies. The multidimensional correlation technique originally shown herein can be applied to other generic optimization problems. The proposed search is verified through a hardware-in-the-loop validation setup and an experimental prototype rated 5 $\mathrm{k}\mathrm{W}$.


I. INTRODUCTION
I SOLATED multiport converters (IMPCs) allow compact solutions with reduced number of components for the interconnection of multiple energy resources. This is valuable in several emerging applications, like, for example, home nanogrids and electrified vehicles [1], [2], [3], [4]. Effective IMPCs can be achieved by extending the concept of the dual-active-bridge (DAB) converter, that is, by utilizing a multiterminal highfrequency transformer to couple ports at different voltage levels [5], [6]. Adding one extra port to the DAB gives the triple-active-bridge (TAB) topology, which is one of the simplest IMPC topologies utilizing a three-terminal high-frequency transformer, as shown in Fig. 1(a) [7]. In order to control the power flow, magnitude, and direction, the phase-shift modulation (PSM) is commonly applied [see Fig. 1(b)]. By the PSM, the three full-bridges generate ac voltages v 1 , v 2 , v 3 with duty-cycle fixed at 50% and phase-shifts φ 2 and φ 3 regulated to achieve specific power flows among the ports [8], [9]. As a disadvantage of this kind of modulation, relatively high conduction and switching losses are typically observed while operating at light load or with significant voltage mismatches among the transformer terminals, that is, while far from the condition V 1 : V 2 : V 3 = n 1 : n 2 : n 3 . To overcome this limitation, penta PSM schemes have been proposed [10]. By such modulation schemes, represented in Fig. 1(c), also the duty-cycles D 1 , D 2  It can be shown that with five modulation parameters D 1 , D 2 , D 3 , φ 2 , and φ 3 the employed three full-bridges can generate 720 different voltage patterns, considering the number of permutations corresponding to the different sequences of rising and falling edges of v 1 , v 2 , and v 3 [5]. To analyze each switching pattern with the aim of finding optimum operation, for example, in terms of losses, may easily result unpractical. The literature describes different approaches reporting converter efficiency improvements [10], [11], [12], [13], [14], [15], [16], [17]. In [10], reduced power losses are achieved by modulating the duty-cycle and the phase-shift (penta phase-shift) based on fundamental component analysis (FCA) of the converter voltages and currents. While analyzes based on only the fundamental components are effective at moderate-to-high transferred power among the ports, the true power loss differs from the fundamental power loss at low-power levels, limiting the effectiveness of the approach. Moreover, fundamental analysis misses most of the insight into the shape of the switching patterns, which gives important information for switching loss reduction. A zerovoltage switching criterion for TAB has been introduced in [11], utilizing penta PSM, and taking into account all the parameters of the TAB converter, including the parasitic capacitance of the switches, the leakage inductance of the transformer, and the switching frequency. However, this approach introduced in [11] neglects conduction loss. In [12], a universal analytical model of the TAB converter is developed utilizing frequency domain analysis (FDA) and then exploited to pursue total loss reduction via particle-swarm optimization (PSO). The PSO searches for minimum rms current while fulfilling ZVS constraints. As in [10] and [11], Li et al. [12] employed a look-up table (LUT) to store the optimum modulation parameters preliminary calculated offline. In [13], a generalized harmonic approximation (GHA) model is developed for the TAB converter, which is then used in an online gradient-descent optimization aiming at minimizing the derived analytical model (AM) of the total rms current. The same approach is also presented in [14] and [15], optimizing the converter overall efficiency based on the GHA analysis.
In summary, a number of solutions are described in the literature for TAB converters efficiency improvements. The complexity of the optimization problem, stemming from the converter behavior and structure and the multiple degrees of freedom available for modulation, commonly brings to complex analytical results and optimization approaches that can barely achieve actual optimal operation. A different approach is proposed herein, in which optimal modulation parameters are pursued automatically by means of an online search method based on ripple correlation control (RCC). The proposed multidimensional correlation technique is model-free, endowing the approach with general validity. In fact, the proposed solution is effective regardless of TAB parameters variations during operation and can adapt modulation parameters automatically online. For example, the approach well tolerates partial or total uncertainties on the parameters of the used switches, parameters of the magnetics and passive components, controller and plant dynamics, etc. Such information is not needed to apply the approach successfully and may be left unknown. The feature just remarked extends the potential application of what presented herein to other similar optimization problems with multiple degrees of freedom. The features of the proposed method compared with other relevant approaches in the literature are summarized in Table I.
In the following, the online search technique based on RCC search method in a three-dimensional space (3D-RCC) is proposed and described to find the optimum duty-cycles for a given set-point of voltage and power levels at the ports of a TAB converter. Specifically, Section II provides the model of RCC establishing the central concept behind the proposed 3D-RCC. Section III presents the 3D-RCC considering the optimal operation of the TAB converter. Section IV discusses stability considerations of the proposed approach. Section V demonstrates the proposed 3D-RCC reporting hardware-in-the-loop (HIL) simulation results and experimental results. Finally, Section VI concludes this article.

II. RIPPLE CORRELATION CONTROL
One-dimensional RCC has been applied for photovoltaic maximum power point tracking [18], [19], [20]. In these applications, the RCC objective is to optimize a cost function x(y), namely, the photovoltaic power, which is dependent on a state variable y, which is typically the panel voltage or current. The technique and related nomenclature are introduced in the following and then extended to the three-dimensional case applied to TAB optimization in Section III.
Given n samples x(k − i), y(k − i), i = 1, . . . , n of the couple of discrete-time variables x(k), y(k), their correlation can be described by the sample Pearson's correlation coefficient [21] as where σ x , σ y andx,ȳ are the sample standard deviations and means of x and y, respectively; for example, for variable x Assume x represents a cost function and y a controllable variable. Three cases can be distinguished for the correlation coefficient in (1), namely, the following.
1) ρ xy (k) > 0: A positive correlation between the variables x and y. This means that the controlled variable y should be reduced to reduce the cost function x. 2) ρ xy (k) < 0: A negative correlation between variables x and y. This means that the controlled variable y should be increased to reduce the cost function x. 3) ρ xy (k) = 0: No correlation between variables x and y. The sign information is enough to describe the correlation between the two variables and decide how to modify the controllable variable y in order to reduce the cost x.
Remarkably, the sign information is solely carried by the numerator of (1), whilst the terms at the denominator are always positive, by definition. It yields Equation (4) can be represented in an approximate way as shown in Fig. 2, where the operation of mean (i.e., average) is implemented by means of low-pass filters (LPFs). In Fig. 2, the sign of the correlation coefficient ρ xy (k) is given as input to an integrator, whose output can used to adjust the value of the controllable variable y to result in lower values of x. For example, assume that the integrator output is used to decrease the value of y. Assume also that a positive correlation is found. In this case the output of the integrator increases and, due to the positive correlation among x and y, x decreases too, moving toward values of lower cost. Eventually, the previously remarked properties of the sign of ρ xy indicate that if a local minimum is reached, the process finds an equilibrium point at that local minimum. In applications, the search is performed by analyzing how a small perturbation of given frequency superimposed to the controlled variable y correlates with the related changes on the cost variable x. The theoretical framework behind the RCC is the perturbing extremum seeking control (ESC) theory, as referred to in the field of control systems engineering [22], [23]. ESC has been studied as an online control search for several fields distinct from power electronics with the same concepts of the RCC described above [24], [25], [26]. Its peculiarity is the property of being a model-free approach for optimization, which allows effective, low-complexity solutions.
The simple principle of RCC can be exploited for the search of optimal operating points even in case of complex systems, that is, relations among cost and controllable variables. The principle, just discussed in one-dimensional terms, as commonly found in the literature, is described in the following in a more general application that considers a three-dimensional problem for the optimization of a TAB converter. This is possible by performing the RCC search by using orthogonal perturbations for the controllable variables. Herein, it is shown the case of using perturbation signals at different frequencies, one for each controlled variable, and separately correlate the perturbation signals with the corresponding ones present in the cost variable.

III. EXTENSION TO 3D-RCC FOR TAB OPTIMIZATION
Consider now the problem of reducing the power loss of a TAB by searching for optimal modulation parameters. Consider also to pursue conduction loss minimization by applying an RCC search having as cost variable the total rms current where weights r p , p = 1, . . . , 3, are the equivalent path resistances of the respective pth port. Notably, the total rms current depends on five modulation variables: two phase-shifts (i.e., φ 2 , φ 3 ) plus three duty-cycles (i.e., D 1 , D 2 , D 3 ). In basic TAB controllers, the phase-shifts are used to regulate the voltages or the power exchanged among the ports [e.g., by proportionalintegrative (PI) regulators], while the duty-cycles are kept fixed at 50%. Differently, more advanced modulators can exploit the three duty-cycles to optimize the converter operation [10], [27], [28]. Such optimization that exploits the degrees of freedom given by the duty-cycles is performed herein by the presented 3D-RCC approach. The core concept of 3D-RCC is analogous to the RCC discussed in Section II, by splitting the three-dimensional problem, with the three variables D 1 , D 2 , D 3 , into three separate problems, each considering a single variable. The separation can be done by employing the orthogonality principle, perturbing each duty-cycle with a different frequency, and evaluating the correlation between it and the cost function.
Three RCC searches can be performed concurrently by using corresponding perturbation signals at different frequencies and by correlating the perturbations of same frequency found on the controllable and on the cost variables. Fig. 3 displays the resulting scheme. Each duty-cycle correlation search has a different frequency (i.e., ω 1 , ω 2 , and ω 3 ) with the same magnitude , chosen small with respect to the full duty-cycle. The forced disturbance of each duty-cycle is correlated separately to the total rms current positively or negatively. By tuning each duty-cycle value to the optimum value till the correlation result is equal to zero, that is, the minimum rms value is reached. In general, as highlighted in Fig. 3 and shown in Section V, different cost variables can be considered, like, for example, the input dc-current, the rms of the fundamental currents, or the total rms current.
As a final remark regarding the implementation of the voltage control-loops, a decoupling matrix, visibile in Fig. 3, is applied herein to slightly adjust the phase-shifts given the duty-cycles computed online. This control refinement was originally proposed in [10], which highlighted a coupling between the two voltage control loops and proposed the use of a decoupling matrix to decompose the multivariable control system into a series of independent, single-loop subsystems.

IV. STABILITY CONSIDERATIONS
While this article focuses on showing the application of the RCC in a multidimensional case in power electronics and presenting the solution to the TAB optimization problem by exploiting this model-free method, in the field of control-systems engineering other papers addressed the analyses and stability study of the class of ESC approaches, to which the 3D-RCC belongs [23], [24], [25], [26]. In addition to what was discussed in the previous sections, a set of constraints should be met to ensure a stable operation of the whole system. Once these conditions are satisfied, the method achieves convergence regardless of the specific converter parameters (e.g., kind of switching devices, amount of conduction loss, etc.), hence, its model-free property. Specifically, to guarantee stable operation of the proposed 3D-RCC, the following relations should be considered for parameters selection.
Hereinafter, ω M and ω m indicate the maximum and minimum values, respectively, of the used perturbation frequencies.
2) Dynamics of the plant to be optimized: the maximum perturbation frequency ω M should be much slower than the plant dynamics. This means that ω M should lay well below the crossover frequency of the voltage controllers (i.e., the PI regulators in Fig. 3). In our case, typical voltage controllers bandwidths are in the order of hundreds of hertz, then ω M may be chosen an order of magnitude lower. 3) LPF cut-off frequency: the cut-off frequency of the LPF in Fig. 2 should be much lower than the minimum perturbing frequency ω m , to well approximate the average operation in (4). 4) Correlation integration gain: the minimum perturbing frequency must be greater than the correlation integrator gain in Fig. 2, namely ω m k i . This means that the optimization variables D 1 , D 2 , and D 3 in Fig. 3 change slowly even with respect to the "slowest" perturbation signal, of frequency ω m . 5) Amplitude of perturbations: the perturbations amplitude in (6) should be chosen to allow small variations around the optimal operating points (e.g., 1% of full dutycycle).

V. VALIDATION RESULTS
In this section, the validation of the proposed 3D-RCC search is reported considering the results from an HIL setup and an experimental prototype. With the aim of demonstrating the effectiveness of the approach considering several different objectives (i.e., cost functions), the approach is demonstrated while considering total true rms current optimization, total fundamental rms current optimization, and input dc current optimization. The use of different validation tools (i.e., HIL and experimental prototype) and cost functions allows an accurate validation of the optimization approach and to show its robustness with respect to various implementations and cost functions.

A. HIL Validation
The real-time simulator PLECS RT-Box1, as shown in Fig. 4, has been used to model the TAB converter operation. The considered parameters are listed in Table II. The 3D-RCC search and converter control and modulation are implemented on an Imperix L.t.d. B-Box RCP controller. These first tests performed on an HIL platform allow to focus on the validation of the proposed technique when deployed in the final digital control platform interfaced to an emulated plant (i.e., TAB converter) that is free from parameters uncertainties and nonidealities. This simplifies the comparison with solutions obtained analytically or in simulation and to test the controller before being connected to the real-hardware. Given the fixed integration time step T step (see Table II) possible with the adopted state-of-the-art emulator PLECS RT-Box1, a scaled-down switching frequency is considered with respect to the one used with the experimental prototype. This allowed us to have T sw /T step = 50 steps within a switching period and then to preserve sufficient accuracy of the obtained results. Of course, to keep the same current waveforms and, then, their rms, leakage inductance values are also scaled up by the same factor as the switching frequency (cf . Tables II  and III).
In order to show the efficacy of the proposed 3D-RCC, the obtained results are compared with the results obtained by a brute-force (BF) search in which, given the voltage levels and the powers exchanged at the ports, the converter operation is systematically evaluated at all the points of a dense mesh of the space D 1 , D 2 , and D 3 . On this basis, the results obtained by the BF search are regarded as the true optimum solutions. Fig. 5 shows the obtained results at V 1 = 400 V, V 2 = 320 V, V 3 = 480 V, and P 2 = 350 W, while port-3 power  P 3 is changing with 150 W step, with total true rms current (5) as optimization objective. Specifically, Fig. 5(a) shows the optimum duty-cycles found by the BF search (i.e., D 1BF , D 2BF , D 3BF ) compared to the 3D-RCC search results (i.e., D 1RCC , D 2RCC , D 3RCC ). The searching techniques give almost the same optimum duty cycles over the considered P 3 variation. Fig. 5(b) shows the total true rms currents corresponding to the BF search findings and 3D-RCC search findings, with the maximum deviation between the two currents of less than 5%.

B. Experimental Prototype Validation
The experimental prototype in Fig. 6, with parameters listed in Table III, has been implemented to verify the proposed threedimensional correlation search. As shown in Fig. 7, port-1 of the converter is connected to a fixed dc-source V 1 = 400 V, port- 2  TABLE IV  3D-RCC SEARCH OPTIMAL RESULTS FOR DIFFERENT TEST POINTS (TP1-TP11 and port-3 are connected to two dc electronic loads R 2 and R 3 , respectively.
The 3D-RCC search together with converter control and modulation is implemented on an Imperix L.t.d. B-Box RCP controller, as done in the previous Section V-A, driving six Imperix PEB8032-A half-bridges. Values of rms quantities are obtained by a signal conditioning circuit based on the integrated circuit LTC1968 by Analog Devices.
A total of 11 points are tested at different loads. The test cases are chosen to represent voltage mismatch case at converter ports with V 1 = 400 V, V 2 = 320 V, and V 3 = 480 V. To obtain different load conditions, port-3 power is changed from low to high power levels while port-2 power is fixed at P 2 = 350 W.
For each set-point the correlation search is run considering the minimization of 1) the total true rms current i rms ; 2) the total fundamental rms current i fund ; 3) the input dc current i dc . Herein, three different cost functions are optimized to validate the generic solution found by the proposed 3D-RCC. It is worth recalling that optimizing the total rms current mainly benefits copper and conduction loss, while optimizing the input dc current allows the optimization of the overall converter loss, including, switching loss, conduction loss, transformer loss, etc. Then, considering the following reported results, input dc-current optimization may lead to lower overall loss while still showing higher total rms current with respect to what obtained by total rms current optimization, which is consistent. The results are summarized in Table IV. The optimal duty-cycles found by the 3D-RCC search are reported as D opt 1 , D opt 2 , and D opt 3 , together with the values of the currents at the found points.
At low-power levels, like at TP1, TP2, and TP3, different final points are obtained as total rms, fundamental, or dc currents are considered as cost variables. This is expected: at low-power levels the optimal duty-cycles are small, which increases the harmonic content of the converter currents. Instead, for highpower levels, like for TP11, the optimization of the cost functions gives almost the same optimal duty-cycles since the optimal duty-cycles lie near the saturation limit, which reduces the weight of the harmonics. The results of optimizing each cost function are discussed in the following.

1) Total True RMS Current Optimization:
The results of the 3D-RCC optimizing the total true rms current [i.e., i rms in (5)] are compared to the optimal points found by a systematic BF search, as done in Section V-A too. Fig. 8 reports the results from both the search techniques. Small differences emerge in Fig. 8(a), (b), and (c) comparing the duty-cycles found by the two different approaches. Fig. 8(d) shows the quantity i rms of the search methods, including also the results by the simple PSM. The proposed 3D-RCC reduces i rms by about 50% in some test cases as compared to PSM. The comparison between the 3D-RCC and the BF results shows the ability of the proposed 3D-RCC to find the true optimum operating point considering the minimization of i rms , with a maximum deviation < 5%. Fig. 9 shows the dynamic response of the 3D-RCC search optimizing i rms for TP1 of Table IV. The algorithm is started at initial duty-cycles values all equal to 0.4 [see Fig. 9 0.23, and D opt 3 = 0.145 with i rms ≈ 4.2 A, thus reducing the total true rms current by about 58% of its original value. Fig. 10 shows the transformer voltage and current waveforms at TP1 with different modulation approaches. Fig. 10(a) and (b) shows the converter waveforms with duty-cycles all at 0.4 before starting 3D-RCC, which, after being activated, brings to the steady-state operation displayed in Fig. 10(c) and (d), the optimal duty-cycles found are reported in Fig. 9(b). Fig. 10(e) and (f) displays the converter waveforms with PSM. Comparing the current waveforms in Fig. 10(b), (d), and (f) with the corresponding results by the 3D-RCC reported in Fig. 10(d), peak current values appear substantially reduced for all the three ports.

2) Input DC-Current Optimization:
The results of 3D-RCC optimizing the input dc current (i.e., i dc ) are now considered. Remarkably, being the voltage at port-1 fixed as well as the power absorbed at port-2 and port-3, such an optimization minimizes the total conversion loss, which includes the sum of conduction and switching loss, of the converter in Fig. 7, because it minimizes the supply current at port-1. Fig. 11(a) compares the obtained input dc-current by 3D-RCC and PSM. At low-power levels, 3D-RCC reduces the input current by about 30% as compared to PSM. Fig. 11(b) reports the percentage reduction of the converter total loss, showing a reduction > 70% for most of the test cases. Fig. 12(a) and (b) shows the dynamics of the 3D-RCC recorded during operation at TP9, similar results would be obtained considering other test points. The search starts with initial duty-cycles values set to 0.4 [see Fig. 12(a)] with input dc-current and total converter loss equal to about 5.15 A and 300 W, respectively [see Fig. 12(b)]. The 3D-RCC finds the optimum point at duty-cycles D opt 1 = 0.37, D opt 2 = 0.44, and D opt 3 = 0.25, where the input dc-current and total loss decrease to approximately 4.68 A and 122 W, respectively. The input dc-current optimization reduces the overall loss by about 60% with respect to the conditions at the starting point.
Although total converter loss minimization does not imply, in general, soft-switching conditions for all the converter switches, a correlation between the two may easily emerge. Fig. 13 shows what was obtained from this respect with the considered converter in Fig. 7, operating at TP9 with PSM and 3D-RCC. With the currents and voltages directions indicated in Fig. 7, soft-switching conditions at devices turn-ON (ZVS) are assumed at a port if i) a rising voltage transition occurs with negative port current ii) a falling voltage transition occurs with positive port  current [29]. Then, Fig. 13(a) shows ZVS for port-1 and port-3 switches, while the conditions are not met for the switches of port-2. Fig. 13(b) reports the results from the proposed 3D-RCC method, showing that the switches of port-2, in addition to those of port-1 and port-3, experience soft switching. Besides, 3D-RCC optimization also reduced the switched current values, which also contributes to decreasing the overall converter loss.

3) Total Fundamental Rms Current Optimization:
The proposed 3D-RCC is used in this part to optimize the fundamental component of the total true rms current (i.e., i fund ). This practice mainly illustrates that the proposed method is generic regarding optimizing different cost functions. Fig. 14 shows the fundamental rms current of the 3D-RCC compared to the fundamental current of the PSM technique, with a reduction up to about 70% in one of the test cases.   In summary, besides the consistency of the obtained results with what expected considering TAB specific application, the experimental validation showed the effectiveness and robustness of the proposed 3D-RCC approach for multidimensional converter optimization. As a final remark, even though various cost functions (i.e., i rms , i fund , and i dc in Fig. 3) are considered herein to show the flexibility of 3D-RCC in solving different optimization targets, dc-current minimization requires only the measurement of dc currents to be performed and should be preferred for overall loss minimization.

VI. CONCLUSION
The operation of the ripple correlation technique has been shown considering the multidimensional problem of minimizing the conversion loss of a TAB converter. Optimal TAB modulation is a complex task, given the many variables concurring in the converter operation. Considering typical modulation schemes, five variables are available for control, that is, two phase-shifts and three duty-cycles. This article shows that a multidimensional correlation search can be adopted to tackle the challenge of finding optimal modulation parameters. The proposed method makes use of an orthogonal perturbation signal for each degree of freedom available for optimization. Then, the modulation parameters are adjusted on the basis of the correlation among the orthogonal signal and the corresponding perturbation found on the variable chosen for optimization. Orthogonality can be achieved by using sinusoidal perturbations of different frequencies. The method-which is performed online and does not require the knowledge of converter parameters, namely, it is model-free-shown general validity in the considered application: it is capable of finding appropriate modulation parameters considering various quantities for optimization, such as total rms, dc, or fundamental currents exchanged at the port of the converter. Experimental results show the effectiveness of the proposed online model-free approach for the optimal operation of the TAB.