Jaime Wladimir Zapata

Samir Kouro

Gonzalo Carrasco

Thierry A Meynard

Step-Down Partial Power DC-DC Converters for Two-Stage Photovoltaic String Inverters

Keywords: Dc-dc converters, string inverter, partial power converters, photovoltaic energy

Photovoltaic (PV) systems composed by two energy conversion stages are attractive from an operation point of view. This is because the maximum power point tracking (MPPT) range is extended, due to the voltage decoupling between the PV system and the dc-link. Nevertheless, the additional dc-dc conversion stage increases the volume, cost and power converter losses. Therefore, central inverters based on a single-stage converter, have been a mainstream solution to interface large-scale PV arrays composed of several strings connected in parallel made by the series connections of PV modules. The concept of partial power converters (PPC), previously reported as a voltage step-up stage, has not addressed in depth for all types of PV applications. In this work, a PPC performing voltage step-down operation is proposed and analyzed. This concept is interesting from the industry point of view, since with the new isolation standards of PV modules are reaching 1500 V, increasing both the size of the string and dc-link voltage for single-stage inverters. Since grid connection remains typically at 690 V, larger strings impose more demanding operation for single-stage central inverters (required to operate at lower modulation indexes and demand higher blocking voltage devices), making the proposed step-down PPC an attractive solution. Theoretical analysis and an experimental test-bench was built in order to validate the PPC concept, the control performance and the improvement of the conversion efficiency. The experimental results corroborate the benefits of using a PPC, in terms of increasing the system efficiency by reducing the processed power of the converter, while not affecting the system performance.

Introduction

Traditional single-stage converters (string and central inverters) have been the mainstream solution for grid-connected PV systems. The most important reasons are the lower capital cost and lower conversion losses compared with a two-stage approach. Nevertheless, due to penetration of larger PV plants, more demanding grid-codes have been introduced, therefore affecting particularly PV systems with central inverters. One of these regulations requires inverters to perform frequency regulation through power curtailment, which forces the PV systems to forfeit power and not operate in its maximum power point [START_REF] Cruzado-Alba | Power Production Losses Study by Frequency Regulation in Weak-Grid-Connected Utility-Scale Photovoltaic Plants[END_REF]. This reduces the dc voltage operating range limited on one side by the minimum voltage needed to perform current control and on the other side to regulate the power curtailment.

An additional dc-dc stage increases the MPPT range, due to the voltage decoupling between the PV system and the dc-link. In addition, the effect of partial shading is mitigated if several dc-stages are used, and the energy yield is increased due to the multiple independent MPPT for smaller strings or arrays [START_REF] Kasper | Classification and Comparative Evaluation of PV Panel-Integrated DC/DC Converter Concepts[END_REF]. This approach is changing the market share of the PV industry, where companies such as Huawei are currently leading the PV industry, by delivering high-efficiency two-stage conversion systems with up to 6 independent strings [START_REF] Huawei | Smart String Inverter SUN[END_REF]. Nowadays, two-stage PV inverters are gaining more interest over central inverters reducing partial shading effects [START_REF] Kasper | Classification and Comparative Evaluation of PV Panel-Integrated DC/DC Converter Concepts[END_REF]. Furthermore, the power density seemed not to be an issue in the past for large-scale PV systems where central inverters are used, but once the two-stage approach was adopted, the reduction of the converter size and weight has become important, due to the benefits in shipping and installation costs.

In large-scale solar PV systems, it is usual to find larger PV strings to reach higher dc voltages, and several strings connected in parallel to increase the power, forming an array. Due to PV voltage isolation limits, traditional string voltages are designed to values under 1000 V. However, larger PV strings around 1500 V are becoming popular [START_REF] Serban | DC-Bus Voltage Range Extension in 1500 V Photovoltaic Inverters[END_REF][START_REF] Serban | Improved PV Inverter Operating Range Using a Miniboost[END_REF], due to reduction of the combiner boxes [START_REF] Gkoutioudi | Comparison of PV systems with maximum DC voltage 1000V and 1500V[END_REF], and also for the losses reduction in the dc-side [START_REF] Inzunza | Development of a 1500Vdc photovoltaic inverter for utility-scale PV power plants[END_REF]. Since grid connection remains below 690 V, larger strings impose more demanding operation for single-stage central inverters. It requires to operate the inverter at lower modulation indexes reducing their efficiency, and it requires higher blocking voltage devices. An alternative, is to change the grid connection step-up transformer, to enable a higher AC voltage operation at the inverter side, but this is not a standard equipment.

On the other hand, adding an additional conversion stage performing buck operation leads to increase the component count, converter's size and conversion losses. This is the motivation to take advantage of the PPC concept, which was initially implemented for PV systems in [START_REF] Kasper | Classification and Comparative Evaluation of PV Panel-Integrated DC/DC Converter Concepts[END_REF], in order to reduce the power losses in the two-stage configurations. The concept of partial power converters (PPC), previously implemented as a stepup stage, has not been fully covered all PV applications [START_REF] Zapata | Step-Up Partial Power DC-DC Converters for Two-Stage PV Systems with Interleaved Current Performance[END_REF]. Therefore, in this work a PPC performing stepdown operation is analyzed and evaluated with a laboratory experimental test-bench.

Step-Down Partial Power Converters for PV Systems

The additional conversion stage in two-stage systems increases the component count, cost and size. Which is translated in higher converter losses to the system. In traditional grid-connected inverters, both converter stages, the dc-dc and the inverter, process all the power flowing from the PV source to the grid, as depicted in Figure 1a. They are known as Full Power Converters (FPC), so that the power losses of each stage, Pdc,l and Pac,l , depend on the converter efficiency, and the amount of power handled by each converter.

However, if the dc-stage is made by directing the power into two power paths, as depicted in Figure 1b, the dc-stage conversion efficiency ηdcs increases. As long as, one of the paths has a higher efficiency than the dc-dc converter efficiency ηdcc. In that case, the power processed by the converter is reduced, leading to a reduction of the conversion losses Pdc,l in the dc-stage. This dc-stage configuration can be named as Partial Power Converter (PPC), as long as the power processed by the converter Ppc is lower than the total power of the PV system Ppv. It can be understood with the analysis of the dc-stage conversion efficiency ηdcs.

Pdc

Ppv -Ppc Ppc ηdcs = Ppv = Ppv + Ppv ηdcc (1)
Simplifying the expression, the dc-stage efficiency is expressed as:

Ppc ηdcs = 1 - P pv (1 -ηdcc ) (2)
As can be seen from ( 2), the dc-stage efficiency does not depend only of the dc-dc converter efficiency ηdcc, but also it is affected by the ratio of the power processed by the dc-dc converter Ppc/Ppv. This power ratio is the main feature of a PPC, and considering this parameter it is possible to make a further analysis of these converters. In essence, the operating principle is the series connection of a voltage source between the PV source and the dc-link at the inverter side. This voltage source is designed to be smaller that the other two sources, then it will process less power. In addition, the PPC is able to indirectly control the PV system output voltage ensuring MPPT. Moreover, the dc-dc converter takes the power from the PV system and because of the connection, the series path also takes the power from the same PV system. For that reason, it is mandatory the use of isolated dc-dc topologies in the PPC configurations to avoid a short-circuit connecting an independently controlled dc-voltage between the PV system and the dc-link [START_REF] Morrison | Partial power DC-DC converter for photovoltaic two-stage string inverters[END_REF]. Considering that the power rating of the converter is much lower than the total power, and high-frequency transformers are used, the isolated topologies requirement does not present a drawback. Furthermore, with the selection of the inverter stage and modulation, as traditional transformer-less PV inverters, a proper grounding of the PV system can be achieved [START_REF] Kouro | Grid-Connected Photovoltaic Systems: An Overview of Recent Research and Emerging PV Converter Technology[END_REF]. The PPC configuration based on a step-up operation has been already addressed in several works for different applications [START_REF] Button | An advanced photovoltaic array regulator module[END_REF][START_REF] Min | A New Topology With High Efficiency Throughout All Load Range for Photovoltaic PCS[END_REF][START_REF] Lee | A Novel Topology for Photovoltaic DC/DC Full-Bridge Converter With Flat Efficiency Under Wide PV Module Voltage and Load Range[END_REF][START_REF] Zapata | Analysis of Partial Power DC-DC Converters for Two-Stage Photovoltaic Systems[END_REF][START_REF] Zientarski | Series-Connected Partial-Power Converters Applied to PV Systems: A Design Approach Based on Step-Up/Down Voltage Regulation Range[END_REF]. Focusing on PPC for microinverters PV applications, the most relevant works are found in [START_REF] Zhao | Analysis of high efficiency DC/DC converter processing partial input/output power[END_REF][START_REF] Zapata | Partial power DC-DC converter for photovoltaic microinverters[END_REF][START_REF] Kim | A high efficiency photovoltaic module integrated converter with the asymmetrical half-bridge flyback converter[END_REF][START_REF] Harfman-Todorovic | A high efficiency PV micro-inverter with grid support functions[END_REF]. They present an improvement in the conversion efficiency, which is commonly a problem in microinverters due to the high voltage elevation required for the gridconnection. In addition, based on the connection evaluated in [START_REF] Marzouk | Analysis of partial power processing distributed MPPT for a PV powered electric aircraft[END_REF][START_REF] Zhou | PV Balancers: Concept, Architectures, and Realization[END_REF], it is also possible to perform step-up operation. Nevertheless, this configuration has a limitation on the voltage elevation, which must be less than the double of the input voltage as is explained in [START_REF] Morrison | Partial power DC-DC converter for photovoltaic two-stage string inverters[END_REF].

As can be noted, there are different step-up PPC configurations used depending on the two-stage PV application. However, step-down PV applications have not been fully analyzed in terms of MPP voltage operation range, volume improvement due to partial power conversion, and the effects produced under solar irradiation changes. For that reason, in this paper is analyzed and validated with experimental results, a PPC with buck operation for larger strings as depicted in Figure 2. connection at PV side.

The main conceptual difference between a step-up and a step-down PPC is the voltage gain Gv = Vdc/Vpv.

A step-up PPC is used for PV applications, where the input voltage must be elevated until reaching the level required for the grid-connection. On the other hand, nowadays new PV strings reach 1500 V insulation, making larger strings a reality, and increasing the need of buck dc-dc stages to properly connect to grid-voltage levels.

Step-down PPCs are oriented to these PV applications, so that traditional large-scale PV applications do not work with two-stage systems due to the high conversion losses. In addition, the MPPT is made with the inverter stage, which is also the responsible for the grid-connection [START_REF] Kouro | Grid-Connected Photovoltaic Systems: An Overview of Recent Research and Emerging PV Converter Technology[END_REF]. It reduces the global efficiency of the system, specially under partial shading conditions, so that the convert does not have a wide operation voltage range. Therefore, the use of a step-down PPC for the dc-stage can be an attractive solution in order to increase the dc-stage conversion system, also increasing the MPPT flexibility, due to the decoupled voltage between the PV and the dc-side. Moreover, some other important benefits come from the reduction of weight and size of the converter, due to the reduced power processed by the dc-stage.

Efficiency Analysis

Due to the connection used to create the PPC configurations, the power processed by the converter is reduced. It decreases the conversion losses Pdc,l in the dc-stage, increasing the dc-stage conversion efficiency ηdcs.

As can be seen from ( 2), the dc-stage efficiency does not depend only of the dc-dc converter efficiency ηdcc, but also it is affected by the ratio of the power processed by the dc-dc converter Ppc/Ppv. This ratio is a parameter defined as partial power ratio Kpr, and it has to be lower than the unity Kpr < 1 in order to operate in PPC mode. Then the dc-stage conversion efficiency can be expressed as:

ηdcs = 1 -Kpr(1 -ηdcc ) (3)
Working with a PPC the efficiency of the dc-stage ηdcs, varies depending on the partial power ratio Kpr. If the converter is designed to work with the half of the rated power Kpr = 50% and it has en efficiency ηdcc = 60%, the efficiency of the dc-stage ηdcs = 80% as depicted in Figure 3. It means that the dc-stage efficiency increases 20% at this condition.

Volume and Power Density Analysis

The power conversion efficiency of the dc-stage, the volume and the power density are related to the power reduction in the dc-dc converter. The simplest theoretical approach to consider a power electronic system is modeling as a cube [START_REF] Kasper | Scaling and balancing of multi-cell converters[END_REF], where the power losses are dissipated by the surface area Ac which scales with the volume V of the cube as Ac = 6V 2/3 . The power losses can only be dissipated as heat over the surface area of the converter Ac, then the power losses dissipation pdc,l in the FPC are:

Pdc,l Ppv(1 -ηdcc ) pdc,l = Ac = 6VFPC2/3 (4)
On the other hand, the power losses dissipation pdc,l in the PPC are:

Pdc,l Ppc(1 -ηdcc ) pdc,l = Ac = 6VPPC2/3 (5)
In order to keep the identical design of the converter and heatsink, holding the same dc-dc converter efficiency, the power losses dissipation pdc,l is considered as equal for both configurations. Then:

PpcVPPC (6) Ppv (1 -ηdcc ) VFPC
Finally, it is possible to relate the volume variation VPPC /VFPC when the power processed by the converter Kpr changes. The equation is expressed as follows:

VPPC = K3pr/2 (7)

VFPC

The result of the volume variation is depicted in Figure 4a. It is possible to notice that the volume variation does not follow a linear trend as the dotted line shown. In fact, by considering the point A, if the converter is sized to handle Kpr = 45% of the rated power, the volume of the partial power converter is VPPC ' 30% VFPC. Thus, the volume of the converter can be greatly reduced by reducing the power handled by the converter.

Moreover, the power density of the converter (ρFPC, ρPPC) is directly related to the power processed per unity of volume, and it can be expressed in terms of the partial power ratio Kpr as:

(8) ρ FPC
As is depicted in the Figure 4b, the power density increases with the reduction of the power processed by the converter. Considering the point B in the solid line, the converter is sized to handle Kpr = 40% of the rated power, the density of the partial power converter is ρPPC ' 1.58ρFPC. .

Operation Range Analysis

The reduced power processed by the PPC comes from the reduction of the voltage at the converter side vpc, due to the series connection of the voltage sources as depicted in Figure 2. However, based on the consideration that the dc-link voltage is fixed, the required voltage for the MPPT algorithm directly depends on the range of the vpc voltage. It means that the lower voltage at converter side, the reduced MPPT voltage range. Therefore, in order to evaluate the voltage variation of a PV cell in terms of solar irradiation it is required to find a model to relate the PV voltage and current. A explicit solution for the PV current and voltage can be derived using the Lambert's W function. It leads to an efficient model of the PV cell, resulting in significantly reduced calculation times and improved robustness of simulation [START_REF] Jain | Solar cell array parameters using Lambert W-function[END_REF]. The Lambert's W function is defined to be multivalued inverse of the function W -1 : x → xe x . Based on this function and the parameters of the single-diode model for a PV cell [START_REF] Shongwe | Comparative Analysis of Different Single-Diode PV Modeling Methods[END_REF], the PV voltage can be expressed as an explicit function of the PV current as:

R

Step-Down PPC Circuit Topologies

The buck operation, required to reduce the PV voltage to a level for the grid connection, can be achieved using the PPC configurations depicted in Figure 2. Nowadays, PV string and multistring applications are gaining more relevance in larger power plants, even though they were initially used for small and medium-scale PV systems. Typically isolated dc-dc converter topologies used for these applications, are based on highfrequency isolated Full-bridge converters and resonant converters. For that reason, in this work the full-bridge isolated topology was selected for the PPC configurations, as depicted in Figure 6, but the same analysis can be expanded to other isolated topologies. Step-down PPC connected at PV-side.

Step-Down PPC Connected at dc-Link Side

Based on the PPC configuration depicted in Figure 2a a step-down operation can be achieved. This connection splits the power at the dc-link side, which allows a voltage reduction at the input side of the dc-dc converter. Then the semiconductor is sized to commute with this reduced voltage.

Partial Power Ratio

In order to define the operation region of the PPC, the partial power ratio Kpr, is calculated as:

VpcIin Kpr = VpvIin Vpv -V Kpr = dc (10) Vpv V Kpr = 1 -dc = 1 -Gv Vpv
The partial power ratio Kpr varies linearly, depending on the voltage gain Gv as depicted in the dotted line in Figure 7a. Where the shadowed area represents the region of partial power operation.

Analysis of the Topology

Based on the configuration depicted in Figure 6a the voltage gain Gvc of the PPC configuration is defined as:

Gv = nT(1 -d) (11) nT(1 -d)+ 1
The voltage gain [START_REF] Button | An advanced photovoltaic array regulator module[END_REF] depends on the duty cycle d and the transformer turns ratio nT. The operation range is depicted in Figure 7a for two different values of the transformer turns ratio, where the shadowed area represents the region of partial power operation. From this picture it is possible to see that the voltage gain is limited by the turns ratio selection.

Considering a voltage gain (e.g., Gv > 0.5), the greater turns ratio allows a higher voltage gain. It means that the PV system is designed to operate close to the dc-link voltage, then the partial power ratio is reduced. As an example, in a design where the converter is sized to handle 40% of the nominal power, the voltage is reduced as much the 60% of the input voltage. In that case, using a transformer with n = 8 turns ratio it is possible to work within a voltage range for the MPPT, between 60 and 90% of the input voltage.

Step-Down PPC Connected at PV Side

Another connection allowed to achieve a voltage reduction is through the PPC configuration depicted in Figure 2b, where the power splitting is made at the PV-side.

Partial Power Ratio

In that case, the partial power ratio Kpr is expressed as: Kpr = ηdcs [START_REF] Min | A New Topology With High Efficiency Throughout All Load Range for Photovoltaic PCS[END_REF] Considering that the dc-stage is highly efficiency ηdcs ≈ 1, the partial power ratio Kpr varies depending on the voltage gain Gv as depicted in Figure 2b. Where the shadowed area represents the region of partial power operation. Notice that in contrast with the step-down PPC configuration connected at dc-link side, this partiality ratio is limited by the voltage gain, as can be seen in the figure the minimum value is limited to the half of the input voltage.

Analysis of the Topology

Based on the configuration depicted in Figure 6b the voltage gain Gvc of the PPC configuration is defined as:

Gv = nT(1 -d)-1 (13) nT(1 -d)
Moreover, the operation range is depicted in Figure 7b, where the shadowed area represents the region of partial power operation. From this picture it is possible to see that the duty cycle and the partiality ratio Kpr are limited by the turns ratio selection and the voltage gain. This is the motivation to select the step-down PPC configuration connected at dc-link side to be experimentally evaluated. Since this configuration presents more benefits for PV systems.

Experimental Validation

The experimental validation is made for the step-down PPC configuration connected at dc-link side, Figure 6a. It comprises: the correct operation of the power topology used for the construction of the PPC, the functionally of converter evaluating the MPPT algorithm, the analysis of partial operation and the conversion efficiency. The theoretical analysis about the partial power ratio and the conversion efficiency, is validated using an experimental test-bench, as shown in Figure 8. Due to laboratory and components availability, the voltage and power ratings have been scaled from typical applications values. It is because the available commercial HF transformers limit the selection of traditional ratings of power and turn ratios for the corresponding PV applications. However, the theoretical analysis and its validation is still respected, because they are evaluated as power ratios, voltage gains and duty cycles. For the emulation of the PV system, a programmable dc power supply Chroma 62050H-600S, with Solar Array Simulator is used. A Full-bridge based PPC is built to validate the step-down PPC configuration connected at dc-link side. The control platform is composed by: a dSPACE 1103 being the responsible to control the dcdc converters, and a FPGA Spartan-3E generating the high-frequency PWM signals. The step-down PPC performs the MPPT algorithm controlling the input voltage, while the output voltage is fixed by a dc power supply Agilent N8762A Technologies, emulating the grid-tied inverter. A resistive load, which is part of this dclink emulator, is used to dissipate the power delivered by the PV system. The voltage measurements are made using a differential voltage probe Keysight N2790A, 100MHz. A Keysight N2783B, 100 MHz probe is used for the current measurements. The parameters of the experimental test-bench are listed in the Table 1.

Operation Performance

The evaluated parameters under constant solar irradiation are the voltage and current in the system. The voltages are depicted in Figure 9a, where the P&O MPPT algorithm presents the classical three levels. Since the voltage at the dc-link vdc is fixed by the inverter, the converter voltage vpc is the difference between vpv and vdc due to the series connection of the PPC. In that case the step-down operation is depicted in the figure ,   where the output voltage Vdc = 167 V is lower than the input voltage Vpv = 230 V. The voltage of the PPC is Vpc = 63 V, which is the blocking voltage withstood by the Mosfets, and it is lower than in the case of traditional FPC. It means that the component size is reduced due to the lower voltage rating. The current results are depicted in Figure 9b, where the mean value of the PV currents is Ipv = 5.3 A and the output current Idc = 7 A.

Based on the connection, the current trough the diode bridge Ipc is the difference between the PV and output current, then Ipc = 1.7 A. This current value is lower compared with a traditional H-bridge based FPC. In order to evaluate the performance of the MPPT algorithm, an irradiation reduction to the 67% of the rated value is made. The result is also depicted in Figure 10, where it is possible to see the current reduction when the change is made. Moreover, due to the implemented MPPT algorithm, the PV voltage start decreasing the value in order to find the maximum power point. The stationary value is around Vpv = 225 V.

Efficiency Analysis

The efficiency is calculated using the experimental results of voltage and current, which are taken from the complete dc-stage (PPC), and the isolated Full-bridge dc-dc converter. The experimental efficiency waveforms are depicted in Figure 11. The global system efficiency ηdcs, is obtained under different power and partial power ratios Kpr. These results are based on the measurements at the output side (Io,Vdc) and the input side (Ipv,Vpv). When the converter is operating below a partiality factor Kpr ≤ 25% of the total power, the dcstage efficiency varies between 90% ∼ 99%, as depicted in Figure 11a. However, the conversion efficiency of the dc-dc converter varies between 72% ∼ 93%, as shown in Figure 11b. The experimental operating points and results, for one of the conducted experiments, are listed in the Table 2. Dc-dc efficiency ηdcc (%)

78.9

From the results, the rated PV power is Ppv = 822 W, but the converter is only handling a Kpr = 13.4% of the total power. In this case, the voltage gain of the system is Gv = 0.84. However, due to the series connection, the PPC voltage is Vpc = 29 V, which means that the isolated converter has a voltage gain GvDC = 0.19. This deeper voltage reduction impacts on the efficiency reached by the full-bridge converter itself ηdcc, which is relatively lower than the global system efficiency ηdcs.

Based on these measurements, it is possible to verify the results using [START_REF] Huawei | Smart String Inverter SUN[END_REF]. Where the measured efficiency of the dc-stage is ηdcs = 97.5%, and the theoretical efficiency is ηdcs = 97.2%. Which gives an error around ≈ 0.3%, validating the hypothesis of the relationship between the increment of the dc-stage efficiency, when the partiality ratio kpr decreases.

It is worth to clarify that in terms of efficiency for the power rating in which they operate, the laboratory prototypes were not optimized in their design. Consequently, it is expected that for industrial developments, which are mainly optimized for a desired power rating, the efficiency results will be improved for the dc-dc converter stages, further improving the global system efficiency.

Partial Power Ratio

The ratio of power processed by the converter, is evaluated taking the measurements of the voltage and current at the Mosfet bridge. Then the mean values of the measurements are multiplied, in order to obtain the converter power Ppc. After that, this value is divided by the input power Ppv obtaining the partial power ratio Kpr.

The experimental result for the partial power ratio Kpr is depicted in Figure 12. The experimental results are shown in the figure with points, and the theoretical value is shown with a continuous line. It can be noticed the linear reduction of Kpr when the voltage gain increases. The behavior fits well with the theoretical expression [START_REF] Button | An advanced photovoltaic array regulator module[END_REF]. Considering that the PV application is sized to a higher value compared with the dc-link voltage, the input voltage vpv will decrease in case of a reduction of the solar irradiation. It means, that the converter will handle lower power in case of partial shading.

Voltage Range of Operation

Finally, in order to obtain the PV voltage variation at different irradiation values, the explicit PV voltage expression is implemented using Lambert's function [START_REF] Morrison | Partial power DC-DC converter for photovoltaic two-stage string inverters[END_REF], and working with a PV module Sunmodule SW 250.

In this evaluation case, it was emulated a PV string made with 7 PV modules, reaching around a Vmpp ≈ 224 V.

From the results obtained by the experiment made in the laboratory, the variation of the PV voltage was ∆Vpv = 48 V. On the other hand, using (9) it was obtained that the PV voltage variation, when solar irradiation changes from (1000-500 W/m 2 ), corresponds to ∆Vpv = 46.86 V and the minimum Kpr = 9.1%. With this information it is possible to conclude that working with the laboratory prototype, a reduction of the 50% of the STC solar irradiation is allowed to ensure the partial power conversion.

Comparison between a Step-Down PPC and a Full Bridge Power Converter

In order to make a comparison between a full power converter vs. a partial power converter, a simulation is carried out for an unique operation point. In order to simplify the analysis, the magnetic losses are neglected and the focus is only in terms of the Mosfet and the diode bridge losses. A simulation is carried out for the step-down PPC depicted in the Figure 6a, and a traditional isolated H-bridge converter for the FPC. In order to work around the same operation point, the duty cycle is imposed for both cases D = 0.2155, the switching frequency is fsw = 80 kHz and the current ripple in the inductor is imposed to ∆I = 10%I. Finally, the PPC is sized to work with a partiality ratio Kpr = 14.1%, and the simulation results are listed in the Table 3. As can be seen in the Table 3, the ratings of the power components are different between both configurations. The Mosfet voltage rate of the FPC is much more higher than the rate of the PPC. It means that the component size will be higher, because it must withstand a higher voltage. Moreover, the power losses PMosfet,l on the FPC will increase, as can be seen in the Table 3. It is because the converter is switching at high frequency, with a higher power rating. The losses results are shown in percentage value, so that it represents the power losses with respect to the nominal power Ppv.

In the diode bridge it is possible to see a similar behavior, so that in this case the rated current increases for the FPC. Finally, the total conversion efficiency ηdcs in a PPC is higher than the efficiency in a FPC, which highlights the advantage working with a partial power convert.

Conclusions

This work presents a step-down PPC for a PV string system, analyzing the range of the partial power operation when an isolated Full-bridge topology is selected to make the PPC configuration. The results show that the converter is capable of retaining the MPPT performance, despite of the reduced amount of power processed by itself. It was evaluated using an experimental prototype, and it was contrasted with the explicit model of the PV cell.

This work shows that the partiality ratio is related to the voltage difference between the PV voltage and the dc-link voltage. Besides, this partiality ratio combined with the turn ratio of the transformer and the selected topology, all of them impact the system efficiency. Moreover, the relationship between the voltage gain and the partial power ratio has been experimentally obtained. It clarifies the benefit working with large PV strings, so that in case of the reduction of the solar irradiation, the PV voltage will decrease reaching a voltage gain close to the unity, which is translated into a lower partial power ratio and a higher efficiency.

The partial power processing technique allows a reduction of the converter volume, also increasing the power density as was analyzed based on the theoretical model. It does not depend of the converter topology, and compared with a traditional FPC, the cost could decrease because of the smaller rated semiconductors and smaller heat-sinks.

Figure 1 .

 1 Figure 1. Diagram of the power flow in a PV system: (a) With a full power converter (FPC). (b) With a partial power converter (PPC).

Figure 3 .

 3 Figure 3. Dc-stage efficiency ηdcs in a PPC in terms of the partial power ratio Kpr.

Figure 4 .

 4 Figure 4. Volume of the dc-dc converter. (a) Working with a full power converter; (b) Working with a partial power converter.

  sh RshIsat e RsIpv (9) Vpv = Vt ph Isat +Isat-Ipv) -RVsht (I pv VtVt Moreover, in Figure 5 it is possible to see the PV voltage reduction under the reduction of the solar irradiation, in this case a variation between 50-1000 (W/m 2 ) is made. Under this condition, the obtained results are a PV voltage Vmp reduction of the 20.9%. This result is required as a limitation for the design of the PPC converter, so that it shows the minimum voltage variation needed to operate as a PPC without affecting system performance.

Figure 5 .

 5 Figure 5. PV voltage variation in terms of the PV current variation under different solar irradiation.

Figure 6 .

 6 Figure 6. Full-bridge isolated dc-dc topology for the proposed: (a) Step-down PPC connected at dc-link side; (b) Step-down PPC connected at PV-side.

Figure 7 .

 7 Figure 7. Operation of the step-down PPC in terms of the transformer turns ratio nT and the voltage gain Gv. (a) Step-down PPC dc-link side connected; (b) Step-down PPC PV-side connected.

Figure 8 .

 8 Figure 8. Experimental test-bench of the step-down PPC connected at dc-link side, using a full-bridge topology for string inverter applications.

Figure 9 .

 9 Figure 9. Experimental results under constant solar irradiation. (a) Voltage waveforms at the PV side, dc-link side and converter voltage; (b) Input and output current waveforms.

Figure 10 .

 10 Figure 10. Experimental results under a solar irradiation reduction. (a) Voltage and current waveforms; (b) Current waveforms in the step-down Full-bridge based PPC.

Figure 11 .

 11 Figure 11. Experimental efficiency curves for: (a) Step-down Full-bridge based PPC, for string inverter; (b) Isolated dc-dc full-bridge converter.

Figure 12 .

 12 Figure 12. Experimental result of the partial power ratio for the step-down Full-bridge based PPC under the variation of the voltage gain.

Table 1 .

 1 Parameters of the step-down PPC with a full-bridge topology.

	Parameter	Symbol	Value
	PV power	Ppv	1.2 kW
	PV voltage	Vpv	230 V
	PV side capacitance	Cpv	330 µF
	Clamping capacitance	Cc	47 µF
	Dc side capacitance	Cdc	330 µF

Table 2 .

 2 Experimental results obtained for a particular evaluation point.

	Parameters	Step-Down	Parameters	Step-Down
	System Power (W)	822	Converter voltage (V)	29
	Dc-dc Power (W)	110.1	Global Voltage gain Gv	0.84
	Kpr(%)	13.4	Dc-dc Voltage gain GvDC	0.19
	PV voltage (V)	182	Global efficiency ηdcs (%)	97.5
	Dc-link voltage (V)	153

Table 3 .

 3 Simulation results for a step-down PPC and an isolated H-bridge FPC.

	Parameters	Step-Down PPC	Step-Down FPC	Parameters	Step-Down PPC	Step-Down FPC
	Ppv	800 W	800 W	V ˆMOSFET	25 V	177 V
	Vpv	177 V	177 V	IrmsMOSFET	4.13 A	3.368 A
	Vpc	25 V	-	V ˆDiode	200 V	177 V
	Vdc	152 V	152 V	IrmsDiode	0.51 A	3.4 A
	Iin	4.52 A	4.52 A	PMosfet,l	0.67%	7.05%
	Ipc	0.74 A	-	Pdiode,l	0.17%	1.24%
	Io	5.26 A	5.26 A	ηdcs	99.16%	91.7%

Funding: This research was funded by CONICYT-PCHA/Doctorado Nacional/2016-21160863, by FONDECYT 1171823, AC3E (CONICYT/Basal/FB0008), SERC Chile (CONICYT/FONDAP/15110019), and CONICYT REDES 170217.

Author Contributions: J.W.Z. conceived, designed and performed the experimental evaluations; S.K. provided insight on the problem definition and conceptual solution, and was responsible of supervision throughout the work; G.C. was part of the development team of the experimental test-bench, and T.A.M. was responsible for guidance and a number of key suggestions.

Conflicts of Interest:

The authors declare no conflict of interest.


Andrija Stupar
email: andrija@stupar.com

Didier Flumian

Basile Gouedard

Thierry Meynard´

Generation and Derivation of Practical Optimization-Oriented Models of Inductors

Magnetic components contribute significantly to the volume and losses of power electronic systems. Their optimal design is therefore crucial for the overall optimization of power density and efficiency of power converters. Recently it has been shown that converters can accurately be modeled using posynomial functions, thus allowing for the use of Geometric Programming, a type of convex optimization problem, to be used to quickly produce globally optimum designs of entire converter families. Existing studies have however treated magnetic components in a case-specific way, deriving posynomial models suited for the given converter or range of operating points. This paper demonstrates and validates methods for the derivation and generation of posynomial models of inductors for an entire catalogue of standard components, allowing a set of models to be generated once and then re-used subsequently, or "plugged into" an overall converter optimization, repeatedly, without the need for re-derivation.

I. INTRODUCTION

Recently it has been shown [START_REF] Ribes-Mallada | Optimization of DC-DC´ Converters via Geometric Programming[END_REF], [START_REF] Stupar | Multi-´ objective optimization and comparison of multi-level DC-DC converters using convex optimization methods[END_REF], [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF], [START_REF] Stupar | Efficiency Optimization of a 7-Switch Flying Capacitor Buck Converter Power Stage IC Using Simulation and Geometric Programming[END_REF], [START_REF] Stupar | Multi-´ Objective Optimization of Multi-Level DC-DC Converters using Geometric Programming[END_REF], that geometric programming [START_REF] Boyd | A tutorial on geometric programming[END_REF], a class of convex optimization problems, can be used to accurately and quickly optimize power converters for multiple objectives. There are two main advantages of using this approach. First, in a feasible convex optimization problem, a local minimum is guaranteed to be a global minimum. Second, many efficient solvers for convex optimization problems exist: in [START_REF] Stupar | Multi-´ objective optimization and comparison of multi-level DC-DC converters using convex optimization methods[END_REF], [START_REF] Stupar | Multi-´ Objective Optimization of Multi-Level DC-DC Converters using Geometric Programming[END_REF] it was shown that complex converter topologies can be optimized for efficiency and volume in under one minute. A Geometric Program (GP) has the following form minimize [START_REF] Ribes-Mallada | Optimization of DC-DC´ Converters via Geometric Programming[END_REF] subject to where x = (x1,x2,...,xn) is the vector of design (input) variables, fo(x) is the objective function to be minimized, and fi(x) and hj(x) are the inequality and equality constraints, respectively, that must be satisfied by the solution. In order for fo(x), fi(x) and hj(x) to be convex functions, x must be a vector of positive non-zero real numbers, as must be the coefficients (ao,ai,aj), while the exponents (ko,ki,kj) can be any real number. Such functions are called posynomials ("positive polynomials") [START_REF] Boyd | A tutorial on geometric programming[END_REF].

Therefore, in order to model a power converter as a GP, each component in the system must be modeled using posynomial functions. As shown in [START_REF] Stupar | Multi-´ objective optimization and comparison of multi-level DC-DC converters using convex optimization methods[END_REF], [START_REF] Stupar | Efficiency Optimization of a 7-Switch Flying Capacitor Buck Converter Power Stage IC Using Simulation and Geometric Programming[END_REF], [START_REF] Stupar | Multi-´ Objective Optimization of Multi-Level DC-DC Converters using Geometric Programming[END_REF], semiconductor models are often already in posynomial form or can be manipulated to be in posynomial form in a relatively straightforward manner. Accurate models of magnetic models, such as those in [START_REF] Muhlethaler | ¨ Modeling and Multi-Objective Optimization of Inductive Power Components[END_REF], [START_REF] Muhlethaler | Improved Core-¨ Loss Calculation for Magnetic Components Employed in Power Electronic Systems[END_REF], are more difficult to put into posynomial form. In [START_REF] Ribes-Mallada | Optimization of DC-DC´ Converters via Geometric Programming[END_REF], [START_REF] Stupar | Efficiency Optimization of a 7-Switch Flying Capacitor Buck Converter Power Stage IC Using Simulation and Geometric Programming[END_REF], comprehensive posynomial models of inductors were not considered. In [START_REF] Stupar | Multi-´ objective optimization and comparison of multi-level DC-DC converters using convex optimization methods[END_REF], [START_REF] Stupar | Multi-´ Objective Optimization of Multi-Level DC-DC Converters using Geometric Programming[END_REF], low-power SMD inductors were considered and modeled in an application-specific way, with a posynomial model that is appropriate to the converters considered for optimization within the operating range envisioned for the studied application. Therefore, these models cannot be easily transferred to a different converter optimization problem, and in most cases, the models would need to be re-derived. In [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF], it was demonstrated that deriving posynomial models of nonSMD custom-built inductors (constructed from standard cores and wires), such as those shown in Fig. 1 was feasible, and a model derivation procedure, based on [START_REF] Hoburg | Data Fitting with GeometricProgramming-Compatible Softmax Functions[END_REF], was presented. However, as [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF] was more general in nature and considered other approaches to model derivation as well, the presented example was only a limited proof of concept that also is not easily transferrable to other applications.

This goal of this paper therefore is to present a method for the derivation of re-usable, "generate-once use-multiple-times" posynomial models for higher-power, custom-built inductors made from standard cores, wires, and bobbins. Such a method allows, for a given catalogue of standard parts (cores of a single type of geometry and from the same magnetic material, and wires of standard gauge from the same conductor material), a set of posynomial functions modeling inductance, peak flux density, total losses, boxed volume, and core and winding temperature to be generated over the entire set of feasible inductors that can be constructed with the considered standard parts. These models can then be reused repeatedly within different converter optimization problems.

In Section II it is shown how such a goal may be attempted to be achieved using analytical equations of inductor geometry based on [START_REF] Forest | Analytic Design Method Based on Homothetic Shape of Magnetic Cores for HighFrequency Transformers[END_REF]. While useful results may be obtained with this approach, it is difficult to accurately model winding losses in gapped cores over a wide range of core geometries and winding arrangements in this manner. Therefore, in Section III, an approach which is a significant extension of the method from [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF], using simulation data from a dedicated power electronics inductor simulation tool, is derived and demonstrated. The accuracy of the simulation tool, and therefore its suitability for use as a source of data for the derivation of posynomial models, is confirmed using experimental measurements. The accuracy of the derived models is evaluated using a set of sample optimization problems. The paper is concluded with Section IV.

II. MODEL DERIVATION BASED ON ANALYTICAL EQUATIONS

Starting from the well-known concept of area product (AeWa -where Ae is the core cross-sectional area and Wa the core winding window area), in [START_REF] Forest | Analytic Design Method Based on Homothetic Shape of Magnetic Cores for HighFrequency Transformers[END_REF], [START_REF] Forest | Design and Com-´ parison of Inductors and Intercell Transformers for Filtering of PWM Inverter Output[END_REF], functions describing several different core shapes are derived, with derived geometry-specific correction factors. The resulting models are used in [START_REF] Forest | Analytic Design Method Based on Homothetic Shape of Magnetic Cores for HighFrequency Transformers[END_REF] in an iterative transformer design procedure, and in [START_REF] Forest | Design and Com-´ parison of Inductors and Intercell Transformers for Filtering of PWM Inverter Output[END_REF] in an iterative inductor design procedure. Most models presented in [START_REF] Forest | Analytic Design Method Based on Homothetic Shape of Magnetic Cores for HighFrequency Transformers[END_REF], [START_REF] Forest | Design and Com-´ parison of Inductors and Intercell Transformers for Filtering of PWM Inverter Output[END_REF] are in fact in, or close to, posynomial form. Key quantities of these models are summarized in Table I. Each considered core shape is characterized such that the core lengths are proportional to , the core areas to , and the core volumes and weights to . Additionaly, a thermal exchange coefficient Htherm is specified as a constant value to be typical either of natural convection, forced convection, or liquid cooling. For details on the expressions for and values of the quantities in Table I, the reader is directed to [START_REF] Forest | Analytic Design Method Based on Homothetic Shape of Magnetic Cores for HighFrequency Transformers[END_REF], [START_REF] Forest | Design and Com-´ parison of Inductors and Intercell Transformers for Filtering of PWM Inverter Output[END_REF].

Therefore, a GP can be formulated where the main free variable is the size of the core, i.e. AeWa, along with the induction at nominal current BDC and the current density at subject to where L is the desired inductance. The first constraint in (2) is the inductance requirement, the second constraint is the saturation limit, and the third the thermal limit, i.e. the total amount of allowed losses. Therefore, (2) minimizes the area product of an inductor built from a specified core material and core shape, for a given upper loss limit and desired inductance. The objective function in (2) could be changed from AeWa to the AeWa-dependent function for losses, weight or volume given in [START_REF] Forest | Analytic Design Method Based on Homothetic Shape of Magnetic Cores for HighFrequency Transformers[END_REF], [START_REF] Forest | Design and Com-´ parison of Inductors and Intercell Transformers for Filtering of PWM Inverter Output[END_REF], or to a weighted sum of two or more of these functions, giving a multiobjective problem.

A loss-volume Pareto front can be constructed by repeatedly solving (2) for different values of Htherm, as shown in Fig. 2. This immediately shows the feasible (theoretically realizable) and non-feasible regions of the inductor design space, showing the minimum volume achievable for different types of cooling. The same can be repeated with different core materials and at different frequencies (as in Fig. 3) or with different core shapes, etc. Since (2) can be solved in a few seconds, it is a powerful tool for the exploration of large design spaces, and can be used to quickly identify which cooling methods, core shapes, core materials, and so on, are appropriate to a particular application or design task.

Unfortunately, this approach is limited when applied to gapped cores. As shown in [START_REF] Muhlethaler | ¨ Modeling and Multi-Objective Optimization of Inductive Power Components[END_REF], a general closed-form analytical solution does not exist for proximity losses due to an air gap's fringing flux, and at least a 2-D summative calculation method must be used instead. The reluctance of an air gap is Fig. 3.

Pareto fronts for a given L for different core material and fs pairs. Fig. 4. The general, structured, successive fitting procedure for deriving posynomial models from data characterizing a family of inductors [START_REF] Hoburg | Data Fitting with GeometricProgramming-Compatible Softmax Functions[END_REF], [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF]. also difficult to encapsulate in a simple analytical equation. Therefore an extension of the approach in [START_REF] Forest | Analytic Design Method Based on Homothetic Shape of Magnetic Cores for HighFrequency Transformers[END_REF], [START_REF] Forest | Design and Com-´ parison of Inductors and Intercell Transformers for Filtering of PWM Inverter Output[END_REF] was developed in [START_REF] Furlan | Homothetic Method to Compute Winding Losses in the Design of Power Inductors[END_REF], using finite element method (FEM) simulations to derive correction factors for ρ that include the effects the air gap on the winding losses. Such correction factors are, however, extremely geometry-dependent and require a large number of FEM simulations to refine. Considering the complexity of the problem, it is simpler to then characterize the entire inductor using a proven, comprehensive power electronics-oriented simulation tool. This different approach is described in detail in the following Section.

III. SIMULATION DATA-BASED MODEL GENERATION

Following [START_REF] Hoburg | Data Fitting with GeometricProgramming-Compatible Softmax Functions[END_REF], [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF] presents a procedure, shown in Fig. 4, to fit posynomial functions to data that has been generated by simulations, measurement, or from the evaluation of nonposynomial analytical models. First, the data is partitioned. The number of partitions determines the number of terms in the fitted posynomial functions. Then, to each partition, a monomial function, shown in [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF], is fitted separately. The results of the fits for each partition are used as an initial guess for fitting a max-monomial function, shown in (4), to the entire data set. The results of the max-monomial fit are then used as an initial guess for fitting the final posynomial function, shown in [START_REF] Stupar | Multi-´ Objective Optimization of Multi-Level DC-DC Converters using Geometric Programming[END_REF], to the data.

h(x) = ealx1kl1...xnkln (3) (4) m n f(x) = eal xikli (5) l=1 i=1
It should be noted that monomial and posynomial functions must be converted to their logarithmic form in order to become convex. Taking the logarithm of both sides in (3) transforms the monomial into a linear (affine) function, the max-monomial in (4) into a max-affine function, and the posynomial in (5) into a softmax-affine function. In the procedure of Fig. 4, the logarithm of the data to be fitted is also taken and fitting is performed to the logarithmic forms of the functions. For more details, the reader is directed to [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF] and [START_REF] Hoburg | Data Fitting with GeometricProgramming-Compatible Softmax Functions[END_REF].

In this section, the method presented in [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF] is extended to generate posynomial models covering an entire family of potential inductor components, that can be built from cores of the same shape and material but different dimensions, and a standard range of copper wires.

A. Experimental Verification

To generate data for fitting, the magnetic simulation software tool GeckoMAGNETICS [START_REF]GeckoMAGNETICS modeling and design tool for magnetic components for power electronics[END_REF], version 1.5.0, was selected. It is based on advanced, published models that have been experimentally verified [START_REF] Muhlethaler | ¨ Modeling and Multi-Objective Optimization of Inductive Power Components[END_REF], [START_REF] Muhlethaler | Improved Core-¨ Loss Calculation for Magnetic Components Employed in Power Electronic Systems[END_REF], [START_REF] Muhlethaler | ¨ Measurement Results (Losses and Temperature) -GeckoMAGNETICS[END_REF], [START_REF] Muhlethaler¨ | Inductance Measurement Results -GeckoMAGNETICS[END_REF]. It contains measurements of core losses with rectangular voltage excitations and DC bias current, allowing for accurate core loss and temperature calculations, and a detailed air gap reluctance model allowing for accurate inductance, winding loss and temperature calculations. To confirm the accuracy of the version of the software used, a measurement setup, shown in Fig. 5, was constructed. The inductor being measured is the device under test (DUT) in Fig. 5 and it is placed between two switching cells. Cell 2 operates in open loop and generates a quasi-constant voltage Vf across the LC filter. Cell 1, switching at switching period Ts, generates a square wave voltage vout ranging from 0 to Vin. It operates in closed loop in order to ensure a given current IDC through the DUT. Therefore by adjusting the operation of the two cells, a current waveform with a peak to peak ripple ΔiL, as shown in Fig. 6 can be imposed on the DUT. In this way the measurement setup can mimic, from the point of view of the DUT, the behaviour of several different buck-and boost-type converter topologies.

The loss measurement was performed in a manner similar to that presented in [START_REF] Muhlethaler | ¨ Modeling and Multi-Objective Optimization of Inductive Power Components[END_REF], [START_REF] Muhlethaler | Improved Core-¨ Loss Calculation for Magnetic Components Employed in Power Electronic Systems[END_REF]. Total DUT losses were measured using a Yokogawa WT3000 Precision Power Analyzer. The total power Ptotal is given as [START_REF] Boyd | A tutorial on geometric programming[END_REF] and was measured across the main inductor winding N1. Additionaly, a second sense winding N2 was added in order to measure core losses PCore only, given as (7) where v2 is the voltage across N2. Winding losses were determined by subtracting PCore from PTotal, and additionally by measuring the DC and AC resistance of the winding using a Hioki IM3533 LCR meter. Measurements were performed on Inductor 1, the characteristics of which are shown in Fig. 7 and the built prototype in Fig. 1(a), and Inductor 2, the characteristics of which are shown in Fig. 8 and the built prototype in Fig. 1

(b). A selection of the performed measurements and their comparison to

GeckoMAGNETICS simulations for the two inductors is shown in Table II and Table III, respectively. It can be seen that the agreement between measurements and simulations is very good. Therefore, GeckoMAGNETICS is an appropriate tool to use to generate data for the derivation of posynomial inductor models. In order to generate posynomial models that can be used over a wide variety of applications, the entire catalogue of TDK EPCOS EE cores (ranging from E13 to E55) made from Fig. 7. Specifications of the Inductor 1 DUT, built from an EPCOS E55 core and N27 ferrite material. Fig. 8. Specifications of the Inductor 2 DUT, built from an EPCOS E32 core and N87 ferrite material. the N87 ferrite was selected for simulation. The cores were not stacked, and only cores with a built-in centre air gap, ranging from 0.04 mm to 2 mm, were selected. To construct the windings, standard American wire gauge (AWG) solid copper wires were used, ranging from AWG10 to AWG44. The dimensions shown in Fig. 9 and Table IV were used to characterize an inductor.

An algorithm was designed to generate a set of feasible inductors over the entire range of the parts catalogue. A maximum allowable current density of Jmax = 4 A/mm 2 was assumed. Fill factor was kept between 0.1 and 0.6. For each core (including air gap), the core reluctance Rtot was calculated using GeckoMAGNETICS. Assuming first N = 1, and using the core material's saturation flux density Bsat, the peak allowable current for the inductor was calculated using .

(8) From this, the smallest wire diameter dmin that can accommodate Ipk,max while respecting the limit on Jmax was calculated as

. ( 9
)
To construct the inductor, the next largest AWG wire was selected. A series of inductors was then constructed with the same number of turns but using every AWG wire size that respected the limits on the fill factor. When the number of possible designs was exhausted, N was incremented to 2 and then the entire procedure was repeated, and so on, until N = 50. For each core in the catalogue, this then created the entire set of feasible designs that could be constructed with AWG solid copper wires with up to 50 turns. At the first attempt this generated a very large number of very similar inductor designs. In order to reduce the amount of designs to simulate, designs using adjacent AWG wire sizes were discarded (for example, designs using AWG10, AWG12, and AWG14 were selected for simulation, but not those with AWG11 and AWG13). This was acceptable since the difference between adjacent AWG wire dimeters is only about 11%. Similarly, an additional turn produces a much larger change inductance when added to a low number of turns than when added to a large number of turns. For this reason, the number of turns was incremented by 1 until N = 10, then by 2 until N = 20, and by 5 thereafter. This finally produced a total of 880 inductor designs, with inductance L ranging from 1 µH to 4 mH. The generated designs therefore spanned the entire set of applications appropriate to this catalogue of cores.

For all simulations, an ambient temperature of 25°C was assumed. Each inductor was simulated using triangular current waveforms, such as those shown in Fig. 6, with 50% duty ratio and different average current IDC and peak-to-peak current ripple ΔiL. For each inductor, IDC was selected to be 0.1, 0.3, 0.6, and 0.8 of the Ipk,max for that inductor. ΔiL was then selected to be 0.1, 0.2, 0.4, 0.6, and 0.9 of the selected values of IDC. Waveform frequencies fs were selected to be 20, 50, 100, and 200 kHz, as that was the range within which the GeckoMAGNETICS tool contained accurate core loss data. Taking together all of the generated inductors, this produced 63360 valid simulations. For each simulated inductor, besides the geometry characteristics shown in Fig. 9 and the previously detailed waveform characteristics, the peak flux density Bpk, total losses PTotal, core temperature Tcore, winding temperature Twind, and boxed volume V olbox were recorded. The simulations took about 15 hours to complete on a low to midrange laptop computer using a single thread of execution.

C. Posynomial Fitting

Following the approach of [START_REF] Stupar | Posynomial Models of Inductors for´ Optimization of Power Electronic Systems by Geometric Programming[END_REF], data was divided into four partitions based on waveform frequency fs. Each partition had the same frequency value. The successive fitting method was used to produce a four-term posynomial for the total losses Ptotal (in mW) and the core and winding temperature Tcore and Twind, respectively (in °C): • Nklw7gklw8fsklw9ΔiLklw10IDCklw11) [START_REF] Furlan | Homothetic Method to Compute Winding Losses in the Design of Power Inductors[END_REF] The exponent values for ( 10) -( 12) are shown in Table V.

4 PTotal =(ealpAeklp1leklp2Wauklp3dklp4sklp5MLTklp6Nklp7 lp=1 • gklp8fsklp9ΔiLklp10IDCklp11) ( 10
)
However, for the inductance L (µH), peak flux density Bpk (mT), and boxed volume V olbox (cm 3 ) it was sufficient to fit just a monomial function to the entire data set:

L = e-5.82Ae0.839le0.062N2g-0.722 [START_REF]GeckoMAGNETICS modeling and design tool for magnetic components for power electronics[END_REF] Bpk = e1.083le-0.269N0.997g-0.785ΔiL1.0026IDC-0.00486

(14)
V olbox =e-22.42Ae-3.08le2.14Wau-1.28d0.528s-0.0156

• MLT8.97N0.243g-0.00294. ( 15
)
L must be a monomial in order to be included into a GP as an equality constraint. Since inductance is physically a function of the number of turns and the core geometry, a monomial function of Ae, le, g, and N can be fitted with high accuracy. Similarly, V olbox is also a function only of the inductor's dimensions, so the variables related to the applied waveform can be ommitted. The form of the function for Bpk was determined by trial and error based on the intuition that it should depend only on the inductor geometry and the peak current.

The fitting was performed using least-squares functions in MATLAB and took about 30 minutes to execute on the same computer which was used to produce the simulation data. As can be seen in Table VI, the produced posynomial functions fit quite well to the simulation data. The highest error, but still less than 10%, is present when fitting Ptot, which is understandable taking into consideration the complexity of accurate inductor loss modeling.

D. Alternative Partitioning Approaches

In addition to the approach of the previous subsection, different partitioning schemes were considered. First, the frequencysorted data was divided into two partitions, with each partition containing two adjacent values of frequency. Then the data was sorted according to Bpk ascending, and divided into four partitions. The data contained nine different values of Ae, so three core geometry-based partitions were created, each containing three adjacent values of Ae. Finally, the successive fitting approach of Fig. 4 was discarded and a posynomial with 2, 3, and 4 terms was fitted directly to the simulation data without any partitioning (the data was sorted by fs ascending). The results of the different approaches are compared in Table VII using the fitting error for the total losses. The fitting errors for Tcore and Twind follow broadly the same trend. The sorting of the data did not have any effect on the fitting results of the monomial functions, as identical outcomes were achieved for them in each case.

Interestingly, the 4-term Bpk partition produces an identical error as the 4-term fs partition and the 4-term non-partitioned data, although with different exponent values. The same relationship holds between the 2-term fs partition and 2-term nonpartitioned results, as well as between the 3-term Ae partition and the 3-term non-partitioned results. This suggests that the number of terms, and not the sorting and the partioning of the data, is most important, with a higher number of terms producing a better fit.

E. Optimization Example

To demonstrate the use of the derived posynomial models, an inductor with L = 100 µH, operating at fs = 36 kHz, ΔiL = 1.67 A, and IDC = 3 A was optimized for losses and volume. Core and winding temperature were limited to be at most 100°C. As in [START_REF] Stupar | Multi-´ Objective Optimization of Multi-Level DC-DC Converters using Geometric Programming[END_REF], the two conflicting objectives were weighted by a factor γ and added into a single objective function. Since loss and volume are calculated in different units, they were normalized to a range between 0 and 1. To achieve this, 9.37% 0.8661 first the volume was minimized in a single-objective problem, and due to the inverse relationship of the volume and losses of Pareto-optimal solutions [START_REF] Biela | Optimal Design of a¨ 5kW/dm3 / 98.5% Efficient TCM Resonant Transition Single-Phase PFC Rectifier[END_REF], the losses of the minimum volume design, Ptot,max, were used to normalize the losses in the multiobjective problem. Similarly, loss was minimized in a single-objective problem as well, and the volume of the minimum-loss solution, V olbox,max, was used to normalize the volume in the multi-objective problem. This gave the final multi-objective GP formulation: Ae = 0.0006963(le 2.683 ), where the design variables -the geometric parameters -were constrained to be within the minimum and maximum bounds found in the generated simulation data. Additional constraints were added to keep the current density and fill factor of the optimized designs within the limits of the simulated data as well. This however was not sufficient to force the GP to produce inductor designs using the desired core catalogue, as in the absence of further constraints, Ae, le, Wau, and MLT are treated as independent variables by the GP solver, potentially producing unrealizable core geometries. Since the winding is wound around the centre core leg in this case, MLT depends on Ae, which is defined by the centre leg. Similarly, le depends on the size of Wau. Therefore two monomial expressions, giving MLT as a function of Ae, and le as a function of Wau, were derived by fitting to the simulation data, and added to the GP as equality constraints. A third monomial constraint relating Ae to le was derived in the same manner in order to force the GP to produce cores with roughly the same proportions as those available in the actual core catalogue.

The GP [START_REF] Biela | Optimal Design of a¨ 5kW/dm3 / 98.5% Efficient TCM Resonant Transition Single-Phase PFC Rectifier[END_REF] was solved in MATLAB using CVX [START_REF] Grant | CVX: Matlab Software for Disciplined Convex Programming[END_REF] and the SeDuMi solver. 21 GPs total were solved, with each corresponding to a value of γ ranging from 0 to 1, incremented by 0.05. Execution time was under one minute. The resulting loss-volume Pareto front is shown in Fig. 10. Three points are indicated on the Pareto front, γ = 0, corresponding to the most compact (minimum volume) design, γ = 1, corresponding to the most efficient (minimum loss) design, and γ = 0.5, corresponding to the halfway compromise between the two. Based on the characteristics of these designs, realizable inductors, shown in Table VIII, were built using the cores and wires of the initially simulated catalogue, and then re-simulated in GeckoMAGNETICS. The results of those simulations are also shown in Fig. 10. Note that the plotted values for the realizable designs are directly from GeckoMAGNETICS simulations, and not from the posynomial models, therefore the difference between them and the optimized curve includes the model error from Table VI. The total losses and boxed volumes given by the GP results, the realizable inductors evaluated using the fitted posynomial functions, and the GeckoMAGNETICS simulations are compared in Table IX. It can be seen that for γ = 0.5 and γ = 1.0, the realizable designs are extremely close to the theoretical optima resulting from the GP. For γ = 0, the volume of the realizable design is very close to the theoretical one, which is most important in this case, as this is the minimum volume design. However, there is a large discrepancy in the losses, with the losses of the realizable design being significantly lower than that of the theoretical design. As can be seen in Table IX, this large discrepancy shows up between the posynomial loss model's calculation and the GeckoMAGNETICS simulation. This is due to a larger fitting error in the area of the data around the selected core. The error reported in Table VI is an average value and there are sections of the data where it is both significantly larger and smaller than the average.

F. Future Improvements

As noted, the error distribution of the fitted functions is not uniform throughout the data set. Therefore, it would be desireable to add a third metric evaluating the goodness of the fit with respect to the error distribution, and to compare the different fits in Table VII accordingly. Also, all of the presented models were derived from simulations using a fixed ambient temperature, and as a next step the ambient temperature should be added to the posynomial functions for PTotal, Tcore, and Twind.

Finally, as noted, all of the simulations were performed using waveforms with a 50% duty ratio. Preliminary investigations performed on the results of the optimization example show that changing the duty ratio changes the loss simulation results noticeably but not significantly. It would therefore be necessary to rigorously quantify the effect of the duty ratio and determine whether is necessary to include it as a variable in the above-mentioned posynomial functions.

IV. CONCLUSIONS

Geometric programming, which operates on functions in posynomial form, is a powerful framework for the multiobjective optimization of power electronics. Previous studies, while showing that converters can be modeled as GPs, have not produced "portable" models of inductors, meaning that a re-derivation of posynomial inductor models was required for each new optimization problem. In this paper, an approach has been presented to mitigate this shortcoming, allowing comprehensive posynomial models of inductors to be generated once from simulation data and re-used in different applications.

Fig. 1 .

 1 Fig.1. Photographs of the built Inductor 1 (EPCOS E55 core, N27 ferrite, 18 turns) and Inductor 2 (EPCOS E32 core, N87 ferrite, 18 turns) used for loss measurements.

Fig. 2 .

 2 Fig. 2. A Pareto front of optimum inductor designs for a given L, fs, core material and shape for different values of Htherm. nominal current JDC: minimize AeWa (2)

Fig. 5 .

 5 Fig. 5. Schematic of the measurement setup used to measure inductor losses to verify the accuracy of the GeckoMAGNETICS software.

Fig. 6 .

 6 Fig.6. The waveform imposed on the DUT by the measurement setup of Fig.5, with a duty ratio of D and a switching frequency fs =1/Ts.

Fig. 9 .

 9 Fig. 9. The physical dimensions used to characterize an inductor.

5 0. 1 ≤

 51 minimize (16) subject to L = e-5.82Ae0.839le0.062N2g-0.722 Tcore ≤ 100• C Twind ≤ 100• C Bpk ≤ Bsat Ae,min ≤ Ae ≤ Ae,max le,min ≤ le ≤ le,max Wau,min ≤ Wau ≤ Wau,max dmin ≤ d ≤ dmax smin ≤ s ≤ smax MLTmin ≤ MLT ≤ MLTmax Nmin ≤ N ≤ Nmax gmin ≤ g ≤ gmax d ≥ 2(((IDC + 0.5ΔiL)/Jmax)/π) 0.Nπ((0.5d) 2 )/Wau ≤ 0.6 MLT = 5.793Ae0.4723 le = 7.196(Wau0.5184)

Fig. 10 .

 10 Fig. 10. The resulting Pareto-optimal set of inductors for the example optimization problem.

TABLE I QUANTITIES

 I

OF THE ANALYTICAL POSYNOMIAL INDUCTOR MODEL fs frequency of the applied square voltage I DC DC component of the inductor current ΔI ΔTmax maximum allowed temperature rise Kw winding factor (inverse of fill factor) Kc1,Kc2,α1,α2,β Core loss model parameters ρ winding resistivity S thermal exchange area proportion factor Z,Y winding and core volume proportion factors J RMS current density, including harmonics of current ripple

TABLE II MEASUREMENT

 II RESULTS FOR INDUCTOR 1

		Waveform			Experimental measurement			GeckoMAGNETICS simulation		Error (Total loss)
	0.2	10	0	0.45	0.28	0.73	0.483	0.336	0.818	12.05	0.088
	0.3	10	0	1.14	0.65	1.79	1.071	0.751	1.822	1.79	0.032
	0.5	5	0	1.56	1.11	2.67	1.271	0.961	2.232	16.4	0.438

ΔB pk-pk (T) fs (kHz) IDC (A) Core loss (W) Winding loss (W) Total loss (W) Core loss (W) Winding loss (W) Total loss (W) % (W)

TABLE III MEASUREMENT

 III

RESULTS FOR INDUCTOR 2

TABLE IV QUANTITIES

 IV OF THE SIMULATION-DERIVED POSYNOMIAL INDUCTOR MODEL

	fs	frequency of the inductor current
	I DC	DC component of the inductor current
	Δ
	Ae
		2)
	g	air gap length (mm)
	MLT	mean length of a turn (mm)

s wire isolation thickness (mm) Wau usable winding window area (mm 2 ) hb,wb bobbin height and width (mm)

TABLE V EXPONENT

 V VALUES FOR PTotal, Tcore, AND Twind FITS

	k11	-2.9014	66.7886	7.7489
	k12	2.1662	-12.7809	-1.9070
	k13	-1.7791	26.8207	3.3015
	k14	0.5600	-0.3203	-0.7791
	k15	0.3944	0.6059	1.3365
	k16	7.2951	-	-19.3796
			174.6128
	k17	2.5932	11.7956	1.2532
	k18	0.2473	-9.0848	-0.9883
	k19	0.6617	0.9039	0.6758
	k110	2.0082	3.4054	1.1878
	k111	-0.0435	8.3293	0.1596
	a2	171.9199	12.1514	-6.3509
	k21	52.8727	5.5936	2.5028
	k22	-6.8350	-1.3211	2.1783
	k23	20.8084	2.2491	-0.0768
	k24	-0.0674	-0.1623	1.3027
	k25	-0.0355	0.1870	-2.0040
	k26	-138.7475	-13.4502	-8.0003
	k27	11.9716	1.3202	2.0799
	k28	-9.1608	-1.1549	0.2497
	k29	1.0854	0.7037	0.4624
	k210	3.5435	1.3332	1.4654
	k211	8.3084	0.0262	0.0447
	a3	-10.3364	19.6335	2.4875
	k31	3.4004	8.6238	-0.1779
	k32	-0.0813	1.0635	0.0316
	k33	0.8497	1.8095	-0.0751
	k34	-0.0999	2.3180	0.0072
	k35	0.2385	-2.9849	-0.0039
	k36	-6.0294	-23.6927	0.4638
	k37	2.2013	2.0271	-0.0164
	k38	-1.7766	0.4142	0.0083
	k39	1.0828	0.3862	-0.0063
	k310	2.2052	1.0735	-0.0121
	k311	0.0031	0.2321	-0.0049
	a4	-6.5900	2.7913	16.1347
	k41	-0.8898	-0.1711	3.7509
	k42	-0.6872	0.0319	-0.6237
	k43	-0.3636	-0.0751	1.4514
	k44	-1.4864	-0.0105	-1.6475
	k45	-0.2330	0.0072	-0.7248
	k46	3.6423	0.4345	-10.2363
	k47	1.1712	-0.0359	0.5925
	k48	0.0047	0.0283	0.0505
	k49	0.0125	-0.0186	-0.1289
	k410	0.0615	-0.0325	-0.1782
	k4	1.9857	0.0020	1.8727
	11
		P Total	Tcore	T wind
	a1	-25.9732	248.0399	27.0296

TABLE VI ERROR

 VI OF FITTED POSYNOMIAL FUNCTIONS WITH RESPECT TO THE

		SIMULATION DATA
	Function	Average	R2
		Error
	Total losses PTotal	9.37% 0.8661
	Core temperature Tcore	2.34% 0.9507
	Twind 1.66%	0.978
		0.62% 0.9998
			L
	Bpk	3.5%	0.9965
	Boxed volume V olbox	4.5%	0.9965

TABLE VIII REALIZABLE

 VIII

				OPTIMUM INDUCTORS
	γ	0.0	0.5	1.0
	Core	E20	E32	E55
	Air gap (mm)	0.5	0.5	0.5
	Wire	AWG18 AWG14 AWG10
	N	28	19	10

TABLE IX COMPARISON

 IX OF OPTIMIZED INDUCTOR DESIGNS: GP SOLUTIONS VS. REALIZABLE INDUCTORS

	γ	0.0		0.5		1.0
		PTotal (mW)	Volbox (cm3)	PTotal (mW)	Volbox (cm3)	PTotal (mW)	Volbox (cm3)
	GP result (ideal inductor)	1328	6.522	326	17.924	167	89.38
	Realizable inductor, posynomial function evaluation	1154	5.508	338	19.943	185	91.48
	Realizable inductor, GeckoMAGNETICS simulation	619	5.807	324	19.321	143	87.93


Mokhtar Aly

Samir Kouro

Thierry A Meynard

Topology and Voltage Balance of Series-Connected T

Keywords: Level shifted pulse width modulation, mediumvoltage drives, multilevel inverter, T-type inverter

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I. INTRODUCTION

The superior performance and wider applicability of multilevel inverters (MLis) have made them favorable over conventional two-level inverters for an increasing number of applications [START_REF] Aly | Design and implementation of adaptive SVPWM algorithm for multilevel inverters in renewable energy applications[END_REF], [START_REF] Ahmed | Multifunctional distributed MPPT controller for 3p4w grid-connected PV systems in distribution network with unbalanced loads[END_REF]. The improved power quality, reduced common mode voltages, and higher efficiency represent the main benefits of utilizing MLis in medium-voltage applications [START_REF] Abhilash | A seven-level vsi with a front-end cascaded threle-evel inverter and flying-capacitor-fed h-bridge[END_REF], [START_REF] Yadav | A hybrid 7-level inverter nsing low-voltage devices and operation with single de-link[END_REF]. The power quality improves with the increase of the number of the output voltage levels. However, increasing the number of levels in the output voltage results in increasing the number of required components. This in turn can impose complexities to other system components such as input transformer design, modulation and controller, capacitor voltage balance, and even increase the system costs [START_REF] Le | A novel six-level inverter topology for mediumvoltage applications[END_REF], [START_REF] Said | Smes-based fuzzy logic approach for enhancing the reliability of microgrids equipped with pv generators[END_REF].

Several MLis topologies have found industrial application in high-power medium-voltage (HP-MV) drive systems, such as cascaded H-bridge (CHB), neutral point clamped (NPC), active NPC (ANPC), flying capacitor (FC), and modular multilevel converters (MMCs), among others [START_REF] Sahoo | Phase-shifted carrier-based synchronized sinusoidal pwm techniques for a cascaded h-bridge multilevel inverter[END_REF]- [START_REF] Gbias | Voltage-balancing method for stacked multicell converters using phase-disposition pwm[END_REF]. Moreover, there are several extensions of these basic structures to develop hybrid MLI topologies [START_REF] Le | A novel six-level inverter topology for mediumvoltage applications[END_REF], [START_REF] Ebrabimi | A new single de source six-level flying capacitor based converter with wide operating range[END_REF].

The CHB and the MMCs present a modular solutions. However, for medium-voltage applications the circulating current problems, complex control, and sensitivity to passive elements mismatch represent the main drawbacks of MMCs [START_REF] Lizana | Control of arm capacitor voltages in modular multilevel converters[END_REF], [START_REF] Liu | Lifetime estimation of mmc for offshore wind power hvdc application[END_REF]. This has hindered the applicability of MMC for medium-voltage drives applications, and has been only used in practice for high-voltage DC transmission applications. On the other hand, CHB topologies suffer from the high number of required isolated de sources based on complex zig-zag transformers [START_REF] Malinowski | A survey on cascaded multilevel inverters[END_REF], and although they have been successfully used in medium-voltage drives, the lack of a real backto-hack regenerative option (a few manufacturers offer an activefront-end based on two-level three phase voltage source inverters instead of passive diode rectifiers) have restricted their application mainly to pumps, fans and other standard mediumvoltage drives. NPC based topologies have been presented in the literature with five and seven level output voltages [START_REF] Cui | A novel hybrid voltage halance method for five-level diode-clamped converters[END_REF], [START_REF] Hotait | Capacitor voltage balancing using redundant states of space vector modulation for live-level diode clamped inverters[END_REF], [START_REF] Wang | A seven-switch fivelevel active-neutral-point-clamped converter and its optimal modulation strategy[END_REF]. However, they lack of modularity and require a high number of components due to the necessary clamping diodes, which increase the cost and reduce the efficiency of the system. Capacitor voltage balance also becomes an issue for NPCs with more than three levels when using passive front ends. Improving on the NPCs, several active NPC (ANPC) topologies have been proposed in the literature for mediumvoltage applications [START_REF] Andler | Switcbing loss analysis of 4.5-kv-5.5-ka igcts within a 31-anpc phase leg prototype[END_REF], [START_REF] Liang | Analysis and design of active npc (anpc) inverters for fault-tolerant operation of bigh-power electrical drives[END_REF]. ANPC-based topologies are successful at eliminating the clamping diodes, which improves the efficiency and reduces the cost, and provides a way to distribute thermal losses across devices eliminating the need of power derating [START_REF] Wang | Ripepl analysis and capacitor voltage balancing of live-level hybrid clamped inverter (51he) for medium-voltage applications[END_REF]. Additionally, hybrid FC and ANPC topologies have been presented in the literature to combine the benefits of FC and the ANPC topologies [START_REF] Yadav | A hybrid 7-level inverter nsing low-voltage devices and operation with single de-link[END_REF], [START_REF] Wang | Ripepl analysis and capacitor voltage balancing of live-level hybrid clamped inverter (51he) for medium-voltage applications[END_REF], [START_REF] Xu | Model predictive control for a hybrid multilevel converter with different voltage ratios[END_REF]. Although, they suffer from the high number of power switches and capacitors. In addition, they possess higher conduction losses due to the increased number of conducting switches. In [START_REF] Dekka | A series-connected multilevel converter: Topology, modeling, and control[END_REF], series connection of NPC modules has been proposed for reaching medium-voltage operation. However, this topology inherits the drawbacks of the tradditional NPC, including the clamping diodes.

It has become clear that the existing medium-voltage topologies can suffer from one or more drawbacks, including in-

ACKNOWLEDGMENT This work was supported by SERC Chile (ANID/FONDAP15110019) and by AC3E (ANID/Basal/FB0008).