1. Introduction

2. Materials and Methods

2.1. Nomenclature

$C$	DC-DC booster capacitance
$C_2$	Inverter filter capacitance $\left[F\right]$
COS	Command of Switch $[-]$
$D$	Switching duty cycle $[-]$
$f$	DC-DC booster switching frequency $\left[Hz\right]$
	Fundamental frequency for the spectral study $\left[Hz\right]$
$f_{\gamma }$	Frequency under study $\left[Hz\right]$
$i$	Indexer $[-]$
$I_D$	Diode current $\left[A\right]$
$I_s$	DC input current from the DC-DC booster or battery pack $\left[A\right]$
	Reverse leakage current $\left[A\right]$
$\overline{I}$	Average current (at zero frequency in the spectral study) $\left[A\right]$
$L$	DC-DC booster inductance $\left[H\right]$
$N$	Number of samples in the register $[-]$
$P_{inv}$	Inverter power $\left[W\right]$
$P_o$	DC-DC booster output power $\left[W\right]$
$P_s$	Input power of the DC-DC booster or supplied by battery pack $\left[W\right]$
pwm	Pulse Width Modulation $[-]$
$r$	Resistance seen from the AC side at the output of the invertir $[\Omega ]$
$R$	Resistance seen from the DC side at the output of the booster $[\Omega ]$
$R_{eq}$	Equivalent resistance with parallel connection between $R_{sw1}$, $R_{sw2}$ and $r$ $[\Omega ]$
$R_o$	Resistance $R$ seen from the DC side at the output of the invertir $[\Omega ]$
$R_{sw1}$	Non-linear resistance at the inverter switch 1 $[\Omega ]$
$R_{sw2}$	Non-linear resistance at the inverter switch 2 $[\Omega ]$
$R_{sw3}$	Non-linear resistance at the inverter switch 3 $[\Omega ]$
$R_{sw4}$	Non-linear resistance at the inverter switch 4 $[\Omega ]$
$r\%$	Percent ripple $[\%]$
$S{W}_0$	Booster switch
$S{W}_1$	Inverter switch 1
$S{W}_2$	Inverter switch 2
$S{W}_3$	Inverter switch 3
$S{W}_4$	Inverter switch 4
$T$	Period or inverse of DC-DC booster switching frequency $\left[s\right]$
	Temperature $\left[°C\right]$
$T_f$	Filtering time constant at the inverter output $\left[s\right]$
$THD$	Total harmonic distortion $[\%]$
$T_s$	Sample time $\left[s\right]$
$\overline{V}$	Average voltage (at zero frequency in the spectral study) $\left[V\right]$
$V_C$	Voltage in the capacitor $\left[V\right]$
$V_D$	Diode voltage $\left[V\right]$
$V_{inv}$	Voltage in the invertir $\left[V\right]$
$V_L$	Voltage in the inductor $\left[V\right]$
$V_o$	DC voltaje in the output of the booster $\left[V\right]$
$V_o^2$	Quadratic value of the DC voltaje at the booster output $\left[V^2\right]$
$V_{rms}$	RMS voltage $\left[V\right]$
$V_{rms}^2$	Quadratic value of the RMS voltage $\left[V^2\right]$
$V_s$	DC voltage at the booster input or at the battery pack $\left[V\right]$
$V_{sw0}$	Voltage in the booster switch $\left[V\right]$
$V_{sw1}$	Voltage in the invertir switch 1 $\left[V\right]$
$V_{sw2}$	Voltage in the invertir switch 2 $\left[V\right]$
$V_{sw3}$	Voltage in the invertir switch 3 $\left[V\right]$
$V_{sw4}$	Voltage in the invertir switch 4 $\left[V\right]$
$V_T$	Voltage by termal effect $\left[V\right]$
$Y_i$	i-th sample in the register
$\Delta I_L$	Absolute variation of DC-DC booster output current $\left[A\right]$
$\Delta V_o$	Absolute variation of DC-DC booster output voltage $\left[A\right]$

2.4. Assessment of open-loop performance

The assessment of the open-loop performance of the source is based on the behavior of ripple in steady-state conditions at both the $5\left[kHz\right]$ switching design frequency and the $120\left[Hz\right]$ inverter system frequency, as well as the $60\left[Hz\right]$ synthesized sinusoidal signal frequency. The study relies on the spectral behavior of the resulting signals at the frequencies of interest and the mathematical framework that enables this assessment.

At the frequency of interest $f_{\gamma }$, the ripple equation allows for quantifying its value. Considering that the frequency at which ripple occurs has two causes—one related to switching and the other associated with load variation subjected to sinusoidal modulation—the design criterion is evaluated at the switching frequency, and the overall performance criterion is assessed at twice the frequency of the synthesized sinusoidal signal. Equation (30) presents a mathematical expression developed to assess ripple with dependence on the frequency of interest in the study.

$$r\%=\frac{4N}{N-1}\cdot \frac{\left|{{\displaystyle \sum }}_{i=1}^N{Y}_i\cdot e^{-j2\pi f_{\gamma }\cdot \left(i-1\right)T_s}\right|}{{{\displaystyle \sum }}_{i=1}^N{Y}_i}\times 100$$

(30)

The equation developed to assess the THD aims to provide the flexibility to assign the sinusoidal frequency, with $f_{\gamma }$ set at $60\left[Hz\right]$ for this study, as it corresponds to the frequency of the local electrical grid. However, the signal sample must be selected when the system is in a steady-state condition to ensure that the study is not affected by transients. The equation developed for this purpose is shown in (31).

$$THD=\sqrt{\frac{\left({{\displaystyle \sum }}_{i=1}^N{Y}_i^2\right)-\frac{1}{N}{\left({{\displaystyle \sum }}_{i=1}^N{Y}_i\right)}^2}{\frac{2N}{{\left(N-1\right)}^2}{\left|{{\displaystyle \sum }}_{i=1}^N{Y}_i\cdot e^{-j2\pi f_{\gamma }\cdot \left(i-1\right)T_s}\right|}^2}-1}$$

(31)

The study of open-loop behavior is conducted considering a time response within an interval of $2.4\left[s\right]$, during which the transient is reduced and the steady state of operation is reached. For the steady-state study, the time beyond 1 second after startup is considered.

Figures 8(a) and 8(b) allow to observe the voltage and current transients of the source during startup. The voltage measured at the capacitor of the boost source in steady-state is $169.70\left[V\right]$, with $208.33\left[A\right]$, which is the current measured in the loop that includes the inductor for a nominal load. It is possible to observe that during the transient, the voltage reaches a peak that exceeds twice the steady-state voltage, while in terms of current, the peak or overshoot exceeds more than 12 times its value compared to the steady-state current.

Figure 8. (a) Output voltage of the $48\left[V\right]$ to $169.70\left[V\right]$ DC-DC source. (b) Current measured in the inductor of the DC-DC source.

Figure 8. (a) Output voltage of the $48\left[V\right]$ to $169.70\left[V\right]$ DC-DC source. (b) Current measured in the inductor of the DC-DC source.

A spectral study allows to determine the value of the magnitude at zero frequency of the capacitor voltage and inductor current belonging to the output boost stage was carried out. Figures 9(a) and 9(b) shown such a voltage and current signals presented in $\left[dB\right]$, when these are normalized with respect to a unit value (taking their mean value as a reference), providing information about the behavior of the signals at frequencies of interest in the study.

In this cases, both for voltage and current, frequencies of $120\left[Hz\right]$ and $5000\left[Hz\right]$ become evident. The former is the frequency of the inverter system for generating the $60\left[Hz\right]$ PWM sinusoidal signal, and the latter is the chosen switching and elevation system frequency and PWM modulation frequency. Equations (30) and (31) allow for assessing performance indicators under statistical considerations, both for the boost output voltage and the inductor current. Table 1 presents average and ripple values obtained through spectral analysis, highlighting that the study at a frequency of $5000\left[Hz\right]$ satisfies design conditions for both voltage and current. The $120\left[Hz\right]$ frequency appears as a condition due to load variation resulting from sinusoidal PWM switching, and that is why operational ripples are observed for both voltage and current.

Figure 9. Spectrum in $\left[dB\right]$ normalized concerning to magnitude at frequency 0 Hz of the: a) output voltage of the $48\left[V\right]$ to $169.70\left[V\right]$ DC-DC source, and b) the output inductor current.

Figure 9. Spectrum in $\left[dB\right]$ normalized concerning to magnitude at frequency 0 Hz of the: a) output voltage of the $48\left[V\right]$ to $169.70\left[V\right]$ DC-DC source, and b) the output inductor current.

Similarly, the assessment of the inverter system at the output of PWM modulation, considering a $5000\left[Hz\right]$ carrier and a $60\left[Hz\right]$ modulator at full load, is carried out using spectral analysis. Figure 10(a) displays the spectral analysis of the voltage signal at the inverter output in [dB], which is normalized based on the spectral voltage value corresponding to $60\left[Hz\right]$. The way to improve the harmonic distortion factor, especially when galvanic isolation is introduced, is by incorporating a transformer with an added capacitive filter. Again, the assessment of the inverter system at the output of the filtering using spectral analysis in [dB] is presented in Figure 10(b), highlighting voltage amplitude components around $5000\left[Hz\right]$ and $60\left[Hz\right]$, but normalized based on the spectral voltage value corresponding to $60\left[Hz\right]$. Table 2 provides the harmonic distortion factor and the rms value of the sinusoidal voltage at the output of the inverter and after filtering.

2.5. Assessment of closed-loop performance

2.5.1. Design of the fuzzy controller

In order to design the fuzzy controller, we obtain a simplified model of the response of the switched boost source. It does not have to reflect the dynamic behavior precisely; it only needs to provide a satisfactory response in terms of timing. Equation (32) fulfills these requirements.

$$G\left(s\right)=\frac{K}{\left(T_1\cdot s+1\right)\left(T_2\cdot s+1\right)}$$

(32)

Considering open-loop performance, adjustments were made to the parameters in equation (32) using the gradient descent method. Figures 11(a) and 11(b) show the evolution of the K, T1 and T2 parameters of the simplified dynamic model when descending the gradient of the cost function in Figure 11(c). Finally, the time responses of the simulated and modeled dynamics of Boost stage are presented in Figure 11(d), named as Data 1 and Data 2, respectively.

Once a simplified model is obtained a PI (Proportional-Integral) control action is implemented which will be a reference model in the fuzzy controller training process. To do this, consider the response of the model system, shown in blue in Figure 11(d), and specific desired response to verify that a classic PI controller can approximate this response when operating on the given model system, as shown in Figure 12(a). Figure 12(b) represents the optimized cost function given by equation (33). Figures 12(c) and (d) show the evolution of the proportional gain and integral time parameters.

$$J=\frac{1}{2}{\displaystyle \sum }_{i=1}^N{\left({ℒ}^{-1}\left\{u_i·\frac{\left(K_p+\frac{1}{T_i·s}\right)\frac{K}{\left(T_1\cdot s+1\right)\left(T_2\cdot s+1\right)}}{1+\left(K_p+\frac{1}{T_i·s}\right)\frac{K}{\left(T_1\cdot s+1\right)\left(T_2\cdot s+1\right)}}\right\}-y_i\right)}^2$$

(33)

With the reference for training weights or singletons of the fuzzy control system, the training scenario is implemented as shown in Figure 13. This model considers the switched process relevant aspects that allow voltage boosting before moving on to the inverter process. The control loop is closed, considering that the duty cycle of the elevator switching is determined by the control output of the fuzzy PI, whose value must always be between 0 and 1. For this reason, it not only needs to be bounded but also needs to have an impact on the integration process to avoid the wind-up effect. Therefore, the system is configured as a fuzzy PI with an anti-windup effect.

Fuzzy PI control is based in 25 rules, which are described as singletons. At first, the singleton weights are set randomly, as shown in Figure 14(a), while Figure 14(b) depicts the surface of the fuzzy PI control output once the singletons are adjusted. Figure 14(c) displays the evolution of the cost function between the system signal with the fuzzy PI controller and the established pattern signal as a viable training criteria. Finally, Figure 14(d) illustrates the 25 singletons, which are gradually adjusted according to the cost function minimization.

Table 3 presents the structure of each of the triangular membership functions that are part of the universes of discourse for both the integral effect and the proportional effect, which are depicted in Figures 15(a) and 15(b), respectively. Figure 15(c) shows the structure of the inference system with its 25 rules that allow obtaining the univalued control output. Finally, Table 4 presents each of the antecedents and consequents associated with each rule, highlighting the value or weight of the singletons with their saturation effect between 0 and 1.

2.5.2. Study of stability based on the limit cycle of the deviation trend

The stability evaluation of a switched system cannot be performed using classical stability techniques such as the Routh-Hurwitz criterion or Lyapunov criteria [16]. The definition of stability itself is satisfied by verifying that, when an input as a limited interval, the resulting output value is also located in a limited interval. However, confirming this stability criterion in control is only possible by conducting a study based on the limit cycle of the deviation trend.

The study of the limit cycle of the deviation trend is a statistical analysis based on smoothing techniques similar to those implemented by Savitzky-Golay, which allows for the evaluation of the behavior of the deviation. In a switched source, the deviation oscillates around an operating point [18]. To apply the suggested method, which consists on the limit cycle of the deviation trend, the first step is to determine the deviation trend based on Equation (34):

$$J_{\left(i\right)}=\min \left({\displaystyle \sum }_{i-\frac{m-1}{2}}^{i+\frac{m-1}{2}}{\left(\left(A\cdot t_{\left(i\right)}^3+B\cdot t_{\left(i\right)}^2+C\cdot t_{\left(i\right)}+D\right)-{\delta }_{\left(i\right)}\right)}^2\right)$$

(34)

The cost function $J_{\left(i\right)}$, given by equation (34), allows us to find the optimal cubic fit around $i$, which serves as a temporal index and accounts for deviation throughout an odd number of samples $m$. This affirmation leads to the prediction and smoothing function presented in equation (35). With equation (35), it can be determined the deviation trend while mitigating the inherent effects of switching in the source.

$${\widehat{\delta }}_{\left(i\right)}=A\cdot t_{\left(i\right)}^3+B\cdot t_{\left(i\right)}^2+C\cdot t_{\left(i\right)}$$

(35)

Once the deviation trend is obtained, the study of the limit cycle will operate on this signal, therefore, it is necessary to determine its first derivative. Finally, as part of the field lines determining the direction of the limit cycle establishment, the second derivative of deviation must be obtained. Equations (36) and (37) show the numerical procedure for determining the first and second derivatives using a 5-term polynomial approximation.

In Figure 16, the deviation of the fuzzy control action is shown in red, and the deviation trend is shown in blue. With the smoothing achieved through filtering, the inherent tendency of the switched system to oscillate around the setpoint is suppressed, as at least third-order differentiability is achieved to apply the limit cycle stability criterion, which requires second-order differentiability of the studied signal.

The study of the limit cycle is achieved by plotting the results obtained with equations (35) and (36), i.e., by plotting the variation or rate of change over time of the deviation trend versus the deviation trend, as shown in Figure 17(a). In order to confirm the trend towards the establishment, the direction of the field lines is determined by considering the derivative of the rate of change of the deviation trend concerning the deviation trend, which is accomplished by implementing the ratio of equations (36) and (37), with the convergence direction being clockwise, seeking the (0,0) point on the limit cycle graph, as observed in Figure 17(b).

2.5.3. Closed-loop behavior under nominal load

The study of closed-loop behavior is conducted by considering a time response within an interval of $2.4\left[s\right]$, during which the transient response fades away and steady-state operation is achieved. The steady-state study begins $1.5\left[s\right]$ after startup.

Figures 18(a) and 18(b) show the voltage and current transient response in the source during startup. The voltage measured at the elevation source capacitor in the steady state is $169.64\left[V\right]$. In contrast, the average current in the inductor measures $265.15\left[A\right]$, considering the inductor's loop, for a nominal load. It can be observed that during the transient, the voltage does not exhibit significant overshoot, and its increase is relatively gradual. On the other hand, the current has a peak that exceeds slightly more than three times its steady-state value.

A spectral magnitude study allows the observation of magnitude at zero frequency, where the signal under investigation should remain constant, and amplitudes of the signal at any other frequency may be attenuated. This scenario is observed in Figure 19(a) and 19(b), corresponding to the spectrum of the normalized signal, presented in $\left[dB\right]$, providing an insight into the behavior of the signals at frequencies of interest in the study. It is worth to note that such figures depict the operating conditions of the Boost stage power source in which the voltage and current signals are affected.

In both cases, frequencies of $120$ and $5000\left[Hz\right]$ are prominent for voltage and current. The former is the frequency of the inverter system for generating the PWM sinusoidal signal at $60\left[Hz\right]$. At the same time, the latter is the chosen switching and PWM modulation frequency for the switching and boost system. Equations (30) and (31) allow for evaluating performance indicators under statistical considerations, both for the output voltage of the boost stage and the inductor current.

Table 5 depicts average and ripple values for output voltage and inductor current obtained through spectral considerations. Notably, for both voltage and current, the study at a frequency of $5000\left[Hz\right]$ satisfies design conditions. However, at the frequency of $120\left[Hz\right]$, there is a condition due to load variation resulting from sinusoidal PWM switching; this is why there are operational ripples for both voltage and current, which differ slightly from open-loop operation due to a slight increase in current ripple at $120\left[Hz\right]$.

The assessment of the inverter system at the output of the PWM modulation, considering a carrier frequency of $5000\left[Hz\right]$ and a modulating frequency of $60\left[Hz\right]$, when DC-DC power supply is in closed-loop PI fuzzy control, is carried out using spectral studies. Figure 20(a) illustrates the spectral analysis in $\left[dB\right]$ of the output voltage signal normalized based on the spectral value of voltage corresponding to $60\left[Hz\right]$. Table 6 presents the value of the harmonic distortion factor and the RMS value of the sinusoidal PWM modulated voltage when the fuzzy PI controller regulates the DC-DC source.

In the same way, a transformer with a capacitive filter is introduced to improve the harmonic distortion factor, especially with the incorporation of galvanic isolation. The evaluation of the inverter system at the output of the transformer with capacitive coupling is performed using spectral studies. Figure 20(b) depicts the spectral analysis in $\left[dB\right]$ of the signal normalized based on the spectral value of voltage corresponding to $60\left[Hz\right]$ of the voltage signal at the output of the inverter, highlighting voltage amplitude components around $5000\left[Hz\right]$ and $60\left[Hz\right]$ when the source is regulated in a closed loop using the fuzzy PI controller.

Table 6 presents the harmonic distortion factor of $0.0632$ and the RMS value of the sinusoidal voltage of $119.82V$ resulting from filtering following expected standard values for this inverter system.

2.5.4. Closed-loop behavior under load variation

Considering that this study corresponds to the dynamics of a rigid system, computational demands are much more stringent. Therefore, the study is critically conducted over $20\left[s\right]$, with load variation occurring over $10\left[s\right]$. The test performed corresponds to a load step starting from the maximum load. Considering that the maximum load occurs at the lowest ohmic value of $1.44[\Omega ]$, this value will be increased by $1.296[\Omega ]$ until it reaches $27.36[\Omega ]$, as depicted in Figure 21(a).

Figure 21(b) displays the deviation per unit (p.u.). After reaching $2\left[s\right]$, the voltage has stabilized, as seen in Figure 22(a), with slight fluctuations in the deviation occurring after the load step, but these variations are imperceptible around the $9\left[s\right]$ mark.

Figure 21(c) shows that the highest control action is when the ohmic value is at its maximum, meaning when the load is minimal. Figure 22(b) illustrates the sensitivity to load variation exhibited by the current variable, i.e., the control action is most pronounced when the load is minimal. However, by design, the current can regulate the system in response to load variation, which is why the regulation acts with minimal control action Figure 22(a) shows the sensitivity to load variation exhibited by the voltage variable of the boost stage capacitor. However, it is significantly lower compared to the sensitivity exhibited by the inductor current.

As performed previously, a spectral magnitude study is presented in Figure 23(a) and Figure 23(b), corresponding to the voltage across the output capacitor and current through the inductor of the boost stage, respectively. The signals are normalized by dividing them by their mean values and expressed in $\left[dB\right]$. This way allows components at frequencies of interest emerge for. In both cases, for both voltage and current, frequencies of $120\left[Hz\right]$ and $5000\left[Hz\right]$ are prominent (red boxes). The former is due to the frequency of the inverter system for generating the PWM sinusoidal signal at $60\left[Hz\right]$. At the same time, the latter is the chosen switching and PWM modulation frequency for the switching, boosting, and PWM modulation system. However, a component associated with the stepped load change at $15.87\left[Hz\right]$ emerges, which affects the inductor current spectrum. Equations (30) and (31) allow for evaluating performance indicators under statistical considerations, both for the output voltage of the boost stage and the inductor current.

Table 7 summarizes average and ripple values obtained through spectral considerations. It is worth noting that, for voltage, the study at the $5000\left[Hz\right]$ frequency satisfies design conditions. However, for current, the ripple increases by $76.88[\%$]. In both cases, for both voltage and current, the component at the $120\left[Hz\right]$ frequency corresponds to the load variation resulting from sinusoidal PWM switching, which is why operational ripples are present for both voltage and current. However, the normalized current spectrum shows a component at $15.87\left[Hz\right]$.

Similarly, the assessment of the inverter system in response to load variation at the transformer's output with capacitive coupling is conducted using spectral studies. Figure 24 displays the spectral analysis in $\left[dB\right]$ of the signal normalized based on the spectral value of voltage corresponding to $60\left[Hz\right]$ of the voltage signal at the output of the inverter, where voltage amplitude components around $5000\left[Hz\right]$ and $60\left[Hz\right]$ are highlighted. Table 8 presents the harmonic distortion factor and the RMS value of the sinusoidal voltage resulting from filtering at the inverter output when a load variation is introduced.

2.5.5. Closed-loop behavior under voltage variation in the battery system

Like the previous section, this study aligns with the dynamics of a rigid system with significantly stricter computational demands. Consequently, the study is critically conducted over a $20\left[s\right]$interval, with a $10\left[s\right]$ variation in the source voltage. The conducted test corresponds to a voltage drop from its nominal value of $48\left[V\right]$, which is decreased until it reaches $24\left[V\right]$, as depicted in Figure 25(a). This value was selected as it represents the maximum allowable $50[\%]$ discharge condition for a lead-acid battery.

Furthermore, Figure 25(b) displays the deviation per unit (p.u.). After reaching $2\left[s\right]$, the voltage has stabilized, as observed in Figure 25(a), with slight fluctuations in the deviation occurring after the source voltage variation. Figure 25(c) shows the highest control action when the source voltage value is at its minimum.

Consequently, Figure 26(a) shows how the control action makes the voltage variable seen at the capacitor of the voltage boost source almost insensitive. However, Figure 26(b) illustrates the sensitivity to source voltage variation showed by the current variable.

As previously mentioned, a spectral magnitude study only allows to observe the zero-frequency component, representing the average values of both voltage and current, belonging to the boost stage. However, once the signals are normalized by dividing them by their mean values and their values in $\left[dB\right]$ are obtained, components at frequencies of interest emerge, as observed in the normalized voltage spectrum in $\left[dB\right]$ for voltage (Figure 27(a)) or current (Figure 27(b)). In both cases, for both voltage and current, frequencies of $120\left[Hz\right]$ and $5000\left[Hz\right]$ are prominent. The former is due to the frequency of the inverter system for generating the PWM sinusoidal signal at $60\left[Hz\right]$. At the same time, the latter is the chosen switching and PWM modulation frequency for the switching, boosting, and PWM modulation system. However, in Figure 27(b), a component associated with source voltage variation at $15.77\left[Hz\right]$ emerges, which affects the inductor current spectrum.

Again, Equations (30) and (31) allow evaluating the performance indicators under statistical considerations, both for the output voltage of the boost stage and the inductor current. Table 9 depicts average and ripple values obtained through spectral considerations, highlighting that the study at the $5000\left[Hz\right]$ frequency satisfies design conditions for both voltage and current. In both cases, for both voltage and current, the component at the $120\left[Hz\right]$ frequency corresponds to load variation resulting from sinusoidal PWM switching, which is why operational ripples are present for both voltage and current.

The assessment of the inverter system, at the output of the PWM modulation, considering a carrier frequency of $5000\left[Hz\right]$ and a modulating frequency of $60\left[Hz\right]$, in response to source voltage variation, is conducted using spectral studies. Figure 28 displays the spectral analysis in $\left[dB\right]$, with the signal normalized based on the spectral value of voltage corresponding to $60\left[Hz\right]$, of the voltage signal at the output of the inverter when closed-loop regulation is active. It highlights voltage amplitude components around $5000\left[Hz\right]$ and $60\left[Hz\right]$. Table 10 details the harmonic distortion factor and the RMS value of the sinusoidal PWM-modulated voltage when the source is regulated by the fuzzy PI controller in the presence of source voltage variation.

3. Discussion

4. Conclusions

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