Abstract
In this article, a real case of a one MW PV substation design problem is described. The impact of the tolerances of power components (mainly inductors and capacitors) on the design is highlighted through two studies using two different control strategies that are selective harmonic elimination (SHE) and selective harmonic modulation (SHM). The obtained results show that the components’ tolerances have a significant effect on the design, especially for the SHE strategy. While the design using SHM control shows a weak sensitivity to the components’ tolerance. In this study, an upgrade of the most sensitive control strategy is proposed in order to improve its sensitivity against components’ tolerance. The results show that the strategy upgrading leads not only to improving the sensitivity but also to generating better power quality.
1. Introduction
Today, PV systems are widely used with other green energies in order to reduce the environmental impacts, especially global warming. However, the use of strong PV stations has a significant influence on the quality of electrical energy generated due to harmonics rejected on the distribution grid. Three key standards and regulations that are IEEE 519 [1], G5/4 [2], and French Decree 2008 [3], are recommended as grid codes to minimize the rejected harmonics.
To be aligned with these grid codes, a lot of works based on Patel and Hoft [4] are proposed. From these works, it is clear that an LCL [5] and an LLCL filter [6] combined with a selective harmonic elimination (SHE) [4] and a selective harmonic modulation (SHM) shall be used. In [7], a real case of a one MW PV is proposed, and a comparison of generated energy quality with LCL and LLCL combined with the SHM and SHE strategy [8] is shown, but until now, there is no research that demonstrates the effect of the suggested tolerance on the design of the PV system. Based on prior research, this work aims to investigate the influence of the PV system’s component tolerances. Furthermore, a lot of design methodologies such as [9] propose an innovative solution for solving and optimizing design problems. For control design strategies combined with filter design among the famous research studies by Patel and Hoft since 1973 [4] and Franquelo et al. [5], it is demonstrated that, for such design problem, SHM design strategy allows to fit the rejected current spectrum under the allowed gauge imposed by standards. However, in this article, it is demonstrated that an upgraded SHE strategy leads to better results. The effort spent to upgrade this control strategy is not important, thanks to mathematical algorithm solvers and powerful calculators. Besides, this effort is compensated by a better power quality generated on the grid.
The structure of this article is as follows. The problem description of a one MW PV system with an LLCL filter is shown in Section 2 [10]. In Section 3, a brief presentation of SHE and SHM strategies is made. The impact of components’ tolerance is analysed through the results obtained from SHE and SHM strategies combined with the LLCL filter obtained in Section 4. In Section 5, a radical solution is proposed to secure the design and make it totally independent from components’ tolerances. Section 6 is devoted to the conclusion.
2. Problem Description of a Real Case of One MW PV Substation
Figure 1 depicts the proposed PV system’s topology.
Figure 2 depicts the analogous circuit for each phase.
The per-phase equivalent frequency diagram after the ignorance of the resistance in front of the inductance impedance is depicted in Figure 3.
This system is already presented in [11]. The filter inductor is integrated with the transformer to lower the overall volume and weight of the system, resulting in cost savings. The tertiary inductor is caused by the leakage inductance of the cables used to link the capacitor’s bank to the tertiary: it is also proved in [11] that this parameter has a significant influence on harmonic rejections.
3. SHE and SHM Theories
SHE and SHM theories are described in the following sections.
3.1. SHE Control Strategy
The selective harmonic elimination suggested by Patel and Hoft consists of solving mathematical equations where all of them must be equal to zero in order to find the position of switching angles.
Considering is the number of switching angles, is the harmonic order, and angles , the following equations were generated, where is the harmonic amplitude of order which is expressed byfor .
Theoretical investigation of the problem aims to zero out a large number of unique odd harmonics using the SHE method [12]. The factors are listed as follows to make up a valid SHE solution:where each is the switching angle at which the inverter must alter the output voltage, with . This indicates that, in order to eliminate harmonics with , a solution must fulfill the following set of equations, which are provided in the following equations:
The required modulation index is :where , and a good solution must adhere to the restrictions of equation (2).
3.2. SHM Control Strategy
The selective harmonic modulation consists of solving the same mathematical equations developed by Patel and Hoft but here inequalities shown should be solved as follows:
Depending on equation (2), as well aswhere denotes the requested modulation index, , , and are the limits which are calculated to guarantee that grid codes should be respected.
4. Tolerances Impact on the Design
4.1. Sequential Quadratic Programming
Optimization of electrical system design [13] is currently a commonly utilized practical method [14]. The SHE [15] and SHM inequalities may be solved using a variety of strategies [16] as a consequence of developments in computer science and numerical concept [17]. We suggest applying optimization techniques, especially a deterministic approach [18], to reduce the size and complexity of the optimization issue [19]. Many algorithms have been studied [20], but the well-known Sequential Quadratic Programming (SQP) algorithm has ultimately been chosen as the current approach for handling challenging nonlinear and restricted programming issues [21].
4.2. Desired Modulation Index
Yearly worldwide irradiation surpasses 2 , the PV substation is meant to run in southern Tunisia. It is assumed that the PV station will normally run in around its nominal mode. The desired nominal value is 0.86.
4.3. Applicable Tolerances
A lot of datasheets from different capacitors and inductors suppliers have been analysed. Almost, a tolerance is considered on inductors. However, a bigger tolerance shall be considered on the leakage inductor introduced by cables. Indeed, a lot of the parameters are involved in the calculation of this leakage inductance that themselves are dependent on the tolerance:(i)A tolerance on cable length(ii)A tolerance on the measuring equipment using in the test bank to estimate the linear leakage inductance of the tertiary cable(iii)A tolerance on the leakage inductance of the transformer tertiary
Considering all of these arguments, a tolerance of at least should be taken into account for .
4.4. Impact on the Transfer Function of the System
Considering the respective tolerances on each component of the PV system shown in Figure 3, the lower response of the transfer function, given in Section 1, can be easily deduced as per the following equation:where , and , , and are defined as
The upper response of the transfer function can be easily deduced as per the following equation:where , , and are defined as
The responses of transfer functions and are shown in Figure 4 as well as the initial value considered in the initial study.
The response function shows a lower resonance and antiresonance frequencies than the initial transfer response than a better filter attenuation behavior at high frequency, especially for the higher harmonics orders.
The response function shows a higher resonance and antiresonance frequencies than the initial transfer function responses, which leads to the worst attenuation behavior of the filter at high frequencies, especially from the harmonic order. Thus, there is a big risk to break grid codes to confirm this risk, the voltage spectrum obtained by SHM and SHE is injected on both and transfer functions, and the current spectrums are compared with grid codes.
Figure 5 shows both harmonic current spectrums obtained by SHM and SHE with and . Indeed, by adopting the SHM method, the harmonic is more concerned with grid standards than the influence of component tolerances on PV design.
So to conclude, grid codes are respected when the SHM strategy is chosen regardless of the components’ tolerances but when SHE is chosen, components’ tolerances have a big impact on the current spectrum tolerances. So the risk to break grid codes is too high.
However, with the SHM strategy even if the grid code is respected, nonzero low-frequency current harmonics are seen and will lead to a high value of THD contrary to the harmonic current spectrum obtained by SHE where low-frequency current harmonics are eliminated except the harmonic that breaks grid codes. For this reason, a radical solution is proposed in the next section in order to mitigate the risk to break grid codes and to avoid the negative impact of component’s tolerances on the design when SHE is used.
5. Tolerances’ Impact on the Design
In the previous section, the advantage of SHE is highlighted. However, with SHE the risk to break grid codes is very high because the tolerances of the PV system could impact the transfer function of the system and move the resonance and antiresonance over a frequency range where the upper and the lower value are related to the upper and lower value of components’ tolerances. In the studied case, the harmonic exceeds the limit of the grid codes where SHE seven switching angles are used.
5.1. Proposed Method for Theoretical Analysis
In order to handle this problem, the proposed solution consists of increasing switching angles to eight in order to eliminate the harmonics and let the filter eliminate the higher current harmonics orders that remain totally uncontrolled. The eight switching angles strategy could be expressed in an (OF) subject towhere all constraints are given from equations (11)–(18).
, , , , , and , are approximated to 0. The obtained current harmonic spectrum is shown and compared to grid codes in Figure 6.
Figure 6 shows that the first seven harmonics are deleted and that the grid codes are followed. Both SHM with seven switching angles and SHE with eight switching angles respect grid codes, but the SHE eight switching angles are more consuming than SHM, but this is justified by the better THD.
5.2. Time-Domain Comparison
The output current curve generated using the combined SHE-LLCL with eight switching angles is shown in Figure 7. With this approach, the THD [22] is .
Figure 8 depicts the current injected into the grid using the SHM-LLCL approach. This technique yielded a THD of .
Both curves produce good results because THD is less than (the most prevalent THD accepted on power networks globally). The combined SHE-LLCL approach with eight switching angles outperforms the SHM-LLCL technique in terms of THD. Indeed, using the SHM method, persistent low-frequency harmonics produce an increase in THD. However, in the suggested method, SHE-PWM eliminates low-frequency harmonics that are damaging to the network, and the optimized LLCL filter attenuates the harmonic, which explains the given findings. This result reinforces the suggested solution’s efficiency in terms of offering greater power quality and performance [23]. The motivation behind moving away from traditional studies persists not only by its high amplitude attenuation of all harmonics regarding grid codes standards but also by its capability of handling several problems that may occur by avoiding the unwanted effect of SHE-PWM taking into account the impact of component’s tolerances. Indeed, the proposed methodology is an efficient strategy, which is based on deterministic optimization, compared to previous works adopted in literature in terms of performance and providing high-quality power.
6. Conclusion
This article shows the components’ tolerance drawback on the design of a one MW PV substation. The comparison of both control strategies that are SHE and SHM considering inductors and capacitors’ tolerances shows that, with the SHM strategy, grid code is respected but with a high THD level. However, with the SHE strategy, the components’ tolerances could lead to break grid codes and this should be anticipated during the design phase by upgrading the SHE strategy in order to remove harmonics from resonance and antiresonance zones. The upgrading of SHE will lead to a much more complicated mathematical problem to solve, but this is easy to mitigate thanks to the optimization algorithm. The proposed research methodology not only complies with grid regulations but also provides a lower THD value and higher quality of generated energy on the grid. The positive point of this work is the real aspect of the photovoltaic power station, which could match with all distribution worldwide. Therefore, it could be considered as a real solar project and a better alternative to be realized in the future.
Symbols
| : | Secondary winding resistance |
| : | Secondary inductance (H) |
| : | Minimum secondary inductance value (H) |
| : | Maximum secondary inductance value (H) |
| : | Tertiary resistance |
| : | Tertiary inductance (H) |
| : | Minimum tertiary inductance value (H) |
| : | Maximum tertiary inductance value (H) |
| : | Tertiary capacitance (H) |
| : | Minimum tertiary capacitance value (H) |
| : | Maximum tertiary capacitance value (H) |
| : | Primary winding resistance |
| : | Primary inductance (H) |
| : | Minimum primary inductance (H) |
| : | Maximum primary inductance (H) |
| : | Magnetic resistance |
| : | Magnetic inductance (H) |
| : | Equivalent grid resistance |
| : | Equivalent grid inductance (H). |
Data Availability
The data used to support the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.