Power System Harmonic Modeling

The document begins by highlighting the Tiwai smelter in New Zealand as a significant example of one of the world's first high-voltage direct current (HVDC) links. This smelter, operating at 450 MW and utilizing a 48-pulse converter configuration, serves as a primary case study for the analysis of harmonic distortion within the power system. The text anticipates a continuation of this trend, with potential expansions to the existing HVDC link, the possibility of a second link, and the electrification of a portion of the North Island's main trunk line. These future developments are expected to further impact harmonic issues in the national grid.

A key concern highlighted is the increasing prevalence of power electronics in various aspects of the New Zealand power system. The widespread adoption of semiconductor controllers in drives, light dimmers, and fluorescent lighting is noted, emphasizing their contribution to the generation of harmonic voltages and currents within distribution systems. The introduction of electric vehicles (EVs) is projected to worsen this problem, as widespread nighttime charging will coincide with periods of lighter loading and reduced system damping. The use of power factor correction capacitors, while intending to improve power quality, could introduce local resonances and exacerbate existing harmonic issues.

The document specifically points out the growing reliance on alternative energy sources such as micro-hydro, solar, and wind power. These sources, being electrically oriented, depend on power electronics for control and contribute to the overall harmonic profile of the power system. The integration of these diverse generation methods presents further challenges in managing harmonic distortion across the entire network. The text implies the need for comprehensive harmonic analysis to manage the impacts of these various sources of harmonic generation in the context of a larger and more complex power system.

Early understanding of power system harmonics in New Zealand significantly advanced during the commissioning of the HVDC link, as documented by Robinson (1966a). This early work, along with a parallel study modeling the French end of the Cross-Channel link using a simulator (Laurent et al., 1962), revealed challenges in accurately reproducing measured results through simulation. A considerable degree of harmonic voltage unbalance was observed in the measured data, highlighting a discrepancy between the simulated and real-world behavior of the system. The difficulties stemmed from the limitations of physical simulators in adequately capturing the strong frequency dependencies inherent in various power system components using discrete RLC components. These early studies emphasized the need for more sophisticated modeling techniques to accurately capture the complexities of harmonic behavior in high-voltage systems.

The analysis extended to the propagation of non-integer harmonic currents, such as those arising from ripple control signals. Studies by Ross (1972) and Densem (1983) investigated the strength of these ripple signals throughout the power system, assessing their ability to operate ripple relays. A crucial aspect was evaluating potential spillover of these signals into other supply authority systems through the transmission network. This highlights the importance of understanding how non-standard frequencies could affect wide areas and the need for robust modeling to account for them in planning and operation.

The text notes that typical power system component modeling assumes passive and linear behavior, treating each frequency independently. However, this simplification is not always accurate, particularly concerning converters which are treated as fixed injections. The document acknowledges that frequency coupling is a factor, highlighting the need to move beyond simplified models. The alternative of dynamic modeling using iterative processes, as employed by several authors (Reeve and Baron 1971; Yacamini and de Oliveira 1980a, 1980b; Harker 1980), is introduced. These methods generally begin by assuming sinusoidal voltages at converters before iteratively updating the AC system response to harmonic current injections. This iterative approach provides a means to address the limitations of treating each harmonic in isolation.

The document identifies unbalanced fundamental terminal voltage as a major factor contributing to non-ideal converter operation (Reeve and Krishnayya 1968; Phadke and Harlow 1968). Such imbalances lead to unequally spaced zero crossings, causing unequal conduction periods in each phase. This results in the generation of all odd harmonics on the AC system and all even harmonics on the DC side. Controller errors also emerge as another source of uncharacteristic harmonics (Reeve and Krishnayya 1968; Reeve et al. 1969). These errors introduce unequal conduction periods and half-wave asymmetry, resulting in a more random generation of both AC and DC side harmonics, potentially leading to issues like converter transformer core saturation. The interplay between converter operation and harmonic generation is further detailed in the text, underscoring the complexity of modeling realistic power systems.

The document highlights the significant problem of harmonic instability in power systems, particularly focusing on third harmonic instability. The experience with the Cross-Channel link is cited as a key example, where a third harmonic instability often rendered the link inoperable (Holmborn and Martenssen, 1966). This problem was successfully resolved by incorporating a third harmonic filter. Based on this experience, a similar preventative measure—the inclusion of a third harmonic filter—was adopted during the design phase of the Sardinian HVDC terminal (Natale et al., 1966; Holmborn and Martenssen, 1966). This underscores the importance of proactive harmonic mitigation strategies in the design of high-power systems to prevent operational disruptions. Temporary mitigation measures, such as decreasing the DC current to reduce the effective size of a converter, are also discussed (Ainsworth, 1967), but these are presented as short-term solutions.

The converter controller is identified as a crucial element in any positive feedback loop leading to harmonic instability. The control strategy directly influences firing pulses and the extent to which an initial disturbance is amplified. Adding control system filters to a conventional constant delay angle controller can help attenuate the loop gain (Ainsworth, 1967), representing a key strategy for stabilizing the system. The relative size of converter currents compared to the AC system is another critical factor influencing harmonic magnification or instability. The short circuit ratio (SCR), defined as the ratio of short-circuit MVA at the terminal AC busbar to the total nominal DC power of the converter station (Arrillaga, 1983), emerges as a practical criterion for evaluating converter behavior. A low SCR indicates a higher susceptibility to harmonic instability.

The study employs an iterative algorithm for harmonic analysis, but notes potential issues with premature convergence. This is addressed by introducing a controller disturbance to excite potentially unstable harmonics, a technique demonstrated as effective in section 4.3.5. The iterative algorithm's convergence is shown to be influenced by several factors, including the level of third harmonic voltage distortion during commutation and the accuracy of the AC system representation. Improving the AC system representation by using a full admittance matrix, while leading to more accurate results for stable solutions, did not significantly change the outcome of unstable runs (Yacamini and de Oliveira, 1980b). Techniques to improve convergence include initially eliminating problematic harmonics, gradually reintroducing them using a scale factor, and adjusting an acceleration factor to control the rate of iterative updates. The New Zealand harmonic regulations (New Zealand Electricity, 1983) provide a benchmark against which the simulation results are assessed.

Further investigation into the impact of third harmonic voltage during the commutation process was conducted, focusing on a simplified scenario with only the third harmonic present and a purely diagonal AC system representation. This involved varying the phase angle of the AC system's third harmonic admittance to alter the phase relationship between injected current and resulting terminal busbar voltage. By keeping the admittance constant and varying the angle from -90 to +90 degrees, the study sought to isolate the effect of the system angle on stability, excluding resonance effects. The experiments revealed the third harmonic voltage during commutation as a primary cause of instability; removing this harmonic allowed for convergence, suggesting the instability was not merely a numerical artifact of the simulation.

The document points out a significant gap in existing harmonic analysis: the limited consideration given to the harmonic behavior of synchronous machines. While it's known that synchronous machines convert negative sequence currents into third harmonic voltages (Clarke, 1950), and generally act as harmonic converters, the complexity of this interaction and the coupling between harmonics have hindered their inclusion in many models. This oversight is problematic because the harmonic conversion process significantly impacts overall system harmonic levels. The text highlights the need for a more comprehensive, three-phase representation of the synchronous machine within harmonic models, rather than simpler, single-phase approaches, to accurately capture these interactions.

To address the shortcomings of existing models, the development of a general Norton equivalent circuit is described to represent the steady-state behavior of synchronous generators in the harmonic space. This approach aims to provide a more accurate representation of the machine's harmonic response. The open-circuit harmonic impedances of this Norton equivalent are shown to reasonably match empirical data, except for resistive damping, which is lower in the theoretical model. These discrepancies are attributed to factors like eddy currents, with the model being further refined for round rotor machines (section 6.4.3). This development improves the accuracy of harmonic analysis, allowing for a better prediction of harmonic behavior in systems with significant synchronous generation.

The analysis extends to the impact of unbalanced loading on synchronous generators, noting that significant harmonic currents and voltages can be generated under fault conditions or large imbalances (Clarke, 1950). While significant unbalance is rare in steady-state operation, the presence of resonance or large imbalances can lead to substantial distortion. The South Island of New Zealand is cited as a region where high third harmonic levels have been observed. The study emphasizes that existing models, which often short-circuit the generator behind the subtransient reactance, are insufficient to detect the harmonic coupling exhibited by the machines. Results from tests with asymmetrical loads and untransposed transmission lines show that harmonics generated by the machine can exceed regulatory limits, highlighting the necessity for more accurate modeling techniques.

The study considers the scenario of harmonic current injection into a system containing synchronous machines, noting that harmonic sources are often geographically separated from the machines by transmission lines. The Tiwai smelter in the South Island of New Zealand, primarily supplied by the Manapouri hydro station (Densem, 1983), exemplifies this. This configuration necessitates a model that captures the interaction between injected harmonics and the synchronous generator, including the conversion of injected harmonics into different harmonic frequencies. The Manapouri power station, comprising seven machines with a total generation of 690 MW, is used as a case study. Conventional harmonic penetration algorithms, which treat each frequency independently, cannot accurately predict the voltage distortion at frequencies other than the injected harmonic. The study therefore highlights the importance of a model capable of capturing the interactions between different harmonics within a synchronous machine and the larger power system network.

The study highlights the limitations of conventional harmonic penetration algorithms, which typically treat each frequency separately. This approach fails to account for the coupling between different frequencies that can occur within a power system, particularly in the presence of synchronous machines. The authors argue that this simplification prevents accurate prediction of voltage distortion at frequencies other than the injected harmonic. This limitation is contrasted with the more comprehensive modeling approach presented in the study, capable of capturing these frequency interactions and providing a more accurate representation of system response to harmonic injections.

The research focuses on analyzing harmonic current injection into a power system that includes a synchronous machine. In real-world scenarios, harmonic sources are often geographically distant from synchronous generators, connected via transmission lines. The Tiwai smelter in New Zealand's South Island, supplied by the Manapouri hydro station (Densem, 1983), is used as a practical example. The study investigates the system's response to harmonic current injection, specifically focusing on the interaction with the synchronous machine. The study predicts that injecting a positive sequence fifth harmonic current will produce significant levels of both negative and positive sequence third harmonic voltages due to the frequency coupling exhibited by the synchronous generator. This highlights the importance of considering the impact of synchronous machines on harmonic propagation.

To assess the accuracy of the proposed model, its predictions are compared to those obtained using a conventional harmonic penetration algorithm (Densem, 1983). This comparison, illustrated in Figure 7.10, reveals significant differences in the predicted harmonic voltage distortion. The conventional method, treating each harmonic independently, significantly underestimates the voltage distortion at frequencies other than the injected harmonic. In contrast, the proposed model accurately predicts the impact of harmonic injection, including the generation of harmonics not directly introduced into the system. This demonstration showcases the superiority of the more sophisticated modeling technique in capturing the realistic behavior of a power system subjected to harmonic current injection.

The study discusses the use of a Transient Converter Simulator (TCS) as an independent method to validate the results obtained from the proposed iterative algorithm. A specific comparison is made considering AC filters and a non-infinite DC system. The AC system is designed to have a parallel resonance near the third harmonic, and harmonic filters are included to assess their influence on the system response. The iterative algorithm simulates a perfectly smooth DC current, while the TCS algorithm incorporates a smoothing inductor to minimize DC current ripple. The results from this comparison help establish the accuracy and reliability of the iterative algorithm, particularly in situations where the solution might diverge, highlighting the benefits of using multiple validation methods to ensure accuracy.