An Analysis on the VSC-HVDC Contribution for the Static Voltage Stability Margin and Effective Short Circuit Ratio Enhancement in Hybrid Multi-Infeed HVDC Systems

Citation: Oliveira, D.; Leal, G.C.B.; Herrera, D.; Galván-Díez, E.; Carrasco, J.M.; Aredes, M. An Analysis on the VSC-HVDC Contribution for the Static Voltage Stability Margin and Effective Short Circuit Ratio Enhancement in Hybrid Multi-Infeed HVDC Systems. Energies 2023,16, 532. https:// doi.org/10.3390/en16010532 Academic Editor: Abdelali El Aroudi Received: 26 November 2022 Revised: 24 December 2022 Accepted: 28 December 2022 Published: 3 January 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article An Analysis on the VSC-HVDC Contribution for the Static Voltage Stability Margin and Effective Short Circuit Ratio Enhancement in Hybrid Multi-Infeed HVDC Systems Diego Oliveira 1, Gustavo C. Borges Leal 1, Danilo Herrera 2, Eduardo Galván-Díez 2, Juan M. Carrasco 2,* and Mauricio Aredes 1 1Electrical Engineering Department, Federal University of Rio de Janeiro, Rio de Janeiro 21941-598, Brazil 2Departamento de Tecnología Electrónica, Universidad de Sevilla, 41092 Sevilla, Spain *Correspondence: [email protected] Abstract: Over the years, high voltage transmission of large energy blocks over long distances has widely developed from the Line Commutated Converter (LCC) technology. However, the continuous expansion of the AC network and the increase in renewable energy penetration leads to the weakening of Electric Power Systems (EPS), causing operational problems for the LCC-HVDC. This paper evaluates the degree of contribution of the VSC-HVDC on the LCC-HVDC dynamic performance when the infeed is in a hybrid multi-infeed HVDC system. Through a steady-state mathematical framework, the new concept of Hybrid Power Voltage Interaction Factor ( h PVIF) is proposed to assess the transient stability improvement of LCC. Additionally, this article introduces two key contributions from h PVIF: the complementary Improved Commutation Failure Immunity Index ( i CFII), as a measure of the effective short circuit enhancement, as well as the apparent line length, which emulates the line lengths adjusting the power dispatch in the VSC-HVDC. PSCAD/EMTDC TM time-domain dynamic simulations are performed to assess the indexes, and the Matlab ® software will be used to assist in mathematical operations. Keywords: hybrid multi-infeed system (HMIF); improved commutation failure immunity index (iCFII); hybrid power voltage interaction factor (hPVIF) 1. Introduction The LCC-HVDC technology has played a crucial role in Brazilian’s National Interconnected System (SIN) in the exchange of large blocks of energy over long distances. The Brazilian scenario in 2022 contains four HVDC links feeding 20 GW in the Southeast area, two 600 kV bipoles of Itaipu, two 600 kV bipoles of Madeira, and two UHVDC 800 kV bipoles of Belo Monte. In the so-called decennial publication SIN Expansion Plan, the government agency responsible for Brazilian energy planning indicates three more ± 800 kV UHVDC links for subsequent years. They are: the bipoles of Graça Aranha–Silvânia link (1460 km, 4000 MW), the Parauapebas–Assis link (1940 km, 4000 MW) to be in operation until 2023, and a third UHVDC link is under study to transport energy from the Tapajós power plant to the Southeast, with 8000 MW for two bipoles (1500 and 2500 km) in the entire operation from 2027 [1]. In the medium-term perspective, systemic insertion of renewable energy sources added to an increase in the AC transmission lines integrating the primary grid [ 2 ], pointing out the consequent weakening of the EPS, thus aggravating issues when LCC-HVDC are connected. Brazil has a large territorial dimension and an energy matrix in which the main generation plants are located far from load centers. Additionally, environmental constraints are reducing the Right of Way (RoW) permission for overhead lines, therefore collaborating to change power systems’ characteristics, becoming more interconnected and reliable, but with new, complex paradigms. This highlights the emergence of some notable Energies 2023,16, 532. https://doi.org/10.3390/en16010532 https://www.mdpi.com/journal/energies Energies 2023,16, 532 2 of 28 technical challenges in LCC-HVDC systems, leading to increased Commutation Failures (CF) susceptibility, for instance. Thus, new actions to meet power requirements, such as higher reactive support, became necessary to overcome problems caused by AC voltage reduction at the inverter side bus. In this context, the literature contains published works analyzing these challenges of LCC-HVDC operation and proposing solutions for enhancing commutation failure immunity and improving the overall reliability of the interaction between AC and DC systems, with special attention given to weak networks. According to the nature of CF, there are three technical routes for CF prevention and mitigation CF. They are improvements to primary equipment, new control strategies for LCC-HVDC and reactive support in the AC side of the inverter. Throughout the literature review, some of the main references in each of these methods will be cited and briefly commented. In [ 3 , 4 ], the authors considered the insertion of voltage control equipment in the vicinity of the LCC-HVDC. The AC system strength and control models assessment was considered in reference [ 5 , 6 ], and employing thyristor-based controllable capacitors to eliminate CF was considered in [7–9]. Advanced control methods for preventing and mitigating the consequences of commutation failure have developed in the second decade of the 21st century, such as those with real-time measurement and calculation of inverter quantities, such as extinction angle, AC commutation voltage, DC current, and overlap area of the thyristor voltage waveforms, in order to anticipate control actions and inhibit CF [10–14]. Modern electrical power systems worldwide have expanded using Voltage-Sourced Converter-based High-Voltage Direct Current (VSC-HVDC) transmission links, which recently was just an emerging technology in the HVDC area. However, its application in transmitting large energy blocks over long distances has increased due to the development of self-commutated devices operating at higher voltage rates and economic competitiveness compared to LCC technology. More recently, the authors of [ 15 ] introduced a control strategy to overcome the disadvantages inherent to conventional LCC systems, combining the VSC and LCC converters. The author detailed the VSC-HVDC link contribution to voltage stability, proposing a flexible power control strategy for transitory reactive power under grid faults. The studies combining VSC and LCC operating in the same AC system and the analysis of the interactions that arise from it contributed to doctoral theses, such as [16–19]. A few recent works are devoted to discussing the VSC-HVDC in its principal function, which is point-to-point power blocks transmission in a controlled way [ 20 , 21 ]. Other publications bring positive results in the performance improvement of the LCC-HVDC when a VSC-HVDC operates in their geographic proximity. Some papers discussed the VSC-HVDC control modes at length, mainly when operating within a multi-infeed system [22–25] , in both transient and steady-state performance. To partially finish the literature review, an influential group of articles forms one of the bases of this work, and the beginning of developing methods for the analysis and simulation of Electric Power Systems (EPS) in the vicinity of the voltage stability limit [ 26 ] will be presented. The study of AC/DC interactions phenomena in low short-circuit capacities systems began with [ 27 ], where methods based on the eigenvalue decomposition technique for analyzing the voltage stability of multi-infeed systems were presented. Similar techniques for power and voltage stability in MIF systems have been established since then and developments continue to this day [28,29]. An exhaustive framework on two essential nodal AC voltage indices applied to the voltage interaction studies in multi-infeed systems was presented in [ 30 ]. The author has compared the respective indices obtained empirically on related literature. It investigated the NVSF and the NVIF and compared them with their counterparts obtained through simulations in electromagnetic transient programs VSF and MIIF , respectively. However, this study is restricted to those MIF scenarios composed purely of LCC-HVDC links. Energies 2023,16, 532 3 of 28 A more innovative approach focuses on examining the relationship between DC power stability and voltage sensitivity of the inverter [ 31 ], calculating the Effective Short Circuit Ratio (ESCR) to predict the maximum power transfer limit in hybrid MIF systems [ 32 ]. Analytical formulation of voltage interactions between inverters in hybrid MIF systems is further explored in [33]. Thus far, journal articles and doctoral theses have been listed, forming this bibliographic review’s basis. However, a valuable group of conference papers is dedicated to dealing with multi-infeed systems, including hybrids and their main issues involving their operation in low ESCR systems and mitigation of CF. Some of them deserve mention here. A large amount of computational electromagnetic simulations is performed in [ 34 , 35 ] to analyze the coupling effect between VSC and LCC, and the commutation failure mechanism of the LCC-HVDC system. The mathematical approach for CF immunity level assessment is also presented in [36–40]. Some primary bibliographic references have been placed from a historical perspective. The other references will be listed and commented on within their respective sections throughout the article. These works address different solutions for quantifying commutation failure immunity levels and enhancements in CF immunity. These publications provided valuable contributions to the field and advanced the state-of-the-art of LCC-HVDC performance improvement. However, such advances were restricted to scenarios of single-infeed HVDC systems or multi-infeed HVDC systems composed purely by LCC links. In addition, some of them use simulation tools as a fundamental basis to reach results, which inevitably take a long time and result in computational burden. Until today, no other work considers hybrid MIF with LCC and VSC links exploring the cross effects resulting from the power–voltage relation between them, especially provided when a tie line transmission line connects these links, thus resembling real scenarios. This work aims to provide a systematic and generally applicable method for stability analysis of hybrid multi-infeed HVDC systems from which the sensitivity of the system parameters on the ESCR in the inverter side can be precisely and effectively determined. By including LCC-HVDC equations into power flow with power injection formulation and based on the eigenvalue decomposition technique, analytical expressions are derived from giving an insight into how inter-inverter interactions influence the transient performance of LCC-HVDC systems. The investigations performed in this manuscript show that strengthening the AC system is highly influenced by VSC operation and location. Geometrically, it is observed that such influence assumes the form of a surface, where the voltage in the LCC and, therefore, its ESCR behave in different ways for the variations of active and reactive VSC powers. The variation of active VSC power produces a parabola with a well-defined global maximum point and an approximately linear characteristic in the direction of the variation of VSC reactive power. Here arises the main motivation for this work. The gradient of a two-dimensional function is applied to map how the VSC impacts the LCC-HVDC voltage stability. The contributions of the work are summarized below: • Hybrid Power Voltage Interaction Factor (hPVIF): As the primary contribution, the factor hPVIF is proposed. From the power–voltage relationship between the LCC and VSC links, and given the length of the transmission line that connects the inverter buses, hPVIFp allows extracting the system operation set point necessary to achieve the maximum ESCR in the LCC link. Knowledge of this point, therefore, includes knowing how far the VSC must be connected from the LCC and setting the values of the active and reactive power coordinates in the VSC for the upper limit for the voltage stability margin in the LCC inverter station. Factor hPVIFq is helpful for the further contribution of the manuscript, i.e., apparent line length; therefore, hPVIFp and hPVIFqcomplement each other to achieve the objectives of this work. • Among the stability problems in LCC systems, commutation failure is the most common and undesirable problem and will be adopted as a predictor instrument of Energies 2023,16, 532 4 of 28 the ESCR gain assessment. Based on CFII, which will be discussed in detail from Section 2onwards, the new metric Improved Commutation Failure Immunity Index (iCFII) is specifically designed to evaluate the ESCR enhancement compared to the situation without the presence of the VSC. • A method for incorporating the equations representing the steady-state response of the LCC-HVDC inverter into the power flow problem formulation is described (Section 3.3). The control mode and its impact on the power system operating conditions are addressed so that it results in an extended Jacobian for hybrid multi-feed systems. • The electrical system operators’ decisions in the operation programming process are carried out daily, monthly, and annually. In all cases, the programming considers systemic aspects, such as the generation and transmission expansion schedule, the level of the reservoirs, load forecasts, analysis of meteorological conditions in the main river basins of the EPS, and forecasts of inflows to the hydroelectric uses. In this context, as these are aspects closely related to medium and long-term horizons and, therefore, with little or no flexibility for short-term changes, the operation agents need to have a reliable instrument that allows emulating a more advantageous value for the ESCR seen by a given LCC-HVDC. From the proper choice of VSC-HVDC set points, the apparent line length concept presented in Section 5.1.2 allows for the emulation of a shorter line length than the actual length, resulting in a higher value of ESCR. The outline of the paper is as follows. After the Introduction, Section 2reviews aspects of the LCC-HVDC operation that influences the CF, in addition to contextualizing the object of study of this article. In the Section 3, a quasi-steady-state mathematical analysis is used to present the main idea with references in the appendix, resulting in the definition of the Hybrid Multi-Infeed Interaction Factor, which is the work’s fundamental contribution. In support of the following steps, the Section 4contains the case study implemented in PSCAD detailing system components and the VSC-HVDC control scheme. The Section 5 shows the simulation outcomes and analysis. Section 6summarizes the research and concludes the paper based on the previous results. 2. Commutation Failure Analysis 2.1. Commutation Failure: Definition and Past Approaches Commutation Failure (CF) is an adverse condition of vulnerability, especially due to weak systems, which can cause sustained overcurrents, damaging equipment. The emergence of the commutation failure theory dates back to the early nineties with a formulation proposed for a CF induction mechanism [ 41 , 42 ]. The CF investigations intensified over the next few years, passing through [ 43 , 44 ], which reveals through simulations that the results from such a previous formulation only match those obtained for a set of specific scenarios. Since fast transient events and unbalances have to be considered in CF studies, the more appropriate way to analyze it is through detailed EMT simulation of the systems [45], and in this way, advances with the most recent works [46,47] were achieved. The CF consists of an adverse event inherent to thyristor-controlled converters. It is a critical operational fault that severely compromises the system performance as it causes temporary interruption of the power transmitted by the converter, current surges in the key thyristors, and can cause significant disturbances in the operation of the AC network and other converters. For these reasons, it requires adequate protective measures to prevent the system collapse. After the end of the commutation process between two thyristors, the off-thyristor must remain with a reverse voltage at its terminals for a longer time than the minimum required to complete the deionization of the semiconductor. Due to operational characteristics, commutation failure is more likely to occur in the inverter station. The dominant causes are large fluctuations in DC current ( Id ), control failures in the thyristors triggering an instant, transient harmonic distortions, and, especially, voltage disturbances in the AC network. Energies 2023,16, 532 5 of 28 Although a commutation failure can be due to many causes, the most common is the voltage drop at the inverter station after an AC fault. In the past, some studies have been conducted to quantify the level of voltage drop that could occur on the inverter AC terminal without causing a commutation failure. They have been used to assess an HVDC system’s immunity to commutation failure [ 48 ]. However, by purely considering voltage magnitude and ignoring phase shifting and wave distortion, it is not enough to accurately determine the critical voltage drop at the inverter AC busbar. 2.2. Minimum Extinction Angle Criteria In this way, it is appropriate to analyze the valve commutation phenomena from the perspective of the extinction angle, described by the extinction angle equation in (1) , which comprises all the process variables. The extinction angle ( γn ) should be greater than ωtq (usually 7 ◦ ) for the thyristor that is not conducting to restore its blocking capacity fully. The time tq corresponds to the safe extinction interval after complete switching of the current Idn under the penalty of commutation failure [49]. γn=cos−1 √2Xcn Idn Vn +cosαn!(1) Considering commutation voltage at bus n ( Vn ) as sinusoidal under normal operating conditions and a ripple-free DC current, the commutating reactance is mainly composed of the short-circuit reactance of the converter transformer given by (2) , in which VnN represents phase-to-ground three-phase AC bus voltage. Xcn =xT(pu) VnN/√3nn √2IdnN/√3=xt(pu)VnN √2nnIdnN (2) Based on the denominator of (1) , it can be noted that the susceptibility of CF occurrence is directly related to the AC network and VSC-HVDC parameters. In ideal circumstances, it would be convenient to choose the VSC’s powers and the transmission line length so that, under a sudden AC voltage sag, γn was not less than a minimum limit, γmin . In other words, for such a line length, if Pk and Qk could be transiently changed during a contingency, and the corresponding critical parameters of (1) could also be determined so that γ>γmin is always possible. The fundamental condition concerns the threshold values to ensure γ>γmin should be reached immediately after a disturbance. However, as will be addressed in Section 2.5, the CF time interval is in the order of µs , and therefore, there is no reason to attempt control over Vk in the first post-fault instants since the controllers and the Phase Locked Loop (PLL) time constants are relatively more extended than the required time intervals for the first fault to occur. 2.3. AC Voltage Interactions: The Critical Indexes The first systematic studies were conducted in the early nineties [ 50 ] and advanced to MIF systems analysis only involving VSC [51]. Among the stability problems in LCC systems, CF is the most common and undesirable adversity in LCC-HVDC systems. It is an operational condition that must be strongly avoided because it leads to a DC voltage short-circuit resulting from the thyristor’s pulse sequence of the same leg (opposite pole), with the consequent interruption in power transmission, in addition to causing harmful overcurrents in the thyristor that has restored conduction. Equation (1) indicates that several events motivate CF, such as sharp commutation voltage sags in Vn , an increase in Id , decreases in β , or even control and triggering failures in the inverter station. However, CF mostly occurs due to the Vn deterioration, which usually justifies the focus on this quantity in this work and other related works. A particular concern in LCC-HVDC in the MIF configuration is the existing electrical interactions between AC and DC systems. The AC faults in the inverter busbar are critical, Energies 2023,16, 532 6 of 28 as well as those in electrically close LCC-HVDC inverters, which will cause the so-called concurrent or remote CF. In this context, the Multi-Infeed Interaction Factor (MIIF) is proposed in [ 52 ] as a metric for estimating the level of voltage interaction between two inverter’s infeed in the same electrical region. The smaller the distance between them, the higher degree of interaction, and the MIIF is used to reflect the effective interactions among any buses of the MIF system. By definition, the index represents the voltage reduction in the busbar n ( Vn ) followed by a 1% sudden drop in voltage of the busbar k(Vk), as described in (3). MIIFnk =∆Vn ∆Vk (3) Essentially, the index denotes how an HVDC link is influenced by another, playing a key role in studying the safe and stable operation of the whole EPS. Generally, for two buses infinitely far apart, MIIF = 0 and MIIF = 1 if two buses are connected to the same AC bus. If two LCC-HVDC have the same ratings, the interactions for MIIF ≤ 0.15 could be ignored, and each link behavior as a single-infeed system. There are some ways of equating the MIIF [ 53 ] considering grid parameters, for which knowledge of the admittance matrix is required. Equation (4) shows the analytical result. The impedance Znk is the interconnection impedance between the buses n and k , and Zkk is the sum of the impedance connected to the bus k. MIIFnk = Znk Zkk  (4) Dynamics simulations could be applied to reach a more precise measure of MIIF [ 54 ], in which the admittance matrix is not required. In practice, a variation of 1% at bus k is performed by switching a shunt inductance, and the resulting voltage Vnis observed. 2.4. The AC System Strength: Short Circuit Ratio Index This section discusses the main index regarding schemes with multiple LCC-HVDC links in the same AC network area. The Short-Circuit Ratio (SCR) is defined in (5) in which Sscn is the short circuit capacity of the AC network at the AC busbar with all remaining converters blocked divided by the MW rating of the n th HVDC scheme ( Pdn ). Once the Thévenin equivalent impedance ( Znn ) is known, the SCRn can still be written in terms of this normalized impedance expressed per unit. SCRn=Sscn Pdn =√3VnIsc Pdn =V2 n/Znn Pdn =1 Znn(pu) (5) In practice, Sscn is the short-circuit power of a symmetrical three-phase metallic fault at converter bus n , and Znn is measured by looking into the AC grid from the commutation AC bus at the fundamental frequency. Another meaningful and widely used indicator of the AC system strength is the Effective Short-Circuit Ratio (ESCR), calculated taking into account the reactive support at the nAC busbar ( QCn ). The ESCR n definition is given in (6). In this case, the capacity of VSC-HVDC (SVSC) can also be considered. ESCRn=Sscn+SVSC −∑QCn PDCn (6) In an MIF scenario where two or more LCC-HVDC inverters are connected in different but electrically close points in a restricted location, planning agencies must pay special attention mainly in low SCR conditions, when interaction amongst the links is likely to occur. In this context, the ESCR acts as a valuable predictor of the degree of the following interactions of concern [55]: • Commutation failure including the effects of its propagation, Energies 2023,16, 532 7 of 28 • Power and voltage instability, • Requirements on recovery control coordination during AC and DC faults, • Transitory overvoltage (TOV), • Resonances and harmonic interaction, • Requirements on coordination between HVDC controls. It is important to emphasize that although this article has adopted the CFII as an instrument for the ESCR gain assessment, other parameters related to the weakness of the EPS are also relevant, such as difficulties in the black start, Maximum Available Power (MAP), recovery times in voltage sags, and TOV in load rejection. 2.5. Contextualization: Background and Motivation Some works have studied the influence of VSC presence on the dynamic support of reactive power when an adverse transient disturbance occurs. In this context, the iCFII and its dependence on the VSC-HVDC operation point will be presented. It shows two parameters’ effects on CF susceptibility for two scenarios. Firstly, Figure 1a illustrates the LCC response to the CF for two Pk values for the same tie line length. When Pk goes from 0.9 to 0.75 pu, the successive CF evolves to a single fault, which denotes a direct improvement in the ESCR. The CF is extinguished by keeping 0.75 pu and increasing the tie line length to 180 km. Figure 1b shows the improvement in CFII when the line length is increased from 70 to 80 km with a fixed power dispatch. Similar behavior can be noted, and the CF is completely extinguished when the line length reaches 100 km. What can be seen is that an immediate improvement in transient state performances is soon observed connecting the VSC at a given operating point M, without any changes to the AC network equivalents in order to maintain the original ESCR. Time (s) I (pu) 12 12.1 12.2 12.3 12.4 12.5 0 1 2 3Pk = 0.9 pu, l = 100 km. i1 i3 i5 12 12.1 12.2 12.3 12.4 12.5 0 1 2 3Pk = 0.75 pu, l = 100 km. i1 i3 i5 12 12.1 12.2 12.3 12.4 12.5 0 0.5 1 1.5 Pk = 0.75 pu, l = 180 km. (a) Figure 1. Cont. Energies 2023,16, 532 8 of 28 Time (s) I (pu) 12 12.1 12.2 12.3 12.4 12.5 0 1 2 3Pk = 0.5 pu, l = 70 km. i1 i3 i5 12 12.1 12.2 12.3 12.4 12.5 0 1 2 3Pk = 0.5 pu, l = 80 km. i1 i3 i5 12 12.1 12.2 12.3 12.4 12.5 0 0.5 1 1.5 Pk = 0.5 pu, l = 100 km (b) Figure 1. Thyristors currents i1 , i3 and i5 as a function of VSC power dispatch and tie line length: ( a ) Thyristors currents as function of VSC power dispatch. ( b ) Thyristors currents as function of VSC-LCC tie line length. Figure 2illustrates the characteristic of the probability of CF. It could intuitively suggest that the faster the vector ~ S excursion towards maximizing Qk , the better the voltage support of the VSC in the AC network, iCFII3>iCFII2>iCFII1 . However, the improvement in control performance is observed only for voltage recovery a few seconds after the event. As will be seen in the discussion based on Figure 3, this fact does not affect the performance of the VSC in terms of contributing to avoiding CF. 𝐿(H) Probability of FC (%) Ponto M Sem VSC 𝑃 = 𝑃0 𝑄 = 𝑄0 𝑖𝐶𝐹𝐼𝐼 3 𝑃 = 𝑃0 𝑄 = 𝑄𝑚𝑎𝑥 Scc (MVA) 𝑖𝐶𝐹𝐼𝐼 1 𝑖𝐶𝐹𝐼𝐼 2 Figure 2. Graphic representation of iCFII gain. Figure 3a shows a window with detail of Figure 1a, containing all the thyristor currents in the transient time interval of the CF when Pk = 0.75 pu for 100 km. It can be seen that commutation failure shows a very short transient. Compared to Figure 3b, which is a one-step in Pk order, it is concluded that no matter how well designed, a power regulator would not be able to prevent the first CF, and it can contribute little or nothing since the resulting closed-loop time constant is 420 ms, that is, more than 70 times greater than the short period of occurrence of the phenomenon. Energies 2023,16, 532 9 of 28 11.995 12 12.005 12.01 12.015 12.02 Time(s) -0.5 0 0.5 1 1.5 2 I(pu) i1 i6 i5 i3 12 12.0055 T3 resumes conduction T5 failure T6 fired (a) 12 12.05 12.1 12.15 12.2 12.25 12.3 12.35 Time(s) 0.6 0.7 0.8 0.9 1 1.1 sk (pu) 12 12.1295 1.02 12 12.12 Times (s) 0.5 Idq (pu) Converter side dq-currents Id Iq (b) Figure 3. Inverter commutation failure simulation. Dynamic detail: ( a ) valve currents under commutation failure. (b) VSC Complex power—~ Sk. These findings indicate that VSC operation and location contribute to strengthening the AC system, which is observed by reducing the probability of CF. In addition, it is more probable that the dynamic contribution of ~ S for voltage support is in the coming stages of the CF, i.e., avoiding successive and remote (or concurrent) CF. In other words, VSC participation is linked to steady-state behavior. It indicates that attention should be focused on how much the VSC can increase the voltage stability margin of the LCC in a steady state. The following sections lead to a clear understanding. The highly transitory nature of CF phenomena may be quite complicated for analytical treatment because of system non-linearities. Added to the need for an exhaustive electromagnetic simulation to at least obtain statistics about the susceptibility of the CF, the search for a method that overcomes the difficulties listed so far and provides more reliable information is justified. From the next section on, this article presents a development that assumes that there are values for the ESCR for each system configuration such that a maximum γ excursion ( ∆γ ) exists following an AC fault that will not cause a CF. If the γ drop at the inverter is greater than ∆γ , the CF is presumed to occur. Figure 4illustrates extinction angle behavior for two situations of Figure 1b and can help with understanding. Such boundary value for ESCR is influenced by control system gains, type of fault and harmonics, etc. The detailed mathematical analysis focuses on determining the maximum ESCR that could be reached in the LCC-HVDC busbar to which the inverter would be CF-protected for a given level of short circuit in MVA. 12 12.2 12.4 Time (s) 0 50 100 150 200 i (Degree) Gamma with CF 12 12.2 12.4 Time (s) 0 5 10 15 20 25 30 i (Degree) Gamma without CF 5 Gamma detail Figure 4. Graphic representation γidynamics following AC fault. Energies 2023,16, 532 16 of 28 3.6. The hPVIF Calculation The strong relationship between Vn and the VSC set point and location has already been discussed, which leads to the hPVIF and iCFII indices being closely related. Mathematically, hPVIF(p) nk is the n th mode eigenvalue of the Jacobian matrix and is equivalent to the cross derivative that establishes a direct relationship between Vn to the electric distance between the buses n and k . For any VSC operating point, ~ Sk provides an expressive approximation of the critical point V2max in time. The Vn characteristic shows that its maximum occurs at ∂Vn= 0 for both hPVIF(p) nk , or in other words, only the numerator of hPVIF(p) nk (denoted by N(hPVIF(p) nk ) ) is enough to study the interesting phenomena. Expanding the numerator results in a polynomial function in the form of (21) , on which a is the set of constructive parameters of the AC grid, while independent variable x represents the operation points, and both will be used as input to the model. N(hPVIF(p) nk )(a,x) = an·xn+an−1·xn−1+. . . +a2·x2+a1·x1+a0(21) Solving N(hPVIF(p) nk )(a , x) = 0 for the dependent variable Vn as a function of the independent variables Pkand Qkresults in second-order polynomials functions (22). V(p) n(a,x) = n4 ∑ i=n aiP2 k+ n2 ∑ i=n3 aiPk+ n0 ∑ i=n1 ai!1/n (22) where n=n1+n2+n3+n4 is half the highest power of Vn , the vector of coefficients is a= [ Gkm Bkm Ysh θkm P2,3 Q2,3 ] and x= [ PkQK] . Bus 3 is the infinite bus, then, V3= 1 pu and θ3= 0 rad , and all the other node voltage and phase-shifting angles are referenced to this one. The scope of this work is restricted to the power system operating at a voltage regular operating point, according to the set of IEEE definitions regarding voltage stability [ 64 ]. In other words, an injection of reactive power ∆Qk> 0 at the VSC-HVDC AC bus leads to an increase (or at least not a decrease) in the voltage magnitude at all other system nodes, except the VSC AC bus itself due to the adopted control mode ( ∆Vn≥ 0, n6=k and ∆Vk= 0). Therefore, the index hPVIF(q) nk does not apply to the analysis on which Vn is calculating through hPVIF(q) nk = 0 since the power system will operate away from the point of voltage collapse (Figure 8). Since the hPVIFnk equations describe a physical system, the following considerations are stated: • By definition, the hPVIFnk corresponds to a critical point. Since Vn goes from increasing to decreasing at a time instant in which ∆Vn=0, this is a local maximum. • Physical constraints force the independent variable Pk∈R∗ +. • The quantities P3 and Q3 are determined by the system operator, being fixed inputs for Vn(a,x)for any line lengths. In summary, the section shows that it is possible to calculate the real and imaginary powers coordinates in the VSC to achieve the maximum ESCR in the LCC link, which means the upper limit of the voltage stability margin. The threshold Vn that provides the maximum possible ESCR (maximum ∆γ ) for each specified tie line length. In other words, the eigenvalue λ = hPVIFnk contains valuable information not only on the maximum ESCR that can be reached for each line length but also delimits the region of instability, calculating the point from which Pk values cause a reduction in SCR. 4. Technical Data of the hMIF System Topology To carry out the simulations to validate the proposed hPVIF and its derived contributions, a typical scheme of a hybrid dual-infeed HVDC system is established in a Energies 2023,16, 532 17 of 28 PSCAD/EMTDC electromagnetic transients (EMT) program. Its configurations and control structure employed in each HVDC subsystem are discussed as follows. 4.1. LCC-HVDC Accurate EMTP models of the LCC and VSC converters are addressed in this section. The LCC-HVDC and VSC-HVDC units’ ratings are summarized in Table 1. The LCCHVDC link adopts the first CIGRÉ HVDC benchmark standard testing system [ 65 , 66 ], widely applied over the decades as a standard reference and effective tool for HVDC system studies [67,68]. Table 1. HVDC constructive and operating parameters. Parameter VSC-HVDC LCC-HVDC System rating (MVA) 1000 1000 Converter AC bus rated voltage (kV) 230 230 DC rated voltage (kV) 630 500 DC side - R = 5 Ω - L = 1.2 H C = 0.25 mF C = 26 µF Rectifier station Switching frequency (Hz) - 50 Capacitor banks and shunt filter Capacity 11th order: 252 Mvar - 13th order: 252 Mvar Capacitors: 125 Mvar Converter transformer leakage reactance (pu) 0.19 0.18 Transformer turns ratio 62.5/13.8 kV 345/213.5 kV AC system 345 kV, 50 Hz SCR - 2.5 at 84◦ Inverter station Switching frequency (Hz) - 50 Capacitor banks and shunt filter Capacity 11th order: 252 Mvar - 13th order: 252 Mvar Capacitors: 125 Mvar Converter transformer leakage reactance (pu) 0.19 0.18 Transformer turns ratio 62.5/230 kV 230/209.23 kV AC system 215.05 kV, 50 Hz SCR - 1.75 at 75◦ 4.2. VSC-HVDC In this VSC-HVDC point-to-point model, the real power dispatch and the AC voltage regulation are achieved using the pq theory [ 69 ], described in (23) . The Instantaneous Power Theory—or p-qTheory—is defined by applying the Clarke Transformation in AC voltages and currents in the time domain. The terms p∗ and q∗ are the power components in the αβ stationary reference frame. The powers p and q are real and imaginary, respectively. Both are defined by average and oscillating components p=¯ p+˜ p and q=¯ q+˜ q , and the instantaneous complex power results from ~ s=p+jq . Since distortions and imbalances in voltages and currents are out of scope, the oscillating components ˜ p and ˜ q will be null, and only the fundamental frequency is considered. Figure 9illustrates the block diagram of the two-layer dq control scheme. The outer power control loop is noticed by gd(s) and gq(s) regulators and receives real power (or a DC voltage) and AC voltage references, according to the selector switch. The output current i∗ d is generated and denotes the average real power ( p ) that should be provided to (inverter) or absorbed from (rectifier) the AC network to ensure the DC link voltage regulation, while i∗ q is associated with necessary reactive power ( q ) required to control PCC voltage. The output voltage components of the inner vector current loop hd(s) and hq(s) pass through a dq -to- abc -frame transformation block to generate the reference ( vabc ) Energies 2023,16, 532 18 of 28 to be synthesized at the AC converter side. Both layers are composed of conventional proportional integral controllers. The DC cable represents an overhead line, whose length is fixed at 1000 km, and the AC line connecting buses k and n , whose length can be varied as desired. All these quantities will be evident in Figure 9. Harmonics and switching problems are out of the scope of this work; thus, the VSC is modeled as a three-phase voltage source in this work, i.e., IGBT devices are not used [70]. Rectifier VSC Model + - + - + - Inverter VSC Model - + - + - + Instantaneous Power Calculation Real power loop selector Control modes: 1 - dc-link voltage control 2 - real power control Figure 9. Grid interface of the VSC model and its vector control scheme. The related parameters of the control system are given in Table 2. Table 2. Control circuit parameters. Parameter Gain Time Constant hd(s)2 617 µs hq(s)0.1 4 ms gd(s)5 7.5 ms gd(s)10 7.5 ms "i∗ α i∗ β#=1 v2 α+v2 βvαvβ vβ−vα·p∗ q∗(23) 4.3. AC Grid Topology The AC voltage source is modeled as a Thévenin equivalent. The LCC link scaled version has source impedance parameters adjusted to achieve the appropriated ESCR. Table 3lists AC network parameters such as the voltage and frequency ratings and power capacities. Table 3. AC grid parameters. Parameter Value Rated power (MVA) 1000 Rated voltage (kV) 230 Rated frequency (Hz) 50 Transmission line Frequency-Dependent (Phase) Model Energies 2023,16, 532 19 of 28 5. Simulations Results Among those transient aspects strongly influenced by the ESCR in LCC systems, the commutation failure susceptibility is considered for the validity verification of the calculation method of the hPVIF. 5.1. Improved Immunity to Commutation Failure The central idea is to study the impacts on the CFII caused by the presence of a VSCHVDC and its direct contribution to the ESCR through a steady-state aspect. By proposing the auxiliary index iCFII, the influence of the VSC-HVDC will be studied according to a steady-state aspect, through the measuring of the static voltage stability margin, which is its operating point, analyzed from the instantaneous complex power vector in the plane. The fault level can be per-unitized by dividing it by the rated DC power of LCC-HVDC. In this work, CFII is per-unitizing the CF fault level and is described in (24). CFII =Worst critical fault (MVA) Pdn ×100 =V2 n ω·Lmin ·Pdc ×100 (24) The inductance Lmin (H) is the smallest fault inductance that never results in a CF, regardless of fault type [45], and ωthe angular frequency in rad/s for 50 Hz. 5.1.1. Evaluation Method of iCFII In [ 71 ], it is demonstrated that CF is influenced not only by the fault level but also by the point on the commutation voltage wave at which the short circuit is applied. Furthermore, it is shown that the higher susceptibility to CF occurs under an inductive three-phase fault than other fault types or impedance. Starting from a small value, the fault inductance is increased in increments of 0.5 mH until it reaches a value where no CF is induced for a period of 20 ms. A set of twelve faults is uniformly distributed in time for each inductance, applying every 1.666 ms for such an inductance value. 360/12 =30◦=⇒30◦×f 360◦=1.6667 ms To demonstrate the sensitivity of iCFII to the length of the transmission line segment, Table 4shows the profile of CFII considering the variation only in the line length and keeping the active power dispatch fixed at 0.7 pu. Although the CFII is in percentage values, the authors chose to show iCFII in MVA. Table 4. Local CFII values in the hMIF system—line length. Length (km) Vn(pu) CFII (%) Pk(pu) Qk(pu) iCFII (MVA) same bus 0.9949 24.159 0.7 0.3261 36.862 25 0.9798 23.666 0.7 0.2685 31.939 50 0.9704 23.258 0.7 0.2283 27.853 75 0.9648 23.162 0.7 0.2045 26.893 100 0.9607 22.436 0.7 0.1910 19.640 125 0.9579 22.557 0.7 0.1850 20.843 150 0.9554 22.318 0.7 0.1808 18.451 175 0.9534 22.054 0.7 0.1771 15.820 200 0.9516 21.968 0.7 0.1761 14.957 225 0.9501 21.925 0.7 0.1791 14.528 250 0.9489 21.883 0.7 0.1746 14.101 275 0.9477 21.897 0.7 0.1664 14.243 300 0.9464 21.826 0.7 0.1683 13.533 325 0.9451 21.505 0.7 0.1725 10.328 350 0.9439 21.355 0.7 0.1768 8.828 In order to verify the VSC performance, Table 5shows the behavior of the iCFII by incorporating the VSC power dispatch to each of the tie line lengths. In this way, the effect of the real power dispatch will be superimposed on every line length, which Pk will be the Energies 2023,16, 532 20 of 28 one related to the point of hPVIF(p) nk = 0. It is also included a column of the hPVIF(q) nk , in which will be a helpful index for the concept introduced in the next section. Table 5. Local CFII values in the hMIF system—VSC complex power. Length (km) hPVIF(p) nk hPVIF(q) nk V(c) n(pu) Vn(pu) 1e (%)2Pk(pu) Qk(pu) CFII (%) iCFII (MVA) same bus 0.003 0.396 0.9969 0.9955 0.148 0.3354 0.3546 24.368 38.960 25 0.009 0.198 0.9806 0.9821 0.152 0.3406 0.2797 23.716 32.439 50 0.012 0.396 0.9769 0.9745 0.766 0.3334 0.2357 23.306 28.336 75 0.012 0.593 0.9656 0.9706 0.506 0.3379 0.2095 23.082 26.099 100 0.012 0.790 0.9673 0.9683 0.098 0.3462 0.1914 22.739 22.670 125 0.011 0.986 0.9659 0.9669 0.105 0.3423 0.1787 22.602 21.297 150 0.011 1.182 0.9641 0.9669 0.287 0.3394 0.1706 22.422 19.491 175 0.006 1.374 0.9706 0.9653 0.547 0.3712 0.1549 22.332 18.599 200 0.004 1.567 0.9650 0.9649 0.003 0.3694 0.1487 22.185 17.128 225 0.002 1.758 0.9698 0.9647 0.528 0.3340 0.1516 22.332 18.599 250 0.001 1.948 0.9653 0.9646 0.075 0.3686 0.1380 22.112 16.399 275 −0.001 2.136 0.9609 0.9645 0.372 0.3337 0.1411 21.982 15.100 300 −0.003 2.323 0.9601 0.9646 0.469 0.3350 0.1360 21.997 15.244 325 −0.005 2.508 0.9593 0.9648 0.569 0.3355 0.1320 21.911 14.385 350 −0.006 2.691 0.9667 0.9650 0.179 0.3379 0.1267 21.812 13.392 cCalculated Vnfrom (22). 1Simulated values of Vn.2Relative error of V(c) n. Figure 10 illustrates the Vn dependence on the tie line length and VSC power dispatch. The impact on CFII in hMIF systems is graphically represented by the hatched area. For every length, the vertical distance between the red dotted line and solid line denotes the real gain of CFII, or iCFII, by just connecting the VSC at operating point Pk = 0.7 pu (scenario under consideration in Table 4). The vertical distance between the solid line and the line with point markers is the iCFII considering those Pk related to hPVIF(p) nk = 0 (scenario under consideration in Table 5). The superscripts with a single and double asterisk mean VSC connected with fixed and variant ~ Sk, respectively. 100 200 300 Tie line length (km) 0.94 0.95 0.96 0.97 0.98 0.99 1 Vn (pu) Vn in hMIF HVDC systems Vn Vn (*) 100 200 300 Tie line length (km) 20 21 22 23 24 25 CFII CFII in hMIF HVDC systems Without VSC CFII(*) CFII(**) Figure 10. Impact of VSC on Vnand CFII in hMIF systems. In addition to the definition of Section 5.1.1, the CF immunity quantified by the CFII is not absolute but relative to a given short-circuit level. If for a given AC bus voltage the LCC-HVDC system is immune to such short-circuit level in MVA, it is also immune to all voltages above this first voltage. Figure 11b,c brings a further graphical representation of Energies 2023,16, 532 21 of 28 the iCFII phenomenon. All voltage values in the range between V(f) n (hatched area) will return CFII equal to and greater than the maximum value itself (V(i) n). 4 6 8 10 12 14 16 18 20 Time (s) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P, Q and S(pu) (p1) (p2) (p3) (p4) V(i) n(l1) V(i) n(l2) Pk Qk-175 Sk-175 Qk-225 Sk-225 (a) V(i) n(l1) V(f) nV(f) n (p1) (p2) hPVIFp > 0 hPVIFp < 0 4 6 8 10 12 14 16 18 Time (s) 0.93 0.94 0.95 0.96 0.97 0.98 Vn (pu) (b) V(i) n(l2) V(f) nV(f) n (p3) (p4) hPVIFp > 0 hPVIFp < 0 4 6 8 10 12 14 16 18 Time (s) 0.93 0.94 0.95 0.96 0.97 0.98 Vn (pu) (c) Figure 11. LCC-HVDC inverter station AC bus voltage as function of ~ Sk : ( a ) VSC-HVDC powers of inverter station. (b) LCC-HVDC AC voltage for 175 km. (c) AC LCC Voltage for 225 km. Figure 11 complements the understanding of what has been presented so far, in addition to supporting the discussion of the next section. 5.1.2. Apparent Line Length through hPVIF Equation (22) can not be used to calculate Vn at any time since the referred equations derive from the boundary condition ∂Vn= 0. Then it is valid only at the maximum Vn . The use of the eigenvalues hPVIF(p) nk and hPVIF(q) nk as complementary indexes is explored in this section. This section introduces the concept of the apparent line length of the transmission line seen from the AC side of the LCC-HVDC inverter. By choosing the proper Qk (and thus Pk ), it is possible to simulate a shorter line length than the actual length, resulting in a higher value of iCFII. This is proved as follows. Energies 2023,16, 532 22 of 28 Considering a given Qk for two line lengths l1 and l2 , with l1<l2 , then Vn(l1)>Vn(l2) , as shown in Figure 11. Therefore, to match the effect of varying power dispatch on different line lengths, the two final voltages of interest must be equal, as in Equation (25). V(f) n(l2)=V(f) n(l1)=⇒∆Vn(l2)=V(i) n(l1)−V(i) n(l2)+∆Vn(l1)(25) Equation (A4) results in the following: Q(f) k(l2)=hV(i) n(l1)−V(i) n(l2)+∆Vn(l1)ihPVIF(q) nk  (−1) l2 +Q(i) k(l2)(26) Or, Q(f) k(l2)=V(i) n(l1)−V(i) n(l2)+∆Qk(l1)hPVIF(q) nk l1hPVIF(q) nk  (−1) l2 +Q(i) k(l2)(27) In (27) , the superscript ( i ) indicates quantities associated with the maximum point, and the subscript ( f ) is related to that point on which the quantity of interest is desired. The value ∆Qk(l2) establishes the necessary reactive power on bus k for l2 that would result in the same iCFII for l1when Qk(l1)is delivered. Since hPVIF(p) nk is null at the maximum point, it could not be directly used in apparent line length. However, it is helpful for hPVIF(q) nk calculation since the latter is nonzero at that point. The practical result of the first term of the equation is to emulate, in a simplified way, a shorter line length by adjusting Qkpower dispatch. Because of the large number of tie line length combinations, the concept of apparent line length will be demonstrated for only two lengths: 175 and 225 km, being therefore extensible to all pairs of line lengths. Figure 11b,c show the tie line equivalence for the lengths chosen when Pk is adjusted from 0.05 to 0.75 pu and Qk values necessary for equivalence is presented in Table 6. Table 6. Apparent line length equivalence between 175 and 225 km. Region Vn(pu) 1Qk(pu) 1Q(c) k(pu) e (%) 2 hPVIF(p) nk >00.9560 0.1901 0.1880 1.126 hPVIF(p) nk <00.9560 0.1616 0.1615 0.0368 1Simulated values of Qk.cCalculated Qkfrom (27). 2Relative error of Q(c) k. 5.2. Discussion and Partial Conclusions Intended for the assessment of voltage static stability in hybrid multi-infeed HVDC systems, this study is vital to the power system and power electronics fields combined since it addresses a valuable open topic related to the LCC-HVDC connection into the grid with low short circuit capacity. The results presented corroborate the mathematical statements and physical conclusions exposed through Sections 3and 5. The CF is an adverse condition associated with a weak system, and the AC voltage deterioration is quite sensitive to tie line length and VSC power dispatch. These results attest to the effectiveness of the index hPVIF(p) nk as a predictor of the maximum ESCR that could be provided for a given line length and operating point. It is also observed that LCC-HVDC is more susceptible to propagating CF when LCC and VSC inverters operate at long electrical distances from each other. As a form of validation, Tables 4and 5show an additional column with the error obtained for the voltage value at the maximum point from e( % ) = (|Vn−V(c) n|)/Vn× 100, where the superscript c refers to the value calculated through (22) . Such errors can be attributed to (1) numerical approximations, mainly rounding errors, since the representation of values relative to state variables was limited to four decimal places, (2) approximations inherent to the bus admittance matrix, and (3) due to model assumptions such as when Energies 2023,16, 532 23 of 28 separately assuming ∆Q= 0 and ∆P= 0 for hPVIFnk calculations. This is valid for applying the gradient concept, but it is not a very accurate assumption for a physical system since to keep Vk=1, ∆Q=0 and ∆P=0 cannot be decoupled. 6. Conclusions This paper introduced a new metric to evaluate the degree of influence of the VSCHVDC on inhibiting commutation failures in LCC-HVDC systems arising from AC faults from the increase in the ESCR. A detailed theoretical analysis is developed considering VSC-HVDC participation factors hPVIF(p) nk and hPVIF(q)nk and its associated contributions intended for the assessment of voltage static stability in hybrid multi-infeed HVDC systems. It is possible to obtain satisfactory results with the proposed method as a function of VSCHVDC operation set point and line length separation. Commutation failures and most of their adverse consequences could be avoided when the LCC-HVDC operates at the calculated maximum short circuit capacity. Oriented computer simulations performed in PSCAD/EMTDC denote the validity of the proposed method. Results indicate close correspondence between the simulationbased results and the predicted results from the analytical expressions. The parametric analysis provided in Section 5was carried out using ranges of tie line length and VSCHVDC power dispatch. To support the findings, Tables 4and 5show that values of iCFII corresponding to hPVIF(p) nk = 0 reflect the ability of the VSC-HVDC to improve AC voltage regulation, enhance the overall system strength and make the LCC-HVDC less susceptible to commutation failure. The presented technique proved efficient at increasing system performance and has become technically convenient as a voltage static stability evaluation tool. In this context, based on daily, weekly, or longer time horizon orders, the system operator can predict the impact on the dynamic behavior of the ESCR when there are changes in the network and when EPS assumes different modes of operation to meet consumer demands. Through proper choice of VSC-HVDC set points, the outcomes of this work show that the apparent line length formulation also reveals considerable accuracy in emulating more advantageous line lengths from an ESCR improvement point of view. It is a valuable concept not only for supporting the new transmission line projects and expansion at the planning stage but also as an enhanced tool for operation engineers dealing with short and long-term decisions, providing elements for the precise application of preventive measures that contribute to the safe integration of LCC-HVDC and VSC-HVDC to very weak AC grids. Author Contributions: Conceptualization, D.O.; methodology, D.O.; validation, D.O. and G.C.B.L.; writing—original draft preparation, D.O., G.C.B.L. and D.H.; writing—review and editing, D.O., G.C.B.L., D.H., M.A., E.G.-D. and J.M.C.; project administration, M.A.; funding acquisition, M.A. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the CAPES (Higher Education Personnel Improvement Coordination—Brazil)—Finance Code 001, the CNPQ (National Council for Scientific and Technological Development—Brazil) and by the European Union—NextGenerationEU—University of Seville, grant number MSALAS-2022-19823. Data Availability Statement: Not Applicable. Acknowledgments: Thanks to the Universidad de Sevilla for the contributions and for providing support from professors and students and to the Federal University of Rio de Janeiro for providing equipment and physical structure. The researchers express gratitude to the national and international authorities for funding the project and the LEMT group from the University of Rio de Janeiro. Conflicts of Interest: The authors declare no conflict of interest. Energies 2023,16, 532 24 of 28 Appendix A. Equations for hPVIF Calculation The hPVIF(p) nk and hPVIF(q) nk Calculation In V -control mode, Vk is maintained constant for any changes in Pk , Vn is nonzero, and Qk will be automatically adjusted to changes in Pk not only to meet the operator’s dispatch but also to meet the stated target of ∆Vk=0. Complete Jacobian matrix considering slack bus (bus 3) in power flow formulation (∆V(3)and ∆δ(3)).          ∆Q(1) ∆Q(2) ∆Q(3) ∆P(1) ∆P(2) ∆P(3)          =L = JQV M = JQδ N = JPV H = JPδ            ∆|V(1)| V0 ∆|V(2)| V0 ∆|V(3)| V0 ∆δ(1) ∆δ(2) ∆δ(3)            (A1) 1—Elimination of bus 3: First Kron reduction In this step, the non-interest AC buses, such as the Thevenin equivalent source buses, have already been eliminated. As ∆V(3)and ∆δ(3)are null, the equation above becomes.          ∆Q(1) ∆Q(2) ∆Q(3) ∆P(1) ∆P(2) ∆P(3)          =         JQV11 JQV12 JQV13 JQδ11 JQδ12 JQδ13 JQV21 JQV22 JQV23 JQδ21 JQδ22 JQδ23 JQV31 JQV32 JQV33 JQδ31 JQδ32 JQδ33 JPV11 JPV12 JPV13 JPδ11 JPδ12 JPδ13 JPV21 JPV22 JPV23 JPδ21 JPδ22 JPδ23 JPV31 JPV32 JPV33 JPδ31 JPδ32 JPδ33                   ∆|V(1)| V0 ∆|V(2)| V0 0 ∆δ(1) ∆δ(2) 0           (A2) In preparation for the elimination of bus 3 (slack bus), the following sequence of operations will be performed for a first Kron reduction: (1) Permutation of the 3rd and 5th rows to place ∆Q(3) and ∆P(3) in the fifth and sixth lines of vector b , respectively. (2) Permutation of the 3rd and 5th columns to place ∆V(3) and ∆δ(3) in the last two lines of vector x , and (3) permutation of the 1st and 2nd columns to place ∆V(2) in the first row of vector x.          ∆Q(1) ∆Q(2) ∆P(2) ∆P(1) ∆Q(3) ∆P(3)          =         JQV12 JQV11 JQV13 JQδ13 JQδ12 JQδ11 JPV32 JPV31 JPV33 JPδ33 JPδ32 JPδ31 JQV32 JQV31 JQV33 JQδ33 JQδ32 JQδ31 JPV12 JQV21 JPV13 JPδ13 JPδ12 JPδ11 JPV22 JPV21 JPV23 JPδ23 JPδ22 JPδ21 JQV22 JPV11 JQV23 JQδ23 JQδ22 JQδ21                   ∆|V(2)| V0 ∆|V(1)| V0 ∆δ(2) ∆δ(1) 0 0           (A3) After the matrix inversion in (A3) , followed by the first Kron reduction, a second matrix inversion is performed, resulting in (A4).      ∆Q(1) ∆Q(2) ∆P(2) ∆P(1)      =       J(r1) QV12 J(r1) QV11 J(r1) Qδ12 J(r1) Qδ11 J(r1) QV12 J(r1) QV11 J(r1) Qδ12 J(r1) Qδ11 J(r1) PV22 J(r1) PV21 J(r1) Pδ22 J(r1) Pδ21 J(r1) PV12 J(r1) PV11 J(r1) Pδ12 J(r1) Pδ11              ∆|V(2)| V0 ∆|V(1)| V0 ∆δ(2) ∆δ(1)       (A4) To formalize the mathematical basis of this article and to make it possible to obtain hPVIFp and hPVIFq directly from (A5) , the coordinate Q will be constrained as a constant Energies 2023,16, 532 25 of 28 parameter so that the directional derivative on the P axis is calculated. Physically, to decouple the effects of the active power variations from those caused by reactive power incremental changes, it is firstly suggested in [ 26 ] to assume ∆Q= 0 and ∆P= 0 separately. With no reactive power injections, the last Kron reduction can be applied. ∆Q(1) ∆Q(2)="J(r2) QV12 J(r2) QV11 J(r2) QV22 J(r2) QV21 #  ∆|V(2)| V0 ∆|V(1)| V0  (A5) As this article deals with a dual hMIF system, from now on, the sub-indices n for LCC and k for VSC ( k= 1 and n= 2) will be adopted. As ∆|V(10)|= 0, considering Equation (A6) and using JRgeneric terms, the system is rewritten as: ∆Q(k) ∆Q(n)=  J(q) Rkn J(q) Rkk J(q) Rnn J(q) Rnk  ∆|V(n)| 0(A6) ∆Q(k)=J(q) Rkn ∆V(n)+0⇒∆V(n) ∆Q(k) =1 J(q) Rkn (A7) hPVIFqnk =1 J(q) Rkn (A8) Similar methodology is applied in (A4) with proper permutations to calculate hPVIFp . ∆P(k) ∆P(n)=  J(p) Rkn J(p) Rkk J(p) Rnn J(p) Rnk  ∆|V(n)| 0(A9) ∆P(k)=J(p) Rkn ∆V(n)+0⇒∆V(n) ∆P(k) =1 J(p) Rkn (A10) hPVFpnk =1 J(p) Rkn (A11) References 1. 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