Assessing the Static Security of the Italian Grid by Means of the N-1 Three-Phase Contingency Analysis ^†

Abstract

The ongoing replacement of synchronous machine generators (SMs) with converter-interface generators (CIGs) is raising the voltage unbalance of power systems, affecting power quality and grid stability. This paper focuses on a key power quality index for power systems, i.e., the voltage unbalance factor. The purpose of this work is twofold. First, it presents the generalization of a three-phase power flow algorithm developed by University of Padova, named PFPD_3P, to assess the voltage unbalance factors of power systems supplied by CIGs. In particular, it is demonstrated that CIGs can be modelled as three-phase PV/PQ constraints embedding their positive-, negative- and zero-sequence admittances. Then, the concept of three-phase contingency analysis is introduced. Indeed, for static security evaluation, the classical single-phase contingency analysis may no longer be sufficient, as it lacks power quality computations, e.g., voltage/current unbalance factors. Numerical simulations evaluating the unbalance factors due to different generation mix scenarios and contingencies are tested on the Italian extra-high-voltage/high-voltage (EHV/HV) grid. The choice of this network relies on its representativeness, as CIGs are the majority of new installations in the Italian generation mix.

1. Introduction

1.1. Motivation

Global warming concerns, growing fossil fuel costs, market liberalization and government incentives [1] are only some of the factors leading towards a massive integration of variable renewable energy sources (vRES) into worldwide power systems. The replacement of traditional thermal plants with converter-interface generators (CIGs), however, could heavily threaten the power quality [2] and the stability of power systems [3] if proper countermeasures are not considered.

Focusing on the static stability, which is the fundamental pillar of AC power system stability analysis, contingency analyses are typically used to list the most severe contingencies/outages that might occur on a specific power system under analysis.

Hence, power flow methods are exploited to verify the security of steady-state operation. This is fundamental, especially in a power system with high penetration of CIGs, where sudden generation variations could happen. However, these analyses typically consider the single-phase positive-sequence electrical quantities, neglecting the real three-phase configuration of power systems. This configuration should be considered, especially in contexts with high presences of CIGs. Differently from SMs, indeed, CIGs are characterized by higher negative-sequence impedances, leading to higher voltage unbalance factors in the network and to potentially dangerous unbalance conditions [4]. Thus, proper contingency analyses should also consider the impact of CIGs on power quality [5,6], especially in terms of current/voltage unbalance factors for the HV and EHV networks. Hence, a three-phase contingency analysis concept should be introduced for a comprehensive steady-state contingency analysis. For this purpose, reliable three-phase power flow algorithms [7,8,9] combined with contingency analysis programs are needed.

1.2. Literary Review

After the flourishing of several power flow techniques in the 1960s of the last century [10], the possibility of estimating the effects of grid outages and contingency was exploited, leading to the achievement of a comprehensive knowledge of the topic in the 1970s [11,12,13]. However, the increasing complexity of power systems, combined with the need for a reliable system, makes the topic still relevant [14,15,16]. These methods, however, do not consider the impact of contingencies on power quality, especially on voltage unbalance. Although CIG access to the grid [17,18] creates other power quality problems, such as the harmonics [19,20] and small signal stabilization problems [21], the problem of voltage unbalance is especially crucial, considering its increase with longer lines and higher loads [22].

To this end, in order to assess the impact on the unbalance factor, a novel AC matrix three-phase power flow algorithm, named PFPD_3P, has been developed and validated by the authors in [9]. This method is based on a hybrid approach; it exploits computations in both sequence and phase frames of reference, and it has recently been generalized to assess power systems in the presence of CIGs [4].

1.3. Contribution

The contribution of the paper is twofold. First, the three-phase power flow algorithm PFPD_3P is generalized to compute the unbalance factors in three-phase grids in the presence of CIGs [4]. In particular, it is given the rationale to model CIGs by means of admittances, by considering their positive-sequence control (PV/PQ control at positive-sequence) and the negative/zero-sequence data. Therefore, a set of new three-phase admittances is defined, and the paper shows how to embed them in the “all-inclusive” three-phase admittance matrix [9]. Then, the concept of three-phase contingency analysis is introduced: this computational concept allows integrating the classical N-1 contingency analysis techniques with the three-phase power flow algorithm PFPD_3P. This is a new way of thinking about N-1 contingency analysis: it shows what the impact is in terms of power quality when one device at a time is removed from the operation. Therefore, contingency analyses assessing the steady-state power quality of power systems can be performed. With PFPD-3P, the power quality is assessed in terms of unbalance factors [9,22]; however, the N-1 contingency analysis can be extended to perform harmonic studies.

Numerical tests are performed on the Italian EHV/HV grid, where steady-state analyses considering four different generation mix scenarios and contingencies are illustrated.

2. Brief Recalls on PFPD_3P

PFPD_3P [9] is the three-phase generalization of the single-phase power flow algorithm PFPD [23]. In the same manner as PFPD, also in PFPD_3P the slack generator is modelled as a quasi-ideal three-phase current source. The approach uses the “all-inclusive” three-phase bus admittance matrix with the inclusion of loads, generators and also the slack generator. In PFPD_3P, the iterative cycle uses both the sequence and the phase frames of reference; for this reason, it is defined as a hybrid power flow approach. The equations from (1) to (5) are iteratively exploited to find the three-phase power flow solution. The formulae from (1) to (5) are represented with reference to the k-th iteration:

$${\underset{\_}{\mathit{u}}}_{\mathit{G}}^{\mathit{ABC}}={\underset{\_}{\mathit{Z}}}_{\mathit{Geq}}^{\mathit{ABC}}\left[{\underset{\_}{\mathit{\Delta }\mathit{i}}}_{\mathit{G},\mathit{c}\text{\_}\mathit{q}}^{\mathit{ABC}}-{\underset{\_}{\mathit{Y}}}_{\mathit{GL}}^{\mathit{ABC}}{\mathbf{\left(}{\underset{\_}{\mathit{Y}}}_{\mathit{LL}}^{\mathit{ABC}}\mathbf{\right)}}^{-\mathbf{1}}{\underset{\_}{\mathit{\Delta }\mathit{i}}}_{\mathit{L}\mathbf{,}\mathit{c}}^{\mathit{ABC}}\right]$$

(1)

$${\underset{\_}{\mathit{u}}}_{\mathit{L},\mathit{c}}^{\mathit{ABC}}=-{\left[{\underset{\_}{\mathit{Y}}}_{\mathit{LL}}^{\mathit{ABC}}\right]}^{-1}{\underset{\_}{\mathit{Y}}}_{\mathit{LG}}^{\mathit{ABC}}{\underset{\_}{\mathit{u}}}_{\mathit{G}}^{\mathit{ABC}}+{\left[{\underset{\_}{\mathit{Y}}}_{\mathit{LL}}^{\mathit{ABC}}\right]}^{-1}{\underset{\_}{\mathit{\Delta }\mathit{i}}}_{\mathit{L},\mathit{c}}^{\mathit{ABC}}$$

(2)

$${\underset{\_}{\mathit{\Delta }\mathit{i}}}_{\mathit{L},\mathit{c}}^{\mathbf{0}\mathit{PN}}={-{\underset{\_}{\mathit{Y}}}_{\mathit{L},\mathit{P}}\odot \left(\mathbf{1}-{\left|{\underset{\_}{\mathit{u}}}_{\mathit{L},\mathit{c},\mathit{P}}\right|}^2\right)/\left({\underset{\_}{\mathit{u}}}_{\mathit{L},\mathit{c},\mathit{P}}\right)}^{*}$$

(3)

$${\underset{\_}{\mathit{\Delta }\mathit{i}}}_{\mathit{G},\mathit{c}}^{\mathit{ABC}}={\underset{\_}{\mathit{Y}}}_{\mathit{Geq}}^{\mathit{ABC}}{\underset{\_}{\mathit{u}}}_{\mathit{G},\mathit{c}}^{\mathit{ABC}}+{\underset{\_}{\mathit{Y}}}_{\mathit{GL}}^{\mathit{ABC}}{\left[{\underset{\_}{\mathit{Y}}}_{\mathit{LL}}^{\mathit{ABC}}\right]}^{-1}{\underset{\_}{\mathit{\Delta }\mathit{i}}}_{\mathit{L},\mathit{c}}^{\mathit{ABC}}$$

(4)

$${\underset{\_}{\mathit{\Delta }\mathit{i}}}_{\mathit{G},\mathit{c}\text{\_}\mathit{q},\mathit{P}}=-j\left[Im\left({\underset{\_}{\mathit{u}}}_{\mathit{G},\mathit{c},\mathit{P}}\otimes {\underset{\_}{\mathit{\Delta }\mathit{i}}}_{\mathit{G},\mathit{c},\mathit{P}}^{\mathbf{*}}\right)\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}/\hspace{0.17em}{\underset{\_}{\mathit{u}}}_{\mathit{G},\mathit{c},\mathit{P}}^{\mathbf{*}}$$

(5)

It is worth noting that the three-phase power flow can be computed only by means of the five iterated Formulas (1)–(5). These equations bring about a lower CPU time for PFPD_3P with respect to the other existing methods and a greater solution accuracy (converging also for power tolerance up to 10⁻¹⁵ p.u.) [9].

In addition, it is important to highlight that PFPD_3P is an open algorithm; it can be easily self-implemented in a common PC by any researcher. Further details can be found in [9].

3. Converter-Interface Generator Modelling

In this paper, it is shown that CIGs can modelled, for power flow purposes, by means of PV or PQ constraints. The choice of the constraint typology depends on the inverter control strategy: if the positive-sequence active and reactive power are controlled, a PQ constraint is adopted; otherwise, if the positive-sequence voltage magnitude and the positive-sequence active power are imposed, a PV constraint is exploited. Due to this approach, the inverters can be included in the “all-inclusive” three-phase admittance matrix [9]. Certainly, the same following reasoning can be extended to loads interfaced by means of converters, such as frequency converter asynchronous motors.

Figure 1 shows how an inverter-based RES is treated in the power flow depending on different control strategies.

For modelling a PQ-type inverter, the following approach must be exploited. The converter control operates to inject the positive-sequence active power P_AC,P and to impose the scheduled positive-sequence reactive power Q_AC,P:

P_PQ = −P_AC,P

(6)

Q_PQ = −Q_AC,P

(7)

It is worth noting that in the power flow, the signs of P_PQ and Q_PQ must be negative, as the powers are injected into the network. The positive-sequence admittance can be computed for each inverter using the following relation:

$${\underset{\_}{y}}_{PQ,P}={\displaystyle \frac{P_{PQ}}{{\left|{\underset{\_}{U}}_{AC,rtd,P}\right|}^2}}-j{\displaystyle \frac{Q_{PQ}}{{\left|{\underset{\_}{U}}_{AC,rtd,P}\right|}^2}}$$

(8)

where U_AC,rtd,P is the positive-sequence rated voltage of the converter AC bus.

By contrast, for a PV-type inverter, the control works in order to introduce into the AC network all the active power generated by the RES, P_{AC, P}.

$$P_{PV}=P_{AC,P}$$

(9)

Furthermore, the electric drive also fixes the positive-sequence voltage magnitude |U_AC,P|.

The positive-sequence admittance can be computed by the following equation:

$${\underset{\_}{y}}_{PV,P}=-{\displaystyle \frac{P_{PV}}{{\left|{\underset{\_}{U}}_{AC,P}\right|}^2}}+j{\displaystyle \frac{Q_{PV}}{{\left|{\underset{\_}{U}}_{AC,P}\right|}^2}}.$$

(10)

In these buses, the reactive power estimation is the same as that for estimating reactive power in synchronous generators [10].

For the negative- and zero-sequence network, instead, the converter’s symmetrical component behavior must be considered (see Figure 2). It is well known that inverters exploit IGBT switches. If the grid rules do not require a negative-sequence current injection, when the voltage unbalance factor at the converter terminals exceeds the limit, the inverter can be represented as an open circuit in the negative-sequence network, in accordance with the approach reported in [4,12].

The zero-sequence network can be also considered as an open circuit, so its admittance can be imposed as a very small value (e.g., 10⁻⁵ p.u.). The VSC converter phase matrix can be immediately computed by means of the Fortescue transformations. Figure 2 shows the positive-, negative- and zero-sequence representations of the inverter.

4. Three-Phase Static Security: PFPD_3P Evaluation on the Italian Transmission Network Scenarios

4.1. Unbalance Factors after the Increasing Share of RES

In this section, PFPD_3P is applied to compute the Italian transmission network unbalance factors. The results are fundamental to knowing the real-world EHV/HV Italian grid behavior. In Figure 3, the red lines represent the 380 kV transmission lines, the green ones the 220 kV lines and the blue and yellow ones the sub-transmission lines (150 kV and 132 kV, respectively); their total length is equal to 72,000 km. The network includes 7800 buses and, in the current generation mix, 728 SMs supply loads. In the loads, the asynchronous share is assumed to be equal to 5%. It is known, in fact, that asynchronous motors help to reduce the negative-sequence voltage magnitude by means of their low negative-sequence impedance values. To perform the power flows of the different scenarios, standard geometrical configurations of the electrical lines are chosen for the different voltage levels.

The tower structures, imposed for the 380 kV and 220 kV lines, are reported in Figure 4. PFPD_3P allows computing the real network unbalance by exploiting the MCA (multiconductor cell analysis) method [13,14,15] for multiconductor line modelling.

From an operational standpoint, the Italian transmission network is split, along the Italian Peninsula, into seven areas. Starting from the current generation mix scenario, the active power generated in the inverter-based RES plants is increased up to their operational limits; at the same time, the active power generated in the synchronous generators is decreased by the same value. This operation is performed for each of the seven areas in the grid in order to avoid line overloads. The scenario in which the inverter-based RES power is increased is named renewable increasing. In Figure 5, the total active power injected by inverter-based RES plants in the two scenarios is represented for each of the seven areas; the blue columns indicate the active power generated in the current generation mix, while the green columns show the active power generated in the renewable increasing scenario.

Figure 5 also reports the active power increase in the inverter-based RES in each of the seven operational areas. The total active power increase is equal to 6411 MW. If in an SM the active power is zeroed, the machine is turned off. The red columns in Figure 5 represent the difference in terms of active generators between the beforementioned scenarios.

The application of PFPD_3P to the Italian transmission network gives the results shown in Figure 6, Figure 7 and Figure 8. The voltage unbalance factors in the transmission sections (380/220 kV buses) are represented in Figure 6, while the same quantities for the sub-transmission sections (150/132 kV buses) are shown in Figure 7. The blue lines represent the current generation mix scenario, whereas the dotted red ones represent the renewable increasing scenario. It is possible to note a dramatic increase of the voltage unbalance factor for both transmission and sub-transmission sections. This increase is due to the SMs turning off. The maximum voltage unbalance factor rises from about 0.9% up to 3.6% in the transmission buses of Figure 6.

The average voltage unbalance factors in each operational area are represented in Figure 8. The diagram in Figure 8 allows the identification of where the voltage unbalance factor increases are located. In the areas where the majority of the inverter-based RES power plants are located, i.e., the southern regions, the average voltage unbalance factor strongly rises. The large number of inverter-based RES, mainly in the southern operational areas, requires turning off many SMs; the disconnection of rotating machines does not allow for draining the negative-sequence currents. The inverter’s negative-sequence circuit, due to its large impedance value, does not help in reducing the voltage unbalance factor. The unbalance factor in the transmission and sub-transmission levels has an impact on the distribution network.

Discussion

The results must be interpreted as follows. The real three-phase configuration of power system devices is not perfectly symmetrical. Non-transposition of the lines or an unbalanced load, for example, are sources of unbalanced current and thus voltage unbalances. This means that such devices absorb positive-, negative- and zero-sequence currents depending on device structure.

Certainly, the power balance between generation and consumption (load and losses) is always valid; however, different sequences are involved. Therefore, the following relation regarding for the computation of complex power is valid:

S = 3E_dI_d* + 3E_iI_i* + 3E₀I₀*

(11)

To determine the value of all the current and voltage sequence components at all the sections, PFPD_3P could be exploited.

The control of the converters could be useful in order to avoid the circulation of negative- and zero-sequence currents.

4.2. Unbalance Factors after Power Quality Mitigation Actions

By considering the scenario with increased RES contributions, power quality mitigation actions are carried out to reduce the voltage unbalance factors. Voltage unbalance factor reduction can be performed by exploiting the following strategies:

• Insertion of synchronous condensers: They drain the negative-sequence currents, due to their low negative-sequence impedance values. The insertion of synchronous condensers to reduce the voltage unbalance factors is not widely investigated in the literature. In fact, they are considered as devices which are able to regulate the voltage and to increase the fault level in transmission network sections. Nine synchronous condensers are connected to the grid; these devices are positioned in the three southern operational areas. Black lines and columns in Figure 6, Figure 7 and Figure 8 show the results of this scenario.
• Insertion of synchronous condensers and phase transposition technique: To the synchronous condenser scenario, line transposition is added: lines longer than 50 km are transposed. This technique symmetrizes the magnetic behavior of the transmission lines. Unfortunately, transmission lines are rarely transposed [16,22]. Green lines and columns in Figure 6, Figure 7 and Figure 8 show the results of this scenario.

Figure 6 demonstrates the effectiveness of synchronous condenser insertion. The black curve presents a reduction in the sections with a high voltage unbalance factor. These sections are positioned in the southern regions, i.e., where the synchronous condensers are installed. The maximum value decreases from about 3.6% down to 2.7% in the transmission buses of Figure 6. In Figure 7, similar behavior is shown for the sub-transmission sections. The average voltage unbalance factors in Figure 8 present a reduction in all the operational areas (see red and black columns); this reduction is felt more in the southern regions, as all the synchronous condensers are installed there. This scenario confirms the validity of synchronous condensers as mitigation devices for the voltage unbalance factor.

The PFPD_3P algorithm can consider also the presence of phase transpositions in transmission lines. This technique reduces the effects of the asymmetric conductor disposition in the AC links. This approach, combined with synchronous condensers, strongly decreases the voltage unbalance factors in all the sections. In Figure 6, a strong reduction in the voltage unbalance factors of the transmission sections can be seen, and a similar behavior is reported in Figure 7 for the sub-transmission buses. The maximum value decreases from about 2.7% down to 1.4% in the transmission buses. In a large number of the Italian transmission network sections, the voltage unbalance factor limit of 1% is respected. Figure 8 confirms this behavior: the voltage unbalance factors further decrease compared with the previous scenario.

5. Three-Phase Contingency Analysis on the Italian Transmission Network Scenarios

In Section 1, the concept of three-phase contingency analysis is introduced. This concept is introduced because the classical contingency analysis tools (e.g., the classical N-1) could yield incomplete results when analyzing power systems. Line contingencies, power outages and network element outages of services could bring about not only overvoltages and overcurrents, but also lower power quality indexes.

By considering the network scenarios of the Italian transmission network described in Section 4, three-phase N-1 contingency analyses (by means of PFPD_3P) are performed to evaluate the voltage unbalance factors.

Figure 9 reports the voltage unbalance factor variations of all the network buses after the outage of the most-loaded electrical line of the Italian network. The evaluation is performed by considering the current generation mix scenario and the two scenarios with the increasing presence of (mitigated) RES (see Section 4). It can be observed that there is a general incrementing of the voltage unbalance factors (ΔUF > 0) with respect to the pre-contingency situation (ΔUF = 0). In particular, after the considered contingency, dramatic voltage unbalance factor increments of almost 3% can be noted with respect to the pre-contingency situation.

Figure 10 shows the voltage unbalance factor variations after the outage of the power plant (SM) with the highest scheduled power. Figure 11 reports the voltage unbalance factor variations of all the network buses after the outage of the most-loaded bus in the Italian network. For all the considered scenarios, this case shows a decrease in voltage unbalance factors (ΔUF < 0) with respect to the pre-contingency situation (ΔUF = 0).

This fact confirms the high variations in voltage unbalance factors that the network could experience during different generation scenarios.

It is worth noting that such variations could be not only positive (power quality worsening) but also negative (power quality enhancement), thus confirming the necessity of the introduction of three-phase contingency analysis to manage the forecasting complexity of network power quality.

6. Conclusions

The paper presents the generalization of PFPD_3P to consider the impact of converter-interfaced generators (CIGs) in real power systems. CIGs are modelled by means of three-phase PV and PQ constraints, according to their specific steady-state set point controls and to their sequence-frame-of-reference circuit models. Extensive applications of PFPD_3P are performed using the Italian transmission network, where a large number of CIGs are going to be installed. In particular, the impact of the increasing amount of renewable energy sources (RES) interfaced with the Italian power system is studied. In these simulations, it is shown that in Italian southern areas, i.e., where the inverter-based RES plants are mainly installed, a dramatic increase in the voltage unbalance factors is observed. It is shown that PFPD_3P can also compute the effects of different mitigation technologies in terms of unbalance factors.

Based on the generalization of PFPD_3P, the paper also introduced the three-phase contingency analysis concept, an algorithm improving on the classical contingency analysis methods. In fact, beyond the indication of overvoltages and overcurrents, indications of power quality, i.e., voltage unbalance factors, can also be carried out. Hence, three-phase contingency analysis could be a powerful tool for the planning and operation of future power systems.

However, the computational methodologies presented in this paper are not complete enough to make an overall power quality evaluation. Another fundamental power quality parameter is, in fact, the steady-state harmonic behavior that, so far, must be evaluated separately. Future works could further extend the three-phase contingency analysis method to add further fundamental evaluations of other power quality parameters e.g., the contingency of steady-state harmonic behavior.