Power system harmonic analysis under geomagnetic disturbances

Geomagnetic disturbances influence the operation of transmission systems by injecting dc like electric currents into the system causing half cycle saturation of iron core magnetic circuits, generation of harmonic currents and excessive heating. Main harmonic generating devices are directly grounded transformers. The level of the half-cycle saturation depends of the level of geomagnetically induced currents and system parameters. GIC induced harmonics will flow to other devices in the system, depending on system topology and parameters and can influence system protection and operation. Accurate computation of these effects is of paramount importance. The paper proposes a high fidelity time domain modeling approach for analysis of the waveform distortion due to GIC and the propagation of the waveform distortion throughout the system. The high fidelity model incorporates the non-linear characteristics of iron core transformers, frequency dependent characteristics of transmission lines and grounding systems and other components of the power system. While GIC does not typically enters generator circuits by virtue of their grounding practice, generators are affected by GIC as harmonics propagate into generating units via the step up transformers.

SECTION I. Introduction

Geomagnetic disturbance (GMD) occurs when ionized particles from solar wind interact with earth magnetic field. Solar activity [1] has an 11 year cycle. During the peak cycle this disturbance intensifies and may influence the normal operation of power systems. Geomagnetically induced current (GIC) is one of the major results of GMD. This current enters the power system at grounding points as the disturbed electric field in the soil creates voltage differentials among the various ground points. In general, areas in high latitude and high earth resistivity are subject to higher level of GIC [2]. The magnetic core of the transformer will experience half cycle saturation, and magnetizing current will drastically increase which is rich in harmonics. The harmonic levels generated by this process can reach levels that may destroy (melt the windings) the transformer as it has happened at the Salem nuclear plant in NJ during the GMD in March 1989 [3]. Prior research has reported harmonics ranging from second to 20^th order with a pattern that includes both even and odd harmonics in descending magnitude with the harmonic order [4], [5]. Reference [4] indicates significant level of even order harmonics with the second harmonic higher than the third harmonic, the forth harmonic higher than the fifth harmonic and so on. All harmonics increase in a nonlinear relation to GIC, although the slope is different for each harmonic. Reference [6] claims that harmonic magnitudes are sensitive to the air core reactance of transformers. When transformers are saturated, substantial levels of flux flow through air and the air path reluctance affects the performance of saturated transformers. Reference [7] claims that, transformers with three-limb cores have harmonics concentrating on lower order, while single phase transformers will have substantial harmonics of higher order. During these conditions, transformers will consume more reactive power as the magnetizing current is distorted, affecting voltage stability of the system [5]. Harmonics may also adversely affect protective relaying algorithms leading to mis-operations. There are documented instances of the effects of harmonics on the protection of static VAR compensators on the Hydro-Quebec system during the March 1989 GMD event [8]. In this case, relay misoperations due to the GMD led to the Hydro-Quebec blackout. Another case of similar effects occurred in 2003; a power outage occurred in Malmö, Sweden during a geomagnetic storm [9]. The major reason was the loss of a 130kV transmission line, which was tripped by an overcurrent relay. Investigation showed that the relay had higher sensitivity at third order harmonic than the fundamental frequency current. In this case, the relay characteristic value for third harmonics was much lower than the fundamental frequency. Another concern is related to current transformers which can saturate during GMD with serious effects on the protection scheme [10]–​[12]. Reference [10] claims that harmonic restrained relays, such as transformer percentage differential relays, may fail to detect faults current during a GIC event. Reference [11] reports usage of harmonics as indicator of GIC impact on transformers. Reference [12] concludes on the basis of simulations, that although there exist CT saturation even in small GIC level, modern relays with proper algorithms are able to discriminate against GIC generated harmonics.

Previous GMD events as well as GMD studies clearly show that the effects can be quite damaging to the power system. For this reason, models that accurately represent the effects of GMD on the power system are important for realistic assessment of the effects of GMDs. While individual components have been extensively studied under GICs, similar comprehensive models for system wide studies are not as well developed. Many approaches use a DC network model to compute the flow of the dc component into the system and then project the effect of the dc on the system. These approaches do not capture the interaction of dc, fundamental and harmonics on a network wide basis. This paper presents a method that does capture these interactions in GMD impact studies. The proposed modeling approach for the power system network integrates dynamic models of transformers, power lines, grounding, generators, etc. and captures the nonlinear magnetics as well as the frequency dependence of the various parameters of power lines, grounding systems, etc. The paper presents the modeling approach and provides results of the levels of harmonics generated by GIC as well as the propagation of the harmonics throughout the system. Comparisons of the proposed model to other methods reported in the literature are provided. These comparisons indicate that the level of harmonics can be miscalculated if simpler models are used.

SECTION II. Modeling of Devices

In time domain simulation, the major problem is to solve the ordinary differential equation (ODE) describing the dynamics of the system. The general form of an ODE system is: \begin{equation*} \dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t), t) \tag{1} \end{equation*} where $x(t)$ is the states of the system and f is the derivative of the states at time $t$. Because of the large-dimension and nonlinearity of f, numerical solvers are needed to obtain the transient of the system. These solvers calculate x(t) at each time step based on solution from previous time steps. Trapezoidal method, a solver commonly used in power system transient analysis, assumes f varies linearly between time steps, so the state x(t) can be directly calculated from the values at last time step, x(t-h). The numerical solver in this paper is quadratic integration method [13]. Instead of linear approximation used in trapezoidal method, quadratic integration assumes the states and functions vary quadratically between time steps. This additional freedom enables the quadratic method to better approximate the dynamics, and a higher accuracy is achieved. This improvement enables the solver to simulate system with highly non-linear devices. In general, the following equations yield the implicit solution in quadratic method: \begin{align*}& x_{m}-\frac{h}{3}f(t_{m}, x_{m})+\frac{h}{24}f(t_{n}, x(t_{n}))=x(t_{n-1})+\frac{5h}{24}f(t_{n-1}, x(t_{n-1})) \tag{2} \\ & x(t_{n})-\frac{2h}{3}f(t_{m}, x_{m})-\frac{h}{6}f(t_{n}, x(t_{n}))=x(t_{n-1})+\frac{h}{6}f(t_{n-1}, x(t_{n-1})) \end{align*}

In which, $x(t_{n}), x(t_{n-1})$ are the states at time step $t_{n}, t_{n-1}$, and $t_{m}$ is the mid-point between them. $h$ is the time step size.

A. Quadratized Dynamic Model (QDM)

An object oriented method [14] is implemented to model the devices in the system. For each device, the dynamic of internal states x(t) is described by the ordinary differential equation sets in equation (1). In which i(t) is the vector of through variables at the terminals. First two equation sets capture the linear behavior of the system, and the third equation set describes the nonlinearities, if present. In case of nonlinearities of order higher than two, additional state variables are introduced to reduce the order of nonlinearities to two. Devices [15], [16] in this standard mathematical form are then integrated to yield a quadratic algebraic companion form (QACF) for each device. Subsequently, the QACFs of all devices are used to form the network model [17]. \begin{align*} \mathbf{i}(t)& =Y_{eqx1}\mathbf{x}(t)+D_{eqd1}\dot{\mathbf{x}}(t)+C_{eqc1} \\ 0& =Y_{eqx2}\mathbf{x}(t)+D_{eqxd2}\dot{\mathbf{x}}(t)+C_{eqc2} \tag{3} \\ 0& =Y_{eqx3}\mathbf{x}(t)+\begin{Bmatrix} \vdots\\ \mathbf{x}(t)^{T}\left\langle F_{eqxx3}^{i}\right\rangle \mathbf{x}(t)\\ \vdots \end{Bmatrix}+C_{eqc3} \end{align*}

Figure 1.

Single phase transformer model

B. Saturable Transformer

Transformer is one of major sources of harmonics during GMD events. GIC flowing through transformer windings will saturate the iron core, forcing flux to flow through air and tank walls. Therefore, much larger magnetizing current is needed when there is DC offset in flux. The distorted magnetization current will have high levels of harmonics. A non-linear magnetizing inductance is needed to represent the saturation of the core. Figure 1 depicts the electric circuit for a single-phase transformer where $l_{fa}$ represents the nonlinear magnetizing inductance. This inductance couples electric circuit and the magnetic circuit together. When the excessive flux saturates the transformer core, reluctance in magnetic circuit increases and the equivalent magnetizing inductance will decrease. Therefore, under saturation condition, the magnetizing current will increase. Equation (2) presents an example of the variable reluctance $R_{x}$ depending on the flux $\phi_{x}$ through the core. A high exponent $n$ represents the nonlinearity of reluctance. \begin{equation*} R_{x}=\frac{i_{0}}{\phi_{0}}\left(\frac{\phi_{x}}{\phi_{0}}\right)^{n} \tag{4} \end{equation*}

Three-phase transformers are modeled in this paper. In addition to electrical circuits for each phase, the magnetic circuit for these transformers should also be considered. For example, for three-phase core form transformers, the windings for phases share the same iron core, so the interaction of flux generated from each phase winding is important. A magnetic circuit describing the phase fluxes and reluctance of limbs and yokes of the magnetic circuit is implemented. Another important device for GIC studies is the autotransformer. Figure 2 presents the equivalent circuit for a single-phase three-winding autotransformer. In autotransformers, a path exists for direct GIC flow between windings. The distribution of GIC will be affected by the parameters of the transformer and the outside circuits connected to the high and low sides of the autotransformer.

Figure 2.

Single phase autotransformer model

Figure 3.

Positive sequence impedance of transmisson line

Figure 4.

Zero sequence impedance of transmisson line

C. Broadband Transmission Line

The parameters of the transmission line are frequency dependent due to skin effect. Alternating current concentrates near the surface of the conductor as opposed to uniform distribution for DC. The equivalent resistance increases while the equivalent inductance decreases as the frequency increases. Figure 3 and 4 presents the equivalent impedance for positive and zero sequence of a three-phase transmission line. There are different methods to model the frequency dependent transmission line. Some analyze the response of line in frequency domain, and transform a series of response back to time domain [18]. These methods are accurate but the implementation is not straightforward since numerical convolution is needed during the transformation. Another approach is to use network of lumped elements to approximate the frequency response of the transmission line, which is used in this paper. To simulate the system with harmonics accurately, we implement a broadband transmission line model based on the geometry of the conductor. Figure 5 presents an example of the model applied to a single conductor. Along the radius, the conductor is divided into several layers. With the assumption of a constant current density in each layer, the impedance network can be formulated using the self and mutual impedance of layers. These impedances are calculated based on physical structure of layers. The current distribution at the various legs of the impedance network depends on the frequency of the currents, so the frequency property of a conductor is reproduced in time domain. This procedure can be extended to all conductors in transmission lines. At frequencies ranging from DC to 20^th order harmonic, this model can reproduce the theoretical results accurately with minimal computational efforts.

Figure 5.

Broadband transmission line model for a single conducor (left: Equivalent circuit model; right: Cross-section of a round conductor)

Figure 6.

Ground impedance

D. Frequency Dependent Grounding System Model

The ground impedance is an important parameter that affects the amount of GIC current that will enter the transformer neutral. The ground impedance is strongly dependent on frequency. This means that it will be different impedance to the flow of dc current and different to the flow of harmonics. The detailed modeling of frequency dependent grounding systems is given in [19]. Figure 6 presents the results for the substation ground impedance of the first transformer.

SECTION III. GMD Testing System

The system in Figure 7 is used to investigate generation and propagation of harmonics due to GMD. The area of interest is the shaded area, which is shown in Figure 8. There are two generators and two loads. Three 115kV transmission lines transfer power from sources to loads. The parameters of the various devices including the saturable core transformers are provided in Table I.

Figure 7.

GMD testing system overview

Figure 8.

GMD study area

The impact of GMD is simulated with equivalent DC sources connected at the ground points of the system as shown in Figure 9. The magnitude of DC voltage between the grounds of two substations is determined by the distance between the two substations $l$, the magnitude of GMD field $M$, and the orientation angle $\theta$. In this paper, we assume a constant magnitude GMD field in a fixed direction. The DC that will appear at the various ground points is computed with this assumption.

SECTION IV. Results

The impact of GMD on the test system is investigated in this section. First, the GIC current flowing through the neutral of transformers is presented in Figure 10 for GMD event 1. As we can see, after GMD starts, the GIC gradually increases with a time constant of about 0.4 seconds. In steady state, the GIC through transformer 4 is 65A, and the GIC through transformer 3 is 40A. In this test case, the direction of the GMD field is aligned with the two transmission lines from BUS1 to BUS4. As a comparison, we have computed the GIC using the DC network of this test system. The GIC through transformer neutral is 86A and 31.2A for transformers 4 and 3 respectively. We can observe that using the DC network model, the ratio of GIC at the two transformers is about 3, which is the ratio of the corresponding line lengths. The linear relationship exhibited with the DC network model is invalid due to the nonlinearities of the system and the different saturation level of the transformers.

Figure 9.

Modeling of GMD at ground points

Table I. Example system parameters

Next, the harmonics in the transformers phases and neutral are analyzed in Figures 11 and 12. The harmonic magnitudes are normalized with respect to the fundamental. The DC component increases as the transformer is in half cycle saturation, and the magnetizing current has a non-zero DC offset. Note the profound generation of even order harmonics due to the GMD. The figures provide the harmonics for three different levels of GMD: 5 V/km, 10 V/km and 15 V/km. Even and odd harmonics exist and the magnitude decreases with the harmonic order. The levels also increase as the GMD increases.

Figure 10.

DC current at neutral of transformers

Figure 11.

Harmonic analysis of transformer3 phase current

Figure 12.

Harmonic analysis of transformer 4 phase currents

Figures 13 and 14 show neutral harmonics for transformers 3 and 4 respectively. The DC component is the largest component. The percentage of second order and forth order harmonics is not large, though there are apparent increase in corresponding component in phase current. The reason is that these harmonics mainly consist of positive and negative sequence, so the sum of three phase harmonics is small.

Figure 13.

Harmonic analysis of transformer 3 neutral current

Figure 14.

Harmonic analysis of transformer4 neutral current

Figure 15.

Symmetric component of harmonics in transformer 3

Figure 16.

Symmetric component of harmonics in transformer 4

Third and sixth order harmonics include more zero sequence components, which can be validated in Figures 15 and 16. This is why the third and sixth harmonics are larger than the other components in the neutral.

Harmonics from GMD may affect the operation of generators. Figures 17 and 18 show the harmonics at the terminals of generators. Since the step-up transformer is connected in $\Delta-\text{Y}$, the DC component and the zero sequence components of the harmonics cannot flow through the transformer into generators. The even order harmonics from saturated transformers do have large impact on the generators. The second order and forth order harmonics rise drastically.

Another influence of GMD is the increased consumption of reactive power. Figure 19 shows the increase of reactive power in generators. In this case the increase is moderate. However, for transmission systems that may exhibit resonances and amplify certain harmonics, the increase may be substantial. More investigations are needed to determine the effects of GMD on generators.

Figure 17.

Harmonic analysis of generator1 phase current

Figure 18.

Harmonic analysis of generator2 phase current

Figure 19.

Reactive power of generators

SECTION V. Conclusions

We presented a method for accurate simulation of the effects of GMD on power systems using a time domain method. The method relies on high fidelity models of transformers, transmission lines and grounding systems for accurate simulation of DC flow and harmonics. The models provide the level of harmonics at transformers lines and generators. The even order harmonics such as second and forth order harmonics are the most obvious phenomenon during GMD. These harmonics can challenge the system operation. The half-cycle saturation of transformers is the major source of these harmonics. Increasing demand of reactive power is also observed in the system, which requires corresponding action to ensure voltage stability.

ACKNOWLEDGMENT

The authors gratefully acknowledge the support of PG&E.