Metrics and Strategies for Design of DC Bias Resilient Transformers

Geomagnetic disturbances (GMDs) give rise to geomagnetically induced currents (GICs) on the earth’s surface which find their way into power systems via grounded transformer neutrals. The quasi-dc nature of the GICs results in half-cycle saturation of the power grid transformers which in turn results in transformer failure, life reduction, and other adverse effects. Therefore, transformers need to be more resilient to dc excitation. This paper sets forth dc immunity metrics for transformers. Furthermore, this paper sets forth a novel transformer architecture and a design methodology which employs the dc immunity metrics to make it more resilient to dc excitation. This is demonstrated using a time-stepping 2D finite element analysis (FEA) simulation. It was found that a relatively small change in the core geometry significantly increases transformer resiliency with respect to dc excitation.


I. INTRODUCTION
T RANSFORMERS, with their ability to change voltage levels and thereby facilitate efficient transfer of electric power, form an integral part of the electric power grid. Solar flare activity causes transient fluctuations in the geomagnetic field which induce earth surface potential (ESP) that can be 3 to 6 volts/km or higher [1]. The ESP generates geomagnetically induced current (GIC) which complete their path through transformer grounded neutral connections. These GICs could have amplitudes of up to 200A and a fundamental frequency range of 0.001 to 0.1 Hz and hence appear as quasi-dc currents in the transformer conductors.
The existence of such quasi-dc currents causes multiple problems in the power grid. The GIC flowing through the transformer windings causes half-cycle saturation due to the dc flux which quickly saturates the highly permeable transformer core. This half-cycle saturation causes a loss of odd half-wave symmetry leading to even harmonics. Indeed, because of the nonlinear magnetics, a rich spectrum of both odd and even harmonics is produced. Further, the reduction of the transformers magnetizing inductance will cause the transformer to appear as a significant inductive load and increase the transformer VAR consumption.
Power transformer saturation reduces the apparent impedance seen by relays, and if this apparent impedance is within the operating zone of the relay unnecessary tripping may occur. Overcurrent ground relays could also malfunction due to the increased zero-sequence current caused by transformer saturation [1]. The effect of harmonics on overcurrent relays has been studied in [2], [3], [4], and [5]. It has been found that the effects of GIC vary with the transformer configuration with the three-legged core configuration being the most resilient. During half-cycle saturation, the leakage flux links with multiple adjacent structural members resulting in excessive heat loss. DC bias and GIC related heating of the transformer core and its structural members has been studied in [6], [7], [8], [9], [10], [11], [12], and [13]. One of the concerns resulting from this heating is that winding insulation adjacent to the structural member may be heated excessively, resulting in thermal degradation of the insulation. The second concern is that an intense, local heat source might rapidly decompose adjacent insulation and generate a free gas bubble in the oil whose existence and mobility could cause or contribute to a dielectric breakdown [1]. The formation of the gas bubble and its role in the dielectric breakdown of the oil has been studied extensively in [14], [15], [16], [17], and [18].
In [1], the authors provide a detailed description of the problems associated with GMD events on the power grid. The authors also provide GIC data at different locations and GMD events with its associated severity during the period from 1969 -1972. In [19], the authors describe the measurements of GIC at various substations within the Kola power grid in Russia. Specifically, it was observed that the neutral currents were rich in third harmonic component. Moreover, it was reported that the GIC results in the saturation of auto-transformer core and harmonic boost which could cause relay operation and/or transformer heating. In [20], the authors summarize the observations of GICs in the high voltage power grid in China. In [20], the authors note that the GIC reached a peak value of 75.5 A at the Ling 'Ao nuclear power plant transformer during a GMD event on November 9th -10th 2004. It was observed that the frequency of these GICs varied between 0.01 -0.0001 Hz. Further observations were made by the authors during GMD events on December 14th -15th 2006. During this period, peak GIC values of up to 13 A and 16.6 A were measured at Shanghe substation and Ling 'Ao nuclear power plant.
In view of the effects of GICs on transformers, efforts have been made to improve transformer dc bias resiliency. In [21], the authors have proposed a strategy to counteract the dc bias flux produced due to GIC. Therein, this is achieved by means of a DC flux blocker (DFB) is connected in series with the transformer tertiary winding to cancel the GIC mmf. The device consists of a low power and high current voltage source and two circuit breakers. THD of one of the primary phases is used to control the voltage of the DFB.
In [22], the authors present an auxiliary winding-based method to cancel the dc bias flux. The proposed topology has been tested with two 10 kVA, 380 V/ 200 V transformers with the auxiliary winding rated for 10% of the secondary winding capacity.
In [23], the authors have proposed a patented transformer design with compensation windings to protect transformers against the adverse effects of GIC. The three compensation windings, corresponding to each phase, are tied together and connected to the ground terminal. The effectiveness of this design has been verified through simulations.
In [24], the authors have proposed a 1 kHz transformer design with a passive negative magnetic reluctance structure (NMRS) to increase its dc bias resiliency. The NMRS consists of a flat copper coil connected to a compensation capacitor and is inserted in the air gap between the cores. The effectiveness of this architecture was tested using a 500 WEE55-based core prototype.
Amongst the studies described above [21] and [22] describe active methods to counteract the effects of GIC, and hence require additional equipment and control strategy.
In [23], the passive compensation windings are a low-cost alternative but will require an increase in the transformer size and these windings remain unutilized in the absence of GIC. While [24] presents an interesting solution it is mainly for high frequency (kHz range) low-power applications. The architectures proposed herein differ from those discussed previously in that they only require a modest modification of the transformer core.
The presence of GIC in transformer windings decreases the lifetime of the transformer and hence there needs to be more resilience to dc current. To calculate how resilient a transformer is to dc current, metrics are needed. In this paper, such dc current immunity metrics are set forth and used to design a dc current tolerant transformer. Time-stepping FEA analysis is used to validate the results.
While three-phase transformers are clearly of the greatest interest, to introduce the proposed metric and the proposed transformer architecture, this work will focus on the singlephase case. The three-phase case will be considered in followon work.
The paper is organized as follows. Section II proposes dc current immunity metrics. Section III provides a baseline transformer design methodology (without dc immunity) and provides the details of a baseline design. Section IV studies a simple gapped core transformer architecture along with its design methodology and explores the effect of dc current tolerance on this study. Section V sets forth a novel transformer architecture, its associated design methodology and studies the effect of dc current on the proposed transformer topology.

II. DC CURRENT IMMUNITY METRICS
The goal of this work is to describe metrics that could be used in optimization-based transformer design to address robustness of transformers with respect to dc bias currents. In order to formulate these metrics, a single-phase transformer will be assumed. Further, it is assumed that the secondary is open circuited for the purposes of formulating a metric.
To proceed, it will be useful to define the following nomenclature. If x represents some quantity, we will decompose it into a constant (dc) term x dc and an ac term x ac which is not sinusoidal in general but has zero time-average value. Thus To proceed, let us assume that using FEA, a magnetic equivalent circuit (MEC), or some other method, the no-load anhysteretic relationship between the primary current and flux linkage may be expressed as Clearly, (2) and (3)  It follows the ac components of these variables may be expressed as v p,ac = r p i p,ac + dλ p,ac dt (5) Since the primary resistance is typically very small, (5) suggests that the ac component of the primary flux linkage is not greatly affected by dc offsets or even the primary current. Thus, the ac component of the flux linkage may be well estimated as where the initial condition is found such that the resulting flux linkage has no dc component. In (6), and throughout this work, time t = 0 is taken to be at some point after a periodic steady state has been achieved. then where λ p,acpk = √ 2V p /ω e . Now let us suppose it is desired to limit the peak primary current to i p,pka . The corresponding peak allowed value of primary flux linkage may be computed as Since the ac component of the ac flux linkage is known, it follows that the corresponding dc value of flux linkage may be expressed where λ p,acpk is the peak value of the ac component of the flux linkage waveform, which is readily found from (6). The dc component of the current for this condition may be expressed The current i p,dc is the dc bias current the transformer can tolerate with an ac flux linkage waveform λ p,ac so that the steady-state peak primary current remains below i p,pka . This primary side dc current can be used either as a constraint or as a design objective to be minimized.
At this point, it is useful to consider an illustrative example. To this end, suppose that with the secondary open circuited, the primary flux linkage and primary current are related by (2), where This characteristic qualitatively looks correct, but the saturation is 'softer' than is typical. It is used here because it is an analytically invertible form (to facilitate a simple example). The parameters L l , L m , and λ sat may be loosely interpreted as leakage inductance, magnetizing inductance, and saturated magnetizing flux linkage, respectively. From (12) we can show where δ = sgn(λ p ) and L p = L l + L m . For this example, the assumed parameters are L l = 1.15 mH, L m = 2.29 H, and λ sat = 1.17 Vs, which are very loosely based on a 208 V to 120 V 5 kVA transformer (selected to be comparable to some studies later in the paper).
For this system, using the rated primary-side voltage, we obtain λ p,acpk = 0.780 Vs. If we, somewhat arbitrarily, take the acceptable value of open-circuit peak primary current to be 1 pu (where here per unit is defined by the peak of the waveform), we obtain i p,pka = 34.0 A, and from (9) λ p,pka = 1.19 Vs. Then from (10) we obtain λ p,dc = 0.412 Vs. Using this value and (8) in (11) we obtain i p,dc = 3.96 A, as the allowed dc current. Fig. 1 illustrates the situation. Therein, the trace labeled 'primary current with no dc offset' represents the normal no-load primary current without dc offset. The waveform is symmetrical. Adding a dc component to the voltage to create a dc offset ('allowed dc offset') causes half-cycle saturation such that the peak primary current is 1 pu (again, per unitized such that the peak is 1 pu). While this level of saturation is no doubt survivable, it is not desirable. Together (9)-(11) provides a process to determine the acceptable dc current for which the peak no-load primary current (essentially the magnetizing current) is below the allowed value. An alternate formulation of the problem is based on the rms value of the primary current rather than being based on the peak value. The rms value of the primary current may be expressed For an allowed rms value of the primary current i p,rmsa , and an ac flux linkage waveform λ p,ac , (14) can be solved for the corresponding value of the dc component of the primary flux linkage λ p,dc which may then be substituted into (11) to calculate the corresponding value of dc bias primary current i p,dc . Returning to the previous example, in which the peak no-load primary current was limited to 1 pu (peak base), the corresponding value of allowed primary no-load rms current i p,rmsa = 9.39 A, or about 0.389 pu (rms base).

III. BASELINE TRANSFORMER DESIGN
The theme of this work is related to the resilience of transformers with respect to dc bias current. To this end, a baseline application will be considered. Although the concern over resiliency is often focused on transmission-scale transformers, here we will focus on low-power transformers to illustrate the principles involved.
In particular, the design of a single-phase, 60 Hz, 5 kVA 208 V to 120 V transformer has been conducted. Some principle design constraints are (i) the no-load secondary voltage must be within 2% of the rated value, (ii) no-load primary current is less than 10% of the base current (inpractice it will not be close to this), (iii) the regulation is less than 5%, and (iv) the peak value of inrush current is 2 √ 2 times the rated rms primary current. The design objectives are to minimize electromagnetic mass and aggregate loss, where the aggregate loss is a weighted loss assuming 10% of the time at no-load, 40% of the time at resistive half-load, and 50% of the time at full-load. The transformer geometry is shown in Fig. 2 and 3. The model and design process used for the study are set forth in [25], which provides extensive details and code. The main analysis is a nonlinear magnetic equivalent circuit. This approach is used for the purpose of expositional efficiency.
There are two updates from [25], however. First (7.8-14) of [25] was changed to k l = 2r c /l c and (7.5-20) of [25] was changed to d xw = d xu (N xud + 1), as suggested in the errata. Finally, the variable r c was set to zero so that a single lamination could be used to produce the transformer.
The magnetic equivalent circuit used in the design of the transformer is set forth in Fig. 4. Specifications are listed in Table 1. Following the methodology of [25], one obtains the Pareto-optimal front shown in Fig. 5. Therein, Design 90 is indicated with a red circle. The name 'Design 90' is from the fact that it was the 90 th design in the order of increasing mass along the Pareto-optimal front. Chief parameters of this design are listed in Table 2. Note that the two primary coils are connected in parallel, as are the two secondary coils. This is the case throughout this work.
Observe that in this baseline design, no provision has been made to address the impact of dc current. In Fig. 6 is applied to the transformer. In (15), r p is the primary winding resistance, and i p,dc is the dc component of the current which must be tolerated. A time-stepping FEA study was conducted using Ansys Electronics Desktop 2021, wherein a 2D model of design 90's primary winding was excited with the voltage waveform described in (15). The secondary was open-circuited. The results are shown in Fig. 6. Therein, in the upper trace, the dc term in (15) is not included and the peak magnetizing current is less than 150 mA; in the lower trace the dc term is included with i p,dc = 7 A, resulting in a peak current of over 40 A. Note the profound difference caused by the rather modest dc excitation. In the next section, increasing the resilience of the transformer is explored.

IV. IMPACT OF DC BIAS IMMUNITY ON TRANSFORMER DESIGN -GAPPED SOLUTION
One method to increase the dc bias resiliency of a transformer is to utilize a gapped core. In this approach, the transformer core is divided into two cores separated by an air gap. The air gap increases the reluctance in the path of the dc and magnetizing flux thereby making the transformer less susceptible to half-cycle saturation at the expense of decreasing transformer magnetizing inductance. To accurately compute the magnetic performance of the transformer with the air gap the original MEC is modified to the one shown in Fig. 7. Therein the reluctance term, R g , represents the air gap reluctance.
It is interesting to conduct a transformer design study using the gapped core for a single-phase, 60 Hz, 5 kVA, 208 V to 120 V transformer as in section III. In addition to the constraints mentioned in section III, one additional constraint that has been introduced in this design study is that the peak primary current, under dc bias condition of 7 A (20% of rated peak primary current), is half the rated peak primary current which in this study is equal to 17.0 A. The constraint on maximum mass was removed -otherwise there were no viable designs. Fig. 8 compares the Pareto fronts of the gapped design study and the baseline design study. It can be observed that the baseline designs, without the dc bias constraint, exhibit a much lower mass for the same aggregate power loss. This is because, in order to satisfy the dc bias constraint, the design algorithm increases the magnetic cross-section to compensate, to some extent, for the increase in reluctance due to the gap. Another important distinction between the designs is that the core of most of the baseline designs utilized M19 steel and a small percentage were made of M43 steel, whereas the gapped core designs utilized of M19, M43, and M47 steel. One of the reasons behind this is the lower relative permeability of M43 and M47 steel compared to M19 steel thereby providing a higher reluctance path to the magnetizing flux. Note the permeability was a function of the specific steels considered, not inherently their loss classification.
Design 248 of the gapped core designs is indicated with a black circle, and its selected parameters are listed in Table 3. An electrical parameter of interest for transformers is the no-load current. It can be seen by comparing the selected baseline and gapped core designs that the gapped core design has a much higher no-load current (2.39 A rms as opposed to 226 mA rms). This is primarily because of the air gap reluctance. Fig. 9 shows the primary current response of the gapped core design 248 with no bias current and with a dc bias current 532 VOLUME 10, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.    of 7 A (the value in the specification). It can be seen that the peak value of the primary current is 19.5 A, which is near the  allowed value of 17A. The reason for the discrepancy is that the design code is based on a 3D magnetic equivalent circuit, whereas Fig. 9 is based on a 2D finite element analysis. In the next section, an alternate transformer design is considered which addresses the dc bias resiliency while maintaining a high magnetizing inductance.

V. IMPACT OF DC BIAS IMMUNITY ON TRANSFORMER DESIGN -MODIFIED CORE SOLUTION
In this section, alternate transformer topologies are considered which offer improved dc bias immunity. Figs. 10 and 11 show two such candidate design architectures. In each case, the transformer core is made up of two different materials. One of these materials is the high permeability transformer steel and the other is a low relative permeability material. The idea behind these architectures is that, upon saturation of VOLUME 10, 2023 the high permeability transformer steel, the low permeability material will share the flux with the high permeability steel thereby reducing the adverse effects of half-cycle saturation.
Herein, attention will be focused on the embedded core architecture, which has an advantage in that less flux passes through the windings under saturated conditions. The MEC for this case is shown in Fig. 12. Therein, the base of the transformer has been split into multiple reluctances to represent the flux paths accurately. The expressions for the additional reluctances are given by l cc w ce µ l cc w ce (21) In (16)- (21), the function µ B () describes the permeability as a function of flux density as described in [26]. Three candidates for the low permeability material are considered herein. The first is air. The second is a CoFe/epoxy composite with an initial relative permeability of 20. This  material is readily put into the main core and allowed to cure. The third material is generic and assumed to have a selectable value of relative permeability. This was done to obtain guidance into other materials that might be appropriate. To this end, the relative permeability of the generic material is a design parameter allowed to vary between 20 and 100. sss.
The design studies are identical to the studies described in Section IV except for the following differences: 1. Two additional design variables relating to the dimensions of the low permeability core have been included. These are the ratio of the width of the low permeability core, w blm , to the width of the transformer window and the ratio of the height of the low permeability core, h blm , to the height of the transformer base-leg as shown in (22) FIGURE 11. Embedded interior core architecture. and (23). The value of r w is allowed to vary between 0 and 1 and r h is allowed to vary between 0.02 and 1.
2. An additional constraint limiting the no-load primary rms current to 2.5% of its full load value has been placed. This constraint results in designs which have low magnetizing current and hence improves the no-load power factor of the transformer. It must be noted that this constraint was not included in the baseline study since the resulting designs already satisfied this constraint. This was not the case with the gapped core design study which resulted in a high no-load magnetizing current. It was found that enforcing this constraint in the gapped core study resulted in no viable designs. Fig. 13 shows the pareto front comparing the results of the three design studies. From Fig. 13, the designs with CoFe/epoxy composite coincide with many of the designs from the variable relative permeability study. These are the designs whose internal core relative permeability is selected to be close to 20. It can also be seen that as the mass decreases the designs for CoFe/epoxy composite study and the variable permeability study do not coincide with each other as closely. This is because the chosen relative permeability of these designs is higher than 20. Furthermore, it can also be seen that   the air interior core designs have a slightly lower power loss as the mass of the transformer increases but suffers from higher  power loss at lower transformer mass compared to the corresponding CoFe/epoxy composite and variable permeability designs. Fig. 14, Fig. 15, and Fig. 16 show Design 274 from the enclosed air study, Design 242 from the enclosed CoFe/epoxy composite study, and Design 232 from the variable permeability design study respectively. Table 4 provides the selected transformer quantities.
The interior air core has the smallest width. This is because of the much smaller relative permeability of air compared to the other interior core materials. Comparing Figs. 15 and 16 the designs are very similar. This is expected from the Pareto front comparison shown in Fig. 13   the voltage excitation which would result in a steady state dc current of 7A to flow in the primary winding. It can be seen from Fig. 17 that the peak of the current waveform is limited to 19.1 A which is close to the maximum allowed value for peak current 17.0 A. As discussed previously, the cause of this discrepancy is the difference between the 3D magnetic equivalent circuit and a 2D finite element analysis in a numerically sensitive situation.

VI. CONCLUSION
This paper introduced two dc current tolerance metrics and presented a dc current tolerant transformer architecture and design methodology. The methodology to calculate the dc current tolerance for a given transformer design and dc bias was found to compare reasonably well to the estimate obtained using time-stepping FEA simulation. It was shown that designing transformers with a low permeability internal core region significantly increased the transformer resiliency with respect to dc currents. Unlike work in the literature, no new transformer parts are required. The proposed approach does not require additional sources and circuit breakers like the dc flux blocker [21], auxiliary windings and controllable source as in [22], compensation windings as suggested by [23], or negative magnetic reluctance structures used in [24]. Since only a modification of the core is needed, this suggests that dc current resiliency could be achieved with a very modest increment in cost using the proposed approach.