Determination of Leakage Inductance Percentage for Gapped Iron-Core Shunt-Reactors with M4 Steel as Core Material

—Leakage inductance component has a significant importance in total inductance value of GISR. Neglecting this component in the design phase, results in an expensive and bulky core structure. Variation of leakage inductance component in percentage is determined and presented as graphical curves for M4 steel by applying energy method. FEA are performed for various GISR with several operating voltages and temperature rise values to determine the leakage inductance component. A design tool with Matlab/Guide is also developed for analytical calculations to obtain the physical dimensions for FEA. Graphical curves introduced to the literature in this work provide manufacturers or design engineers to perform fast, reliable and economical GISR design with an alternative material and offer variety.


I. INTRODUCTION
HUNT REACTORS are widely used in power grids for reactive compensation, harmonic filtration etc. They mostly have gapped iron core with distributed air-gaps along the limb. Equivalent circuit for a Gapped Iron-core Shunt-Reactor (GISR) is given in Fig. 1, where L l is leakage, L c is core and L g is total air-gap inductance, whereas R w ,R c and R g are the resistances representing winding, iron and gap losses [1]. therefore this component is mostly neglected in the design phase and analysis. Influence of core gap in design of current limiting transformers is presented in [2] but leakage inductance component is not considered. Leakage inductance is neglected in another shunt reactor design study [3]. Although the leakage fields are considered in [4] determination of leakage inductance is not studied. Leakage inductance component is usually neglected in the studies on design of GISR including influence of dimensional parameters, design optimization, stress characteristics, etc. [5][6][7][8]. Finite Element Analysis (FEA) is widely used in GISR studies [9][10][11][12]. Determination of inductance components including leakage inductance by FEA is performed in [1] and [12], where calculation is based on energy method. A family of graphical curves called nomographs representing the variation of leakage inductance component in percentage against reactive power in terms of four different operating voltages and three different temperature rise values are presented in [12,13] for M330-35 AP Non-Grain-Oriented (NGO) steel (will be called M33 steel hereafter) by FEA. The proposed design criteria in this study is minimum Present Value Cost (min PVC). Presented results show that ratio of leakage inductance in the total amount may be significant. Therefore, neglecting the leakage inductance component in the design phase will result in a bulky and costly core.
In this work, leakage inductance components for M120-27S Grain-Oriented (GRO) steel (will be called M4 steel hereafter) by the method proposed in [12,13] are calculated and presented as graphical curves. A design tool for single-phase GISR in Matlab/Guide is developed for analytical calculations. As a result, physical dimensions of the GISR under design are obtained for FEA. FEA are performed for various GISR with several operating voltages and temperature rise values to determine the leakage inductance component.
Since such information in the literature exist only for M33 steel as core material, the results presented in this work are beneficial for the design of GISR with M4 steel as an alternative core material. In addition, the developed design tool provides a practical and accurate design of such reactors.  Figure 2. The input data such as target inductance value, operating current and line-to-line voltage, minimum and maximum values of number of turns with incremental step, magnetic field density, ambient temperature, tolerances for temperature rise, leakage inductance percentage, operating frequency, type of core material, stacking factor, distances and clearances between coils and yoke, insulation material thickness, per-kg prices for core and winding materials, number of air-gaps and number of parallel foils for the GISR under design are to be entered by user. Extra data for labor and general expenses and extra cost of additional material can also be added. Similar assumptions, for each set of operating voltages, such as • Clearances  between yokes  between windings  between tube and windings • insulation thickness • tube thickness • temperature rise values (60, 80 and 100 K with ±2 K tolerances) • number of air-gaps in the core (40 in both the design and FEA analysis) • aluminum foil as winding material • operating magnetic flux density as 1.1 Tesla • prices of iron and aluminum are taken to be 2.5 and 3.7 € per kg, respectively • optimization criteria (min PVC) defined in [12] are considered in the design phase. This will provide comparison of the leakage inductance percentage results in two different core materials, M33 and M4. Magnetization and loss curves for M4 steel provided by the manufacturer are given in Figure 3. B-H curve required for FEA analysis software is defined from the magnetization curve in Figure 3. Core loss calculation in the analytical design phase is performed as defined in Equation (1) which is obtained via curve fitting of the total loss characteristic given in Fig.3.

II. METHODOLOGY
After completing data entry, by pressing Min PVC button on the software screen given in Fig.2 The following design example can be considered for better understanding of the design process. A 250 kVAR 13800 V (808.66 mH, 31.38A) shunt reactor is to be designed. Input data and post-calculation results are as in Figure 2 and also summarized in Table I. The core material for this design example is M4 steel and the winding material is Aluminum foil. Desired value of operating magnetic flux density is 1.1 Tesla, minimum and maximum temperature rise values are set at 78 and 82, respectively for 80 K temperature rise. Leakage inductance percentage is selected as 20 percent. Setting the initial number of turns (N) to 500 up to 800 with 20 incremental step, the software starts calculation with all data input given in Figure 2. After all iterations completed in almost a minute, the software will determine the optimum reactor satisfying the optimization criteria among thousands of reactors (N*a*t*J = 16*10*20*50=160000). At each iteration step firstly physical dimensions of the reactor, secondly core and winding losses, then Present Value Cost (PVC) and finally temperature rise are calculated. Since the design criteria is Min PVC, the reactor with minimum PVC among the reactors satisfying the desired temperature rise limit is determined as the optimum one and its calculated physical parameters are displayed on the screen The results show that the optimum for defined assumptions; has 680 turns, 39200 mm2 cross-sectional area (limb width 140 mm and depth 280 mm, ratio of depth to width is 2). The software rounds the value of limb width to multiples of ten to satisfy the realistic steel dimensions used in practice. Prices of iron and aluminum are taken to be 2.5 and 3.7 € per kg, respectively. It is worth to note that the proposed design software provides all the input data and design criteria to be modified according to the special design under consideration. The analytical results obtained from design software will be used as physical dimensions for modelling the reactor with Ansys/Maxwell software in determination of leakage inductance percentages of GISR.

B. Finite Element Analysis to Obtain Leakage Inductance Percentages
Firstly, physical dimensions and parameters of the reactor under design are obtained with the aid of the design software (Fig.2) and then the reactor is modelled in Maxwell 3D software to calculate inductance components by energy method as defined in [1,12]. Table I represents target reactor specifications (design data), analytical calculation results obtained via the design software and FEA results for a set of iterations which are performed to obtain the leakage inductance percentage values for various power rating at 13.8 kV operating voltage, and at 80 K temperature rise. Each column in Table I represents an individual iteration, thus an individual reactor design phase. Since the estimated leakage inductance value changes at each new iteration, the physical parameters differs from each other.
Since the leakage inductance percentage is not known, design starts with an initial estimate. For the design of 250 kVAR 13.8 kV (808.66 mH, 31.38 A) GISR, initial estimate of leakage inductance at first is set to 5% as given in the first column of Table I. Then the analytical calculations with this value for the reactor are performed via the design software (Fig.2). After the physical parameters of the target reactor are obtained from analytical calculations, the reactor is then modelled in Ansys/Maxwell 3D software. Once FEA is performed, inductance parameters are obtained by the energy method. FEA results, shown at the end of the first column of Table I, show that the leakage inductance percentage and the inductance decline for this design iteration are 19% and 17%, respectively. However, two criteria  leakage inductance percentage value close to the initial estimate  inductance decline value smaller or equal to 5% should be simultaneously satisfied for the iteration to be accurately completed; Since the solution is not satisfying, a second iteration with the information given in the second column of Table I is performed for the same GISR. Now, initial leakage percentage is estimated to be 10% and defined as input data into the design software for analytical calculations. With the new set of physical parameters obtained for this case, the same procedure is repeated. Post-simulation results in the second column of Table I show that the inductance decline and the leakage inductance percentage criteria after the second iteration have not been met yet. Finally, a new design iteration and FEA are performed with 20% leakage inductance percentage estimation. Criteria to finalize the design for this individual reactor are finally satisfied in this third iteration, as being 18.18% leakage inductance and 2.34% inductance decline as shown in the third column of Table I.   Table I shows only a few iteration to explain the procedure in determining the leakage inductance percentage values. The process is repeated for each set of operating voltages, temperature rise values and power rating. As a result, hundreds of design and simulation iteration are performed to obtain leakage inductance percentages as graphical curves given in Figure 5. Calculation of inductance percentages in FEA are performed by energy method [1,12]. The energy (co-energy may also be used for linear operation which is the case in general) in each volume; occupied by iron, air-gaps, and the total volume surrounding the core is calculated by integrating the energy density after post-simulation via the fields calculator of the software as shown in Fig.4. After having the resultant energy value, the inductance of the related volume is obtained by (2). By this way, all the inductance components of the GISR are obtained. Calculation of energy in each volume and plot of magnetic field density for 250 kVAR 13.8 kV GISR with 20% leakage inductance and 80 K temperature rise are given in Figure 4.

C. Leakage Inductance Percentages as Graphical Curves
Variations in leakage inductance percentage against reactive power in terms of operating voltage and temperature rise for M4 steel are graphically represented in Figure 5. In addition, the graphs for M33 steel are given in Figure 5  Graphical results show that both steel has its specific leakage inductance percentage values. Any change in core material during the design phase therefore requires new set of graphical curves for reliable results.

III. CONCLUSION
Neglecting leakage inductance in the design phase of GISR results in an expensive and bulky device as shown by presented simulation results. Graphical curves representing leakage inductance percentages for M4 steel in addition to the existing curves for M33 steel, will offer variety to the literature in design of GISR. Results presented in this work will provide manufacturers and designers to implement a fast, accurate and economical design by taking the effect of leakage inductance into consideration. The proposed design software for analytical calculations is a valuable design tool and provides design of GISR in a wide range.