Abstract: This paper introduces a method for improving output voltage in cascaded multi-unit high-voltage frequency converters using the midpoint drift method to solve unit faults. The handling measures under different fault modes and the advantages and disadvantages of this method are analyzed. The differences between symmetrical cut-off methods are compared and analyzed. Keywords: High-voltage frequency converter, drive, midpoint drift algorithm 1 Introduction With the popularization of high-voltage frequency converters in China, cascaded multi-unit multi-level frequency converters are increasingly widely used due to their flexible modular structure, especially the price advantage of low-voltage IGBT devices. For example, based on the modular structure of 580V standard units, by changing the number of cascaded units from 3 units, 6 units to 10 units, the drive needs of 3kV, 6kV and 10kV motors in China can be fully met. In practical applications, it has been found that when multi-unit frequency converters are running, due to the large number of semiconductor components contained in the device, occasional unit faults can lead to shutdown or derating. How to maintain motor operation without stopping when one or a few modules fail, waiting for maintenance opportunities, is a major concern for frequency converter manufacturers and users. Modular frequency converters offer numerous ways to provide redundancy, such as modular redundancy when cascading units. This method relies on the premise that all components within the unit do not cause serious accidents like fires or electrical short circuits under fault conditions. It uses the unit's bypass system to isolate the faulty unit's function while fully utilizing the functions of units in hot backup mode, ensuring full system output. This method does not explicitly locate the backup unit; its effectiveness depends on the total output capacity of the frequency converter system. For example, in a 6kV system using eight 580V units with two redundant modules, reliability is increased by 2.6 times. If there are no redundant modules, midpoint drift can be used to increase the system output voltage even under partial unit failure conditions. Other methods include increasing the module's DC voltage and injecting zero-sequence components. Currently, due to cost factors, most frequency converters on the market do not choose modular redundancy, making midpoint drift the preferred method for increasing the device's output voltage when some units fail. This article analyzes and compares these methods. 2. Basic Principle of Midpoint Drift Method In 1998, a foreign inverter company proposed a PWM control algorithm based on midpoint drift, the basic idea of ​​which can be represented by Figure 1. Figure 1: Basic Principle of Midpoint Drift Method As a flexible three-phase AC power supply, the inverter is connected to the motor without a neutral line. Therefore, it is sufficient that the line voltages at the motor terminals are symmetrical three-phase. As shown in Figure 1, the three-phase line voltages form an equilateral triangle. Cascaded multi-unit multi-level inverters use a star connection, and the phase voltages are represented by the three line segments a, b, and c in Figure 1. If they are equal, the midpoint is located at the center, centroid, and orthocenter of the equilateral triangle. If some units are damaged, causing the three line segments to no longer be equal, the midpoint drift method can be used to change the angle between the phase voltage reference signals of the PWM control, achieving symmetrical output line voltages. Only symmetrical line voltages can form a circular rotating magnetic field within the motor, thereby reducing torque pulsation caused by reverse sequence and vibration caused by the radial magnetic pull of the motor. Since inverters are generally controlled by microcontrollers with relatively weak computing power, it is relatively difficult to calculate the midpoint drift strategy when different units are faulty online. Using the lookup table method to generate the modulation scheme when the fault occurs is a simple solution. Angle calculation is mainly based on simple geometric relationships, such as the cosine theorem: where a∈[0,NA], b∈[0,NB], c∈[0,NC], NA, NB, NC are the number of remaining intact units in each phase. Let's assume that NA, NB, NC are arranged in descending order. It is not difficult to see that discarding the units in phases A and B that are more than phase C and considering symmetrical output with NC is the simplest method, called symmetrical cut-off. However, the maximum output voltage of the inverter is reduced to (NC/N)×100% of the original, which affects the available maximum output voltage. There are 7 variables in formula (1), [mc,a,b,c,α,β,γ], and 4 equations. The constraint condition can be that all variables are non-negative real numbers. The goal of optimization is to maximize the equivalent phase voltage mc. To simplify the analysis, we can fix a, b, and c as NA, NB, and NC, removing the three unknowns, thus solving the equation. The first step is to remove the variable mc, resulting in: Let x = cos a and y = cos b. Considering that the values ​​of a, b, and g are all within [0, 1800], the cosine value is monotonic, and the sine value is non-negative, we have: Further simplification yields: This simplifies to solving the last univariate nonlinear equation, with the usable variable range being x ∈ [-1, 1], but a solution is not guaranteed. For example, when [NA, NB, NC] = [6, 5, 0], a solution may not be found unless [a, b, c] = [5, 5, 0]. In the above analysis, it was assumed that the midpoint after floating is within the triangle to satisfy the requirement of maximizing the equivalent phase voltage. Without loss of generality, the midpoint after floating may also be outside the triangle, as shown in Figure 2. Figure 2 shows a comparison of midpoint drift. Assuming phases b and c have the same number of remaining units, and phase A has the most, then point P in the figure is the drifted midpoint. Draw a circle with radius Nb or Nc, the length of AP equal to Na, and point A fixed. Draw two rays with a 30° angle to AP, with A as the endpoint. These rays intersect the previous circle with two, one, or no points respectively. When there are two intersections, triangles AB′C′ and ABC are obtained. For triangle ABC, the midpoint has drifted outside the triangle. If b = c, and the equivalent phase voltage is m, then according to the cosine theorem: Clearly, 2b ≥ a has a real root, and when a = 2b, there is only one root. The aforementioned circle becomes the incircle of the two rays, and b is the radius of the incircle. Classifying the cases according to the relationship between a and b is as follows: As shown in Figure 2, in the third case, there are two possible outputs, which differ greatly. The midpoint after the drift is not inside the triangle, resulting in a small output amplitude. The active power will exchange between different phases, having only pure mathematical value. When the number of remaining intact units in the three phases is not equal, in addition to calculating according to equations (1) to (5), the following method can also be used. Let a > b > c, and after synthesis, it is equivalent to m. Then the floating midpoint P is located at the intersection of three circles with radii a, b, and c as vertices A, B, and C, as shown in Figure 3. Figure 3 Drift when the number of remaining units in the three phases is not equal. Using the coordinates in the figure, let P(x, y), then we have: 3 equations, 3 unknowns, simplified to: When solving, as long as y is in [0, b], the result can be quickly calculated using Newton's method. However, the solution is not unique. Taking the number of remaining units [6 5 4] as an example, two results can be obtained, as shown in Figure 4. The third equation of formula (8) has two zero crossings with respect to y, resulting in two drift results with different magnitudes. The output voltage obtained when the midpoint is inside the triangle is larger, and the output voltage obtained outside is very small. In addition, there is power exchange between the phases of the frequency converter, so it is generally discarded. Figure 4. Two results obtained from drifting with 6, 5, and 4 unit numbers . 3. Midpoint drift calculation results. Applying the above formula, taking a 6-unit system as an example, the required angles and equivalent results for different remaining unit numbers are shown in the attached table. Symbols are shown in Figure 1. There are a total of (7+6+5+4+3+2+1=28) possible effective combinations. Subtracting the 12 cases (a>b+c), there are 16 remaining combinations with 23 results. The extra 7 combinations have midpoints that have drifted outside the triangle and are discarded. For more systems, such as 10-unit systems, the calculation is not difficult by referring to the above formula, and will not be elaborated here. 4. Comparative Analysis Compared with the above method, most manufacturers currently use the symmetrical cut-off method, that is, after a unit fails, the corresponding units of the other two phases are cut off. This results in wasted functional units, thus reducing the output voltage. Taking [6,6,5] remaining units as an example, the line voltage obtained by the two methods is compared as shown in Figure 5. Figure 5 shows the midpoint drift and symmetrical cut-off methods. When all units are in good condition (ABC), the area is largest; A′B′C is the area with the midpoint drift method, and the area with the symmetrical cut-off method (A″B″C) is the area with the smallest. As can be seen from the figure, the midpoint drift method makes full use of the remaining units and is an effective method. The voltage angles after midpoint drift are shown in the attached table, where the electrical angles that are no longer 120° apart have very irregular data. If the influence of this factor cannot be effectively removed during PWM modulation, it will produce a reverse sequence in the output voltage due to asymmetry, endangering the normal operation of the motor. Generally speaking, the total number of units cut off due to faults should not exceed 20% of the total number of units. For example, in a 6-unit inverter, if more than 4 units are damaged, the machine should be shut down immediately or its capacity reduced. 5. Conclusion This article analyzes the basic principle of the midpoint drift method, obtains the specific drift angles and issues that need attention, and verifies that the data is accurate, providing guidance for improving the performance of cascaded multi-unit multi-level inverters.