Abstract: The design of filtering devices is one of the key technologies in high-voltage direct current (HVDC) transmission. This paper proposes a design method based on multi-objective programming, taking into account the inherent characteristics of HVDC systems. A comprehensive mathematical model of the filtering device is established, which considers both the physical meaning of the filtering device and the economic efficiency of engineering investment. Applying a genetic algorithm to this model can solve for an economical, practical, and effective filtering device scheme. Simulation experiments verify the correctness of the parameters of the filtering device designed using this mathematical model. Keywords: Multi-objective programming, high-voltage direct current, dual-tuned filter, genetic algorithm 1 Introduction High-voltage direct current (HVDC) converter stations use semi-controlled thyristor devices and utilize phase control for both AC-DC and DC-AC conversions, generating a large number of high-order harmonics. Currently, HVDC converter devices generally use a 12-pulse converter bridge, which generates 12n±1 order current characteristic harmonics on the AC side of the converter station, where n is a natural number; and 12n order voltage characteristic harmonics on the DC side. Various asymmetries (such as unequal interval trigger pulses, bus voltage asymmetry, phase-to-phase commutation reactance asymmetry and transformer excitation current) will generate a small amount of additional non-characteristic harmonics. After the harmonic current on the AC side of the converter station enters the AC system, it will cause the system voltage waveform to be distorted and cause adverse effects and hazards. The harmonic voltage on the DC side of the converter station will distribute harmonic voltage and current on the DC line, causing interference to the nearby communication lines. Please log in to: Power Transmission and Distribution Equipment Network to browse more information. Filtering devices can suppress the above harmonics. HVDC uses a large number of filtering devices with high voltage levels and large equivalent capacity, and they are generally outdoor. Filtering devices account for a considerable proportion of the investment and land area of ​​the converter station. Among them, the cost of filtering devices accounts for about 10% to 15% of the total investment of HVDC[1]. A typical HVDC topology is shown in Figure 1. The rectifier station and the inverter station generally have a symmetrical structure. In HVDC systems, smoothing reactors are first used on the DC side to reduce harmonic components of voltage and current in the DC line; however, the smoothing reactors alone are insufficient to meet harmonic control requirements, necessitating the installation of additional filters. Traditional HVDC systems primarily use passive filters (PFs) targeting characteristic harmonics. 2. Performance Evaluation Standards for DC-Side Filtering Devices When HVDC uses overhead transmission lines, communication interference is a serious problem. Due to the significant difference in relative transmission power levels between power lines and communication lines, and the overlap between the characteristic harmonic frequency band of HVDC and the ordinary line communication frequency band, there is a noticeable interference with call clarity. Harmonics also seriously endanger the safe operation of other equipment in the converter station. Currently, HVDC transmission projects in various countries primarily assess the harmonic level of the line based on the degree of communication interference, often using the equivalent interference current Ieq index. Ieq is a single-frequency (800Hz or 1000Hz) current equivalent to the harmonic currents on DC transmission lines. The interference it generates can be equivalent to the interference generated by the harmonic currents. It is generated by the harmonic currents of the rectifier station and the inverter station, and reaches its maximum value at the outlet of the rectifier station and the inverter station. Its definition formula is: where m is the highest harmonic number considered, which is usually taken as 100 for HVDC systems; In is the effective value of the nth harmonic current; hn is the coupling coefficient of the nth harmonic; Pn is the frequency weighting coefficient. The correspondence between hn, Pn and frequency is shown in reference [2]. When the DC system is in bipolar, balanced operation, the allowable value of Ieq is divided into: high standard (Ieq is 100~300mA); medium standard (Ieq is 300~1000mA); low standard (Ieq exceeds 1000mA). For DC systems operating in unipolar mode, this standard can be increased by 2 to 3 times. In recent years, with the popularization of optical fiber communication, the above standards have also gradually relaxed. 3 DC side filter device 3.1 Smoothing reactor The design of the smoothing reactor needs to meet the following requirements [3]: (1) The smoothing reactor can maintain the continuity of the current under the condition of small current in the DC line. When the trigger delay angle is 10.1°＜a<169.9°, its inductance is given by the formula where Ld is the inductance of the smoothing reactor, H; Idlj is the average value of the DC critical current; angular frequency w=314rad/s; k0 is a fixed coefficient, k0=0.023 for 12 pulse converter and k0=0.015 for 24 pulse converter; Udo is the rated DC voltage. (2) When a fault occurs in the DC power transmission circuit, the smoothing reactor can suppress the current rise rate, thereby preventing secondary commutation failure. At this time, its inductance is given by the formula. In the formula, the inverter commutation fault time, the system frequency f=50 Hz, βc is the rated lead trigger angle, δmin is the minimum turn-off angle, ΔId is the maximum allowable current increment in the time Δt, and is the change in DC voltage of the inverter in the time. The lower limit of the inductance value of the smoothing reactor can be determined by formula (2) and formula (3). (3) Reduce the harmonic pulsation component on the DC side. As can be seen from the above analysis, the larger the inductance Ld of the smoothing reactor, the better; however, if Ld is too large, the overvoltage Ld(di/dt) generated on the smoothing reactor when the current changes rapidly will also be larger. In addition, as a delay element, if Ld is too large, it will be detrimental to the automatic regulation of DC current; therefore, under the premise of meeting the above requirements, the inductance of the smoothing reactor should be as small as possible. The engineering cost of the smoothing reactor is mainly related to its structure and capacity. The smoothing reactor used in HVDC has high voltage and large equivalent capacity, and is often oil-immersed, hollow and magnetically shielded. The formula for calculating the equivalent capacity sLd of the smoothing reactor to the 50Hz AC excitation reactor is [4] where Id is the rated DC current, A; the unit of SLd is kvar. Under the condition that the basic structure of the smoothing reactor remains unchanged, the engineering cost TLd is determined by the capacity, and its calculation formula is where KL is the unit capacity cost of the smoothing reactor, yuan/kvar. 3.2 Passive filter The DC side passive filter does not undertake reactive power compensation, but is only used for filtering. Its parameters are determined by the line voltage, filtering requirements and economy. Passive filters are usually connected to the back end of the smoothing reactor and can be single-tuned filters, double-tuned filters, C-type filters and triple-tuned filters. For economic and floor space considerations, HVDC systems use double-tuned filters more often, and its function can be equivalent to two parallel single-tuned filters. (1) Dual-tuned filter The actual model and frequency impedance characteristics of the dual-tuned filter are shown in Figure 2 [5]. The total impedance of the dual-tuned filter is Z1 (W) of the dual-tuned filter reaches a minimum value at two characteristic frequencies W1 and W2. In general, the rectifier bridge can be regarded as a harmonic voltage source, and its amplitude depends on known quantities such as the trigger delay angle, the arc extinction angle, and the commutation overlap angle. Assume that the impedance of the filter branch of the ideal passive filter is zero for all harmonic frequencies. The harmonic voltages all drop onto the smoothing reactor. Un is the nth harmonic voltage output by the rectifier bridge. At this time, the harmonic current is the filter capacity of capacitor C1 when the DC voltage level is determined. In the cost of the passive filter, capacitor C1 accounts for a large proportion. Therefore, the engineering cost of the passive filter is Kc is the unit capacity cost of the DC filter capacitor; T1 is the cost of other components that make up the dual-tuned filter. (2) High-pass filter [6] The actual model of the second-order high-pass filter is shown in Figure 3. The total impedance of a high-pass filter is that of a second-order attenuated high-pass filter. Within the passband W > W0, there is a relatively low impedance frequency range, where W0 is called the cutoff frequency (W0 = 1/R³C³). Another important parameter is that the cost of the smoothing reactor and the passive filter are contradictory in the total cost of the filtering device. Increasing the inductance of the smoothing reactor will increase its filter capacity, but it can reduce the required filter capacity of the passive filter, and vice versa. This paper uses a genetic algorithm to solve a multi-objective programming problem. The core of the genetic algorithm is to continuously improve the current solution group until the requirements are met; that is, the core of the algorithm is improvement rather than moving towards the ultimate evolutionary goal. 5 Design Example and Simulation Verification Figure 4 shows a simulation model of the rectifier station of a 900MW ±500kV HVDC system operating on a single pole, using a standard 12-pulse rectifier bridge. Considering the symmetry of the system, the transmission line and inverter station are simplified to a 278Ω resistive load based on the system power. The rated voltage is 500kV, the rated current is 1.8kA, the minimum current limit is 10% of the rated current, and the rated trigger delay angle is 5°～17°. The rectifier station filter device is composed of smoothing reactor, double-tuned filter (12th/24th order) and high-pass filter. The engineering cost of the filter device in the simulation model is required to be economical, and the maximum equivalent interference current value is guaranteed to be less than 1A. The design steps are as follows: (1) Calculate the compensation capacity of the filter based on the harmonic data of the HVDC system. This paper establishes a simulation model of the HVDC system on the PSCAD (Power Systems Computer Aided Design) software platform, obtains the harmonic voltage and harmonic current values ​​of the rectifier bridge output based on the system operation data under rated operating conditions, and calculates the compensation capacity required by the system. (2) Establish a multi-objective programming model for the filter device. Establish a multi-objective programming model based on equations (15)～(19). The objective function minimizes the sum of the 36th (W3=11304rad/s) and 48th (W4=15072rad/s) harmonic impedances. Other parameters in the objective function are selected from the bidding data of HVDC projects in recent years: KL is 200-300 yuan/kvar; Kc1 is 27-32 yuan/kvar; T0=8*106 yuan. The above parameters only affect the calculation results and will not affect the performance of the algorithm. The feasible region of the problem is determined based on actual experience. According to the minimum current limit of 180A, the critical current Iklj=90A is obtained. The trigger delay angle a=17° is taken, and Lmin=119.0mH is determined. Therefore, the feasible region is (3) Solve. The evaluation function is constructed by using the linear weighted sum method [7], so that the multi-objective planning problem is transformed into a single-objective minimum problem. How to reasonably determine the weight coefficients according to the characteristics of the problem is an important part of the solution. First, the objective functions need to be uniformly dimensional: the objective functions are positively quantified in the feasible region, the minimum value of each objective function is then calculated and normalized, a new objective function is constructed and assigned a corresponding weight coefficient. In this way, the magnitude of each weight coefficient can fully reflect the importance of its corresponding objective in multi-objective planning and design, without being affected by the relative size of the objective value. The methods for determining the weight coefficients include the a-method, the mean difference sorting method, the old method, the judgment matrix method, etc. [7]. The old method relies on experience evaluation and combines statistical processing to determine the weight coefficients, which is simple and practical. In this paper, the old method is used to select the weight coefficients a, b, c, d as 10, 5, 5, 1. Different weight coefficients reflect different intentions of the decision-makers, and the solutions obtained are also different. The single objective minimum problem is solved by using a simple genetic algorithm program to obtain multiple solutions (i.e., filter device parameters). One of them is selected based on engineering experience: high-pass filter m=0.5, L3=6.2mH, R3=550Ω. It should be noted that the multi-objective planning problem has multiple sets of feasible solutions. The simple genetic algorithm can only obtain one set of solutions each time it is calculated. Therefore, it is necessary to calculate multiple times to obtain multiple sets of feasible solutions with similar performance. The niche genetic algorithm can overcome this problem [8]. (4) Simulation verification. The designed filtering device was applied to the HVDC simulation system. The effective values ​​of each harmonic before and after filtering are shown in Table 1. In the table, Ihn is the amplitude of the nth harmonic current at the rectifier station, and Iln is the amplitude of the nth harmonic current of the DC transmission line after filtering. The maximum equivalent interference current (Ieq=643mA) calculated according to formula (1) meets the engineering filtering requirements. 6 Conclusions Based on the characteristics of high voltage DC transmission, this paper establishes a multi-objective programming model for the integrated design of smoothing reactors and passive filters. Its objective function considers both filtering requirements and the economic efficiency of the device, which has certain practical engineering significance. The genetic algorithm was used to solve the problem and a relatively ideal result was obtained. Computer simulation verified the rationality and feasibility of the model. References [1] Asplund Q, Zhang WY. Active DC filters for HVDC systems[J]. ABB Review, 1995, 6(7): 17-21. [2] Xia Daozhi, Shen Zanxun. 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