Research on harmonic power flow calculation of novel urban distribution network considering the switching power supply harmonic and the correlation of impedance-frequency

Harmonic power flow calculation is the basis of harmonic analysis, tracing, and treatment. This paper focuses on the harmonic issues associated with the implementation of the “dual-carbon” plan in novel urban distribution networks. Due to the widespread integration of renewable energy generation into urban electrical grids, an increasing number of power electronic devices are being connected to the grid, which has a certain impact on harmonic power flow. Firstly, the harmonic components of 6-pulse and 12-pulse rectifiers are analyzed for the switching power supply model widely used in the distribution network. Secondly, the equivalent parameter models of the generator, the transformer, and the lines are reviewed. Finally, from the perspective of power balance, a harmonic power flow model considering the terminal switching power supplies and the correlation between impedance and frequency is proposed. Simulation verification is conducted to demonstrate the feasibility and accuracy of the proposed method.


Introduction
Harmonic power flow calculation is an important analytical tool for studying harmonic tracing, mitigation, and other related aspects [1] [2].With the help of the harmonic power flow calculation results, the harmonic characteristics of the power grid can be understood from the mechanism, and the index can be quantified, enabling the development of corresponding assessments for harmonic responsibility quantification [2] and the effective measures of control.Compared with fundamental power flow, the harmonic power flow calculation is more complicated, with its basic principle being based on Kirchhoff's equations, and the direction of the harmonic current injected into the grid is from the load side.In addition, due to the strong correlation between impedance and harmonics in traditional power flow calculations, the harmonic impedance of the power components must be considered in the harmonic power flow calculation.
According to different solving methods of harmonic power flow, it can be decomposed into the iterative method and the decoupling method.The premise of the iterative method is that the harmonic current of the harmonic source is a variable controlled by its voltage or a function of the fundamental voltage and harmonic voltages [4].With the help of nonlinear equations such as Newton-Raphson iteration method or Gaussian iteration method, alternate terminal voltages can be obtained to solve the harmonic power flow.The decoupling method assumes that the harmonic source is the constant current source [5], and its data setting derives from the measurement of large data analysis results, experiments, or planning calculations.The method is simple and efficient.In [6][7][8], the power flow nodes are divided into slack nodes, PV nodes, PQ nodes, and non-linear nodes, and on this basis, the unified power flow equation is constructed.The iterative method is adopted to solve both the fundamental power flow and the harmonic power flow simultaneously.This method has high theoretical accuracy but a slower solving speed.To address this problem, Sun et al. propose a fast noniterative harmonic power flow calculation method [1], which treats the harmonic sources at the base frequency as constant power loads and then calculates the operating parameters and models of the harmonic sources according to the fundamental frequency results.By simultaneously solving the harmonic admittance equations and the harmonic source model equations of the system, the harmonic voltages of all nodes in the system can be obtained without the process of iteration.
Currently, in China, there is a strong push towards achieving the "dual-carbon" goals, the transformation of energy consumption structure is continuously deepening, and vigorous development of sustainable energy as a substitute for traditional energy has become one of the effective solutions [9] [10].Especially in the urban distribution network, due to the large-scale application of electricity, the harmonic problems caused by the power electronic switching power supply and other devices have become one of the performance evaluation indicators of the power grid [10].The switching power supply is composed of multiple rectifying elements, which achieve the control target formed by the control device through rigid short-circuit or power interruption methods and generate a large number of harmonics.The measured current waveform distortion rate of power electronic devices such as computers, energy-saving lamps, air conditioners, and washing machines using power electronic devices is 100% [10].From the perspective of harmonic waves, the residential load types are diverse, and the harmonic generation characteristics of various electrical appliances are quite different.In previous studies, the constant current source model is mostly used for analysis.However, the voltage fluctuation in China's low-voltage distribution network can reach nearly 10%, and the voltage distortion can reach up to 5%.The constant current source model does not consider the influence of voltage fluctuation and distortion, and when there are multiple loads, the accuracy of the collective harmonic evaluation is reduced.To address this issue, Sun et al. adopt the Markov chain Monte Carlo method [12], combining the electrical model and behavioral model of each load, utilizing the idea of bottom-up layered modeling to propose a collective harmonic assessment method of residential load, which improves the accuracy.Yong and Xiao present the harmonic admittance model of energysaving lamp load [13], which requires the circuit structure with known load, but the specific circuit structure of most household electrical appliances load is complex and the parameters are unknown, so it is difficult to establish this model according to the circuit structure of electrical appliances.
Furthermore, power electronic devices such as electric locomotives, aluminum electrolysis cells, and charging devices are widely used in urban industrial and transportation systems.The thyristor rectifier device used in these applications employs phase-shift control, and the current absorbed from the power grid is a sine wave with a missing angle (non-integer period), thus leaving another part of the sine wave with a missing angle to the power grid, leading to the presence of a large number of harmonic waves in the power grid [14] [15].The statistics show that the harmonic waves generated by the rectifier devices account for nearly 40% of all the harmonic waves, which is the largest harmonic source in the power grid.Wang et al. take into account the trigger angle offset and phase overlap angle of the phase rectification circuit during voltage imbalance [16], and the harmonic coupling admittance matrix of the 12-pulse rectifier under the action of positive-sequence and negative-sequence voltage is derived to establish the frequency domain harmonic coupling admittance model of 12-pulse rectifier under unbalanced voltage conditions.With the advancement of power electronic technology and cost decline, the usage of the phase-control switch power supply becomes prevalent, leading to a large influx of harmonic components into the power grid, so the study of switching power harmonic is particularly important.

Modeling of power electronic devices in the harmonic domain
Power electronic devices exhibit strong switching characteristics due to their high-frequency switching behavior, which inevitably generates a significant amount of harmonic interference during their operation [17].Additionally, with the progress of power electronic technology, the power capacity of the device is increasing, resulting in larger associated harmonic distortions.The widespread use of thyristor rectifiers in various applications such as electric locomotives, aluminum electrolysis cells, charging equipment, and switch-mode power supplies has caused significant harmonic disturbances in the power grid.Thyristor rectifier devices operate in a phase-controlled manner, absorbing a portion of the power grid's sinusoidal waveform with missing angles and leaving behind another portion of the sinusoidal waveform with missing angles, which contains a lot of harmonics.
A rectifier is a power electronic device used to convert an AC power supply to a DC power supply.A 6-pulse rectifier typically consists of six thyristors, while a 12-pulse rectifier refers to the addition of a set of 6-pulse rectifiers after the input, with the inclusion of a phase-shifting transformer, which allows the DC bus current to be rectified by 12 thyristors, hence it is also called a 12-pulse rectifier.For commonly used rectified bridges, according to Fourier analysis, most of the harmonic waves are the odd-order ones such as 3 rd , 5 th , 7 th , 9 th , 11 th , and 13 th , and the amplitude of each harmonic decreases with the increase of harmonic times.In the case of a 6-pulse rectifier, the 5th harmonic has the highest content, while for a 12-pulse rectifier, the 11th harmonic has the highest content.
Below, as shown in Figure 1., a 6-pulse and a 12-pulse rectifiers are simulated by MATLAB/Simulink platform, and the harmonic components of the output voltages are analyzed by Module FFT Analysis, with the results illustrated in Figure 2. The simulation results illustrate that the output of the 6-pulse rectifier contains a significant amount of 5 th and 7 th harmonic components, while the output of the 12-pulse rectifier contains a significant amount of 11 th and 13 th harmonic components.

Generator model
The harmonic content produced by the generator is relatively small, and the generator in the power grid can be regarded as the ideal generator whose potential can be considered purely sinusoidal, without any harmonic component.In the harmonic power flow analysis, the generator can be treated as a load, and the composition of the generator's harmonic impedance is shown in Figure 3. , the harmonic impedance of the generator can be expressed as Further considering the skin effect, when the frequency of the power system increases, the current in the conductor tends to concentrate near the surface layer, resulting in an increase in AC resistance, and the harmonic impedance of the generator can be expressed as In the steady-state operation of the AC system, the voltage and current of each harmonic contain positive-sequence, negative-sequence, and zero-sequence, respectively, which are independent of each other, and there is a certain relationship between the number of harmonics and the phase sequences.When the symmetrical component method is applied to study the three-phase asymmetric harmonic power flow, the harmonic impedance parameters of different phase sequences are required, and the three-sequence harmonic impedance of the generator can be uniformly expressed as where 0 X is zero-sequence reactance, d X ϒϒ is sub-transient reactance, and 2 X represents negative-sequence reactance.
The positive-sequence harmonic reactance of the generator is similar to the negativesequence harmonic resistance, so the harmonic reactance can be expressed uniformly by the form of positive-sequence harmonic.Generators are typically connected to the power grid through a delta-connected transformer, where the zero-sequence path is open-circuited.Therefore, in calculations, a sufficiently large reactance is used for equivalence.

Transformer model
In the fundamental power flow calculation, the excitation branch of the transformer is usually ignored.It has nonlinear characteristics due to the core saturation, and the degree of nonlinearity changes with the magnitude of the applied voltage.When the external voltage is low and the core is unsaturated, the excitation branch shows linear characteristics with low harmonic content, which can be ignored in the calculation of harmonic power flow.However, when the external voltage is high, the saturation degree of the core increases, and the excitation branch has nonlinear characteristics with high harmonic content, which can be equivalent to the harmonic source in the power flow.To ensure high economic benefits, the transformer generally works in the nonlinear core saturation region, thereby generating some harmonic waves.The harmonic impedance equivalent circuit of the transformer is shown in Figure 4.When the harmonic current flows through, the skin effect in the excitation winding and eddy current losses in the core increase, then the harmonic impedance can be expressed as () When studying the three-phase asymmetric harmonic power flow, the connection mode of the transformer will lead to different zero-sequence loops, so it is necessary to consider the influence of the transformer connection modes in the calculation of harmonic power flow.When the connection mode is not YN/YN, the zero-sequence path between the primary and secondary sides of the transformer is interrupted, and a sufficiently large reactance is utilized as a replacement in the calculation process.Moreover, the connection modes of the transformer are also related to the phase shift angle of the current and voltage.In symmetrical power flow or when constructing positive-sequence and negative-sequence networks, the influence of the phase shift can be ignored.

Transmission line model
In the fundamental power flow calculation, the transmission lines are generally simulated with the ο -type equivalent circuit per unit length.However, lumped-parameter equivalent circuits are only suitable for lines with short lengths, and the errors associated with lumped-parameter models increase as the length of the line grows.In such cases, multiple ο -type equivalent circuits are used to simulate.While in the harmonic power flow calculation, ο -type equivalent circuit can also be used, which is shown in Figure 5., but the difference is that the distributed characteristics of the line under harmonic conditions will be more obvious than that of the fundamental frequency.As a result, the effective representation length of each equivalent circuit is significantly reduced.In such a situation, a ο -type equivalent circuit represented by a hyperbolic function will be more convenient.The parameters in the equivalent circuit can be expressed as: ( ) sinh where h Z κ represents the wave impedance of the h-th harmonic, h φ is the propagation constant of the h-th harmonic, and l is the length of the transmission line.
h Z κ and h φ can be calculated from the following equation where 0h z and 0h y represent the unit-length impedance and admittance of the transmission line at the h-th harmonic wave, respectively.

Load model
In the calculation of harmonic power flow, the composition of the load is complex and random, which is difficult to model accurately, so the equivalent circuit of typical loads' harmonic impedance is generally adopted in the calculation process.The typical loads in the power system can be divided into harmonic source loads and non-harmonic source loads in the harmonic power flow calculation.Harmonic source loads refer to nonlinear loads such as power electronic equipment, which are equivalent to the harmonic source in the calculation.For the non-harmonic source loads, they can be mainly divided into impedance loads and motor loads.
For individual or series impedance loads, as shown in Figure 6(a), assuming that the power of the load is S p jQ < * , then the fundamental resistance 0 R and reactance 0 X can be obtained: The harmonic impedance can be expressed as follows: , then the fundamental resistance 0 R and reactance 0 X can be obtained: Since it is more convenient to use the harmonic admittance to represent the parallel load, the harmonic admittance is For the motor-type loads, the effect of the rotating magnetic field generated by the harmonic current on the motor is similar to that of the negative-sequence current.Considering the skin effect, the harmonic impedance of the motor can be expressed as where R and X are the negative-sequence resistance and reactance of the motor at the fundamental frequency respectively.

Harmonic Power Flow Calculation
Harmonic power flow calculation usually requires considering the basic components such as harmonic sources, transformers, transmission lines, and loads.On the source side, considering the distributed harmonic sources, the equivalent infinite constant current harmonic sources are adopted.The impedance of transformers and transmission lines is closely related to frequency, which has been discussed in the aforementioned part of this paper.Moreover, the zero-sequence harmonic can only pass through when the transformer is type YNyn.However, in the general power grid, the transformers are usually connected in YNd or Yd modes, so the influence of the zero-sequence harmonic component can be disregarded.
In practical engineering applications, the levels of harmonic current and voltage are important technical indicators [18] [19].For example, when assessing the impact of harmonic on the safety of the equipment, it is common to calculate harmonic voltage or current values rather than harmonic power.Therefore, harmonic power flow equations generally use current balance equations instead of power balance equations.
Then, from the analysis above, the harmonic power flow can be expressed as the network equations in matrix form, as follows: where m Y represents the admittance matrix of the nodes, m I is the harmonic constant current source, m U is the node harmonic voltage, m represents the node number, and N represents the total number of nodes.
It can be seen that to obtain a reasonable harmonic voltage, the harmonic impedance value under the corresponding operation mode is needed, especially the resistive resistance value and inductive resistance value which are closely related to the frequency.
In general, harmonic power flow calculation methods can be divided into iterative methods and decoupling methods.In this paper, the decoupling algorithm of the harmonic power flow calculation is employed, which considers the coupling relationship between the fundamental power flow and the harmonic flow.In the calculation process, the former has a great influence on the latter, while the effects of the latter are relatively small on the former.Therefore, when calculating the fundamental power flow, the impacts of harmonic power flow can be temporarily disregarded.The fundamental power flow and harmonic power flow can be calculated separately, with the fundamental power flow solved first, followed by the harmonic power flow.This approach achieves complete decoupling between the fundamental frequency and harmonic power flows.Because in the process of calculation, only one complete solution of the harmonic network equations is required, and there is no need for multiple iterations.This method is characterized by its simplicity, lower memory usage, and faster computation speed.It also avoids suffering convergence issues.Although its computational accuracy and precision may be lower than the iterative method, it still meets the accuracy requirements of practical engineering applications.Therefore, it is widely used in practice.
To calculate the harmonic power flow, the following assumptions are made: (a) Harmonic sources are assumed to be independent of each other and are equivalent to singlephase constant current sources.Electrical equipment with non-linear characteristics is considered the primary source of harmonics.The magnitude of harmonic currents depends on the characteristics and operating conditions of these harmonic source devices and is independent of the power grid parameters, making them akin to constant current sources.
(b) The impedance parameters of the power grid are strongly frequency-dependent.
(c) The electrical quantities of the power grid satisfy linear superposition.(d) Sinusoidal signals act on nonlinear electrical loads.The current from such loads is nonsinusoidal, and its harmonic components result in sinusoidal voltages in the system.
(e) It is assumed that harmonic sources contain a certain amount of the 4 th , 7 th , and 11 th harmonic components.
(f) Amplification or resonance phenomena related to harmonics are neglected.The main flow of the harmonic power flow calculation method of the distribution network based on the general component is shown in Figure 7. Initially, according to the Newton-Raphson iteration method, the fundamental power flow is solved.After obtaining the results, combined with the input harmonic sources and the harmonic impedance equivalent models of the electrical elements, the corresponding harmonic admittance matrixes of each number are constructed.Finally, based on the harmonic power flow equations, the harmonic voltages and currents at various nodes in the network are computed.Further calculations are then conducted to determine the voltage distortion or harmonic distortion rate.

Analysis of the Simulation Results
To verify the reliability of the proposed harmonic power flow calculation algorithm, the simulation was constructed.Based on the MATLAB/Simulink platform, the IEEE 33-node system is used.The harmonic sources are set at nodes 12, 22, 25, and 29.The current amplitude of the harmonic sources at each number of harmonics is shown in Table 1 Through simulation, the following results can be obtained.The harmonic voltage amplitude and distortion rate at each node in the system are shown in Figures 8 and 9, respectively.It can be seen from the figures that the node harmonic voltage near the harmonic sources (nodes 12, 22, 25, and 29) has a large amplitude and a serious distortion rate, especially at node 25, where the harmonic voltage reaches the maximum value.The diagram shows the amplitude of the harmonic current in Figure 10.As can be seen from the chart, the branches directly connected to the harmonic sources (nodes 12, 22, 25, and 29) contain large harmonic currents, and the harmonic currents will spread to the external power grid through the line or transformer, thus affecting the power quality of the power grid and reducing the reliability of the power grid.
As can be noticed from Figure 11, it is evident that active power losses in various branches of the system are higher in places where the amplitude of harmonic currents is significant.Therefore, it is essential to mitigate harmonics to reduce active power losses in the power grid and improve efficiency.

Conclusions
This paper focuses on urban power distribution networks and considers the correlation between harmonics generated by rectifiers or switching power supplies and frequency impedance.It proposes a corresponding harmonic power flow calculation algorithm.The algorithm is based on the decoupling method and takes into account the harmonic impedance models of various electrical components in the power grid and the harmonics generated by rectifiers or switching power supplies.It comprehensively forms the harmonic admittance matrix of the power grid, enabling fast and accurate calculation of harmonic power flow in the network, which provides data support for further addressing harmonic issues in the power grid and maintaining its stability.The feasibility of the proposed harmonic power flow calculation method is further validated through simulation experiments conducted on the MATLAB/Simulink platform.

Figure 3 .
Figure 3. Generator harmonic impedance equivalent circuit Assuming that the fundamental impedance of the generator is Z R jhX < * , sinc e the reactance is 2 X fL ο <, the harmonic impedance of the generator can be expressed as

Figure 4 .
Figure 4. Transformer harmonic impedance equivalent circuit.When the harmonic current flows through, the skin effect in the excitation winding and eddy current losses in the core increase, then the harmonic impedance can be expressed as

Figure 5 .
Figure 5. Harmonic impedance equivalent circuit of the transmission line.

Figure 6 . 7 For
Figure 6.Harmonic impedance equivalent circuit of the impedance loads 1.(a) Series impedance load; (b) Parallel impedance load

Figure 7 .
Figure 7. Flow chart of the proposed harmonic power flow calculation algorithm.

Table 1 .
: The Harmonic Current Ratio of Each Harmonic Source.