Probabilistic Voltage Optimization Considering Load Static Voltage Characteristics

In the conventional power flow calculation, the load is mostly treated as a constant power node. When the voltage deviation occurs, the static voltage characteristics of the load will change, resulting in the change of the power flow calculation. This situation is more likely to occur in the distribution network with distributed power supply. Therefore, it is very important to consider the static voltage response characteristics of the load in the power flow calculation. In the existing literature, when evaluating and optimizing voltage stability, deterministic indicators are usually used to evaluate the voltage level of the system. In the active distribution network with uncertain new energy output, this method has its limitations. Based on the above analysis, this paper uses the power flow calculation method considering the static voltage characteristics of the load to solve the node voltage distribution of the distribution network system. Considering the uncertainty of new energy output and load, the semi-invariant method is used to analyze the probabilistic power flow. An evaluation index considering the digital characteristics of voltage deviation distribution is proposed. The voltage optimization model is established according to the proposed index, and the genetic algorithm is used to solve the problem. The IEEE33 node standard example is simulated to verify the correctness and feasibility of the proposed method.


Introduction
Voltage quality is a key index to comprehensively reflect the planning, construction and operation level of distribution network.With the large-scale access of distributed photovoltaic and wind turbines to the power system, the distribution network has evolved from a one-way passive network to an active network with multi-factor coexistence of source, network, load and storage [1].The uncertainty of wind power and photovoltaic output has brought new challenges to the voltage stability of the distribution network [2].
In the conventional power flow calculation, the load is mostly treated as a constant power node [3].When the voltage deviation occurs, the static voltage characteristics of the load will change, which leads to the change of the power flow calculation [4].Therefore, it is very important to consider the static voltage response characteristics of the load in the power flow calculation.In the existing literature, when evaluating and optimizing voltage stability, deterministic indicators are usually used to evaluate the voltage level of the system.In the active distribution network with uncertain new energy output, this method has its limitations [5].
Based on the above analysis, this paper uses the power flow calculation method considering the static voltage characteristics of the load to solve the node voltage distribution of the distribution network system.Considering the uncertainty of new energy output and load, the semi-invariant method is used to analyze the probabilistic power flow.An evaluation index considering the digital characteristics of voltage deviation distribution is proposed.The reactive power optimization model is established according to the proposed index, and the genetic algorithm is used to solve the problem.The IEEE33 node standard example is simulated to verify the correctness and feasibility of the proposed method.

Source-load Stochastic Model
The uncertainty of active distribution network mainly includes distributed generation and load.The distributed power supply is mainly photovoltaic power station and wind farm.

Photovoltaic Stochastic Model
The light intensity in a short period of time usually satisfies the Beta distribution [6], and its probability density function is as follows: Where, α and β are related parameters of Beta distribution; r and rmax are the actual light intensity and the maximum light intensity.
The relationship between photovoltaic output PM and light intensity r is obtained by the following formula: Where, A is the total area of photovoltaic panels, η is the photoelectric conversion efficiency.The random output model of photovoltaic can be obtained by ( 1) and (2). ) Where, Pmax is the maximum output power of photovoltaic.

Stochastic Model of Wind Turbine Output
The wind speed distribution in a short time obeys the Weibull distribution [7], and its probability density function is as follows: Where, V is the wind speed; k and c are distributed shape parameters.The relationship between fan output Pw and wind speed V is usually obtained by the following formula: Where, PN is the rated active power of the fan, Vin is the cut-in wind speed, VN is the rated wind speed, and Vout is the cut-out wind speed, k1 and k2 are linear coefficients.It can be seen from ( 5) that when the wind speed is at the cut-in wind speed and the cut-out wind speed, the fan output and the wind speed are approximately linear.

Load Stochastic Model
The uncertainty of power system load is usually characterized by the positive distribution model [8]: Where, P and Q are the active power and reactive power of the load, μP and μQ are the mathematical expectations of the active power and reactive power of the load, σp and σq are the standard deviation of the active and reactive power of the load.

Load Static Voltage Characteristic Model
Conventional power flow calculation, based on the assumption of constant power load, ignores the relationship between load and voltage.This processing cannot accurately predict the system response under abnormal voltage or other disturbances.Therefore, the power flow calculation considering the static frequency and voltage characteristics of the load has significant advantages [9].
The load static voltage characteristic model used in this paper is as follows: Where, P and Q are the actual active load and reactive load , PN and QN are active and reactive loads under rated voltage, a and b are the sensitivity indexes of load to voltage change, UN is rated voltage.

Stochastic Power Flow Calculation based on Semi-Invariant
Probabilistic power flow analysis method based on semi-invariant is widely used in power flow analysis of distribution network because of its simple calculation and high calculation accuracy.This method first calculates the semi-invariants of each order through the random distribution model of distributed power output and load, and then uses the linearized AC power flow equation as a bridge to calculate the semi-invariants of each node voltage [10][11][12].
The AC power equation is as follows: () Where, w is the power injection value of each node, x is the system node voltage and phase angle, y is the branch power flow, f and z are the system power flow equation and the branch power equation, respectively.
The AC power is first-order Taylor expanded at the system operating point: Further written in linear form of output variables and input variables: Where, J0 is the Jacobian matrix of the last iteration of Newton-Raphson method, Z0 is the first-order partial derivative matrix of the branch power equation.
Equation ( 12) is the transformation formula of each order semi-invariant and each order origin moment.Due to the limitation of space, only order 4 is given here, where γ is the semi-invariant of each order; α is the origin moment of each order.Through (12), the digital characteristics of the node voltage can be easily obtained.

Voltage Optimization Algorithm Considering Uncertainty
The existing voltage stability evaluation indexes are mostly based on deterministic power flow analysis, and there are few studies considering the uncertainty of source and load.Firstly, a voltage deviation evaluation index considering the digital characteristics of voltage distribution is proposed, which is used as the objective function to establish the voltage optimization model.The model is solved by genetic algorithm.

Probabilistic Voltage Stability Index
The minimum voltage deviation index, also known as voltage stability index, is an important index to evaluate the voltage stability of power system.The minimum index of voltage deviation is used to describe the change range of node voltage in power system.The smaller the value is, the better the voltage stability of power system is.The voltage deviation index ΔUi of a single node is obtained by the following formula: Where, i is the node number, Ui is the actual voltage of node i, UN is the rated voltage of node i.There are a large number of nodes in the distribution network, and the voltage deviation index of a single node is difficult to reflect the overall level of the system voltage.In this paper, the minimum value of the voltage deviation index of all nodes in the network is used as the voltage stability index to reflect the overall voltage level of the network, as shown in (14): (14) Considering the uncertainty of photovoltaic, wind power output and load, the voltage stability index is no longer a certain value.In order to evaluate the voltage stability of each node in the distribution network more comprehensively, the mean, variance, skewness and kurtosis indexes of the voltage deviation ΔU are introduced, as shown in (15): (15) Where, k1, k2, k3, k4 are weight coefficients; μ is the mean value of the voltage deviation index, which can directly reflect the voltage level of the system, σ is the standard deviation of the voltage deviation index, which can be used to evaluate the stability of the voltage deviation index.s is the skewness of the voltage deviation index, reflecting the asymmetry of the index, k is the kurtosis of the voltage deviation index, reflecting the peak degree of the index.These indicators provide a comprehensive view of the voltage deviation indicators.

Probabilistic Voltage Optimization Model
According to the defined probability voltage deviation index, the following objective function is established: Constraints usually include power flow equation constraints, state variable constraints and control variable constraints.
Power flow equation constraints: Where, PGi and QGi are the active power and reactive power injected into the i node respectively, PLi and QLi are the active power and reactive power flowing out of node i, respectively.Gij and Bij are the conductance and susceptance between node i and node j; Ui and Uj are the voltage amplitude of node i and node j; θij is the phase angle difference between node i and node j.
Control variable constraints: Where, Qi is the switching capacity of the group i shunt reactive power compensation capacitor; Qimax and Qimin the upper and lower limits of reactive power compensation capacity; ki is the transformer ratio; Kimax and Kimin are the upper and lower limits of transformer ratio.
State variable constraints: Where, Ui_max and Ui_min are the upper and lower limits of node i voltage, respectively.

Model Solution
Genetic Algorithm (GA) is a search optimization algorithm that simulates natural selection and genetic principles.It is inspired by the mechanisms of natural selection, crossover, mutation and heredity in the process of biological evolution, and searches for the optimal solution in the solution space by simulating these processes.The main advantage of genetic algorithm is that it has strong global search ability and can deal with complex, nonlinear and discontinuous optimization problems.In addition, because it is a population-based search, it can be processed in parallel, thereby improving computational efficiency.Therefore, this paper uses genetic algorithm to solve the model.

Parameter Setting of Example
The IEEE-33 node system is selected, and the network structure and line parameters of the test system are shown in [13].The load of each node obeys the normal distribution, the expected value is the original data value, and the standard deviation is 30 % of the expected value.
The 600kW wind farm and photovoltaic electric field are connected to the network nodes 18 and 33 respectively, and the constant power factor is 0.98.The rated wind speed of the fan is 15m/s, the cut-in wind speed is 3m/s, and the cut-out wind speed is 25m/s.The wind speed distribution satisfies the Weibull distribution with scale parameter c=10m/s and shape parameter k=2.The total area of the square array of solar panels in photovoltaic power station is 2000m 2 , and the photoelectric conversion efficiency is 0.15.The light intensity satisfies the Beta distribution, and the shape parameters α= 0.4, β=9.5.
Parallel capacitor banks are installed at nodes 6, 17, 30 and 31, each with a capacity of 1000kvar.An on-load voltage regulating transformer is installed between node 0 and node 1, the ratio adjustment range is 0.9 to 1.1, and the number of taps is positive and negative 5.

Results Analysis
The unoptimized and optimized systems are simulated and tested respectively, and the voltage deviation related indicators before and after optimization are compared.The calculation results are shown in table 1.It can be seen from table 1 that after optimization, the expected value of the voltage deviation probability index decreased from 0.0838 to 0.0316, a decrease of 62 %.The standard deviation σ decreased from 0.0084 to 0.0057, a decrease of about 47 % ; the skewness s decreased from 0.1578 to 0.0270 ; kurtosis changed from-0.0863to-0.0082 ; in general, the overall voltage level of the optimized system has been significantly improved in terms of average performance, stability and data distribution : the decrease in mean indicates a significant reduction in voltage deviation ; the decrease of standard deviation indicates the decrease of system voltage fluctuation.The change of skewness and kurtosis indicates that the data distribution is closer to the normal distribution, and the voltage outliers and extreme cases of the system are reduced.These changes show that the optimization measures have achieved good results, so that the power quality of the system has been improved.Figure 1 shows the expected value distribution of nodes before and after system optimization.It can be seen from figure 1 that before optimization, the voltage distribution of nodes 6-18 and 26-33 is lower, the overall level of system voltage is lower, and the voltage deviation is larger.After optimization, the voltage distribution is no longer beyond the lower limit, the difference of system voltage distribution and the voltage deviation of each node of the system are significantly reduced, which shows the effectiveness of the optimization method proposed in this paper to improve the voltage quality.exceeding the lower limit.After optimization, the voltage probability density curve of the node has been greatly improved, and there is almost no over-limit.

Conclusion
In this paper, a probabilistic voltage optimization model considering the static voltage characteristics of load is proposed for the active distribution network with distributed energy.The voltage stability probability index is minimized by optimizing the control transformer tap and reactive power compensation device.The optimization results show that the reasonable control of the number of taps and the configuration of reactive power compensation capacity can significantly improve the voltage level of the distribution network.At the same time, the voltage quality evaluation index considering the uncertainty of distributed power output and load is adopted.After optimization, the influence of uncertainty on the voltage of distribution network can be reduced.

Figure 2
Figure2is the voltage probability density curve of node 33 before and after optimization.It can be seen from figure2that the voltage of No.33 node before optimization has a great probability of

Table 1 .
Comparison of results before and after optimization.The results of reactive power optimization configuration are shown in table 2.

Table 2 .
Reactive power optimization configuration.