A probabilistic power flow calculation method considering the uncertainty of the static frequency characteristic

1 Introduction

As the power demand rapidly grows with the global economic development,power systems are being developed towards forming a super large power grid with the features of long distance transmission,high voltage class,and AC/DC mixed connection,increasing the severity of safety and stability problems[1].Changes in the operation environment may induce random disturbances in power systems.A large number of uncertain operating conditions in practical operation pose greater challenges to the safety analysis of power grids.Therefore,a new method is required to accurately analyze and evaluate system operating conditions and ensure the safety and stability of the system.

Various random factors of system operation,such as load fluctuation [2],equipment failure [3],and intermittent energy output fluctuation [4-7],can be simulated using probabilistic power flow.In addition,it may provide macro statistical information under stable system operating conditions through the probability statistic method.Therefore,probabilistic power flow has become an important tool for the analysis of economic operation,reliability,safety and stability of power grid system [8].The probabilistic power flow calculation model mainly includes the DC model,linear AC model,multi-point linear model,and retaining-nonlinearity AC model.The calculation method comprises simulation method [9-11],estimation method [12-14],and analytical method [15-18].The existing probabilistic power flow algorithm mainly considers the random fluctuation characteristics of new energy sources,such as wind and photovoltaic power,and calculates the probability distribution of the node voltage and the branch power flow [19-20].However,because the frequency modulation characteristics of the unit and load are not considered,the imbalanced power caused by fluctuations of the new energy power is entirely absorbed by the slack machine,which is inconsistent with the actual power grid regulation mode,and the calculation results cannot reflect the operating characteristics of the system [21-23].In [22],the probabilistic power flow algorithm takes into account the multi-machine balancing strategy,which considers the influence on system power flow distribution incurred by the conventional system frequency modulation characteristic as well as conventional energy and intermittent energy complementary strategies.However,the state of the system is complex and variable in the actual operation of the system under disturbance because the response of the load and unit is affected by many factors [24].In particular,when the system encounters a significant disturbance,the uncertainty of load and unit response characteristics becomes more prominent,hindering subsequent frequency modulation control and system recovery.

This paper presents a probabilistic power flow method,in which the primary frequency response uncertainty of the load and the unit is simulated through the randomness of the static frequency characteristic coefficient under significant disturbance of the power system.A probabilistic power flow model is established.The method combines Monte Carlo sampling and the semi-invariant method for calculation and the Gram-Charlier series expansion method for estimating active power flow distribution after the fault.The improved IEEE-30-buses system is taken as an example to analyze the influence of response conditions on the power flow distribution.The results of the example case are compared with those of the Monte Carlo simulation method to verify effectiveness.

2 Uncertainty analysis of system frequency modulation characteristic

The static frequency characteristics of a power system is related to the load and the unit response as the system is subjected to unbalanced active power due to disturbances,which will be transferred to a new state through frequency modulation.In solving the existing probabilistic power flow,the unbalanced power generated by the disturbance is always handled by the slack bus alone or some controlled units are selected to handle the disturbance as a fixed partition.Power frequency characteristic curves are shown as Fig 1,which are not straight lines.The power frequency characteristics of a steam turbine have the features of curve 2,where each segment represents the role of one control valve according to the control position characteristic.A water turbine has only one valve,so the frequency characteristic is similar to curve 3.

Fig.1 Practical and ideal static frequency characteristic curves of generators

The frequency characteristics of the unit are affected by many factors,such as the unit load rate,unit heat storage,primary frequency modulation mode,heating mode of the heating unit,and mode of auxiliary machine.Furthermore,some plants cancel the primary frequency modulation mode for obtaining more benefits.The frequency characteristics of the load is relevant to the proportion related to the frequency response,and these factors are still unknow and uncertain.If the speed regulating parameters of the unit is not measured and the frequency response coefficient of each load is not identified,the frequency response results of original parameters are positively exaggerated,and the primary frequency regulation of the unit becomes unsatisfactory in actual operation.Therefore,it is very important to simulate these uncertain factors of the load and the unit in the probabilistic power flow calculation through the randomness of the static frequency characteristic coefficient,which can more comprehensively reflect the practical operating conditions,and to identify possible risks in the system.

In addition to the primary frequency modulation,there are the secondary frequency modulation,which usually refers to the generator active adjust based on automation generation control(AGC)and tertiary frequency modulation,which usually refers to the generation plan according to the foresting load based on the economic and safety criteria.Nevertheless,these modulations are not considered in this study.Only the following conditions are considered:

(1)when power imbalance occurs in the system,there should only be primary frequency modulation action;

(2)the uncertainty of the primary frequency modulation characteristic is reflected by the random distribution of the static frequency characteristic coefficient.Therefore,the power flow analysis considering the uncertainty of the system frequency modulation characteristics can be regarded a type of probabilistic power flow,and use the cumulants method to solve.

3 Probabilistic power flow model considering the uncertainty of frequency modulation characteristic

3.1 Unbalanced power distribution model

The following unbalanced power distribution model is established based on the analysis and assumption above.Assuming that the unit regulating power at node i is KGi,the load regulating power is KLi,and the random distribution models thereof are known.

The system active power flow equation is:

In the equation,n is the system node number,PGi and PLi respectively represent the active output of the unit and the load corresponding to node i,is the active power of node i expressed by the state variable,V represents the voltage amplitude,and θ represents the voltage phase angle.

When the system is subjected to disturbance,the unbalanced active power is expressed as:

where is the total unbalanced active power of the system,and PLoss is the total active power transmission loss in the system.

Ignoring system transmission loss,the unbalanced active power can be expressed as:

The unbalanced active power is jointly borne by all units with regulation capabilities and the load.Set the share ratio of the node i as ,which must satisfy:

can be determined from the aspect of equality,capacity,capacity margin,or active power frequency characteristic coefficient.In this paper,the fourth parameter is selected such that it can meet the trend of bulk power grid interconnection and the increasing amount of unbalanced active power,which is more consistent with the actual situation.

Then the active power equation changes to:

After the occurrence of disturbance,the degree of disturbance is known and the redistribution of disturbance frequency is determined according to the frequency change calculated:

where K∑ represents the unit power regulation of the system,KGi and KLi respectively represent the unit and load participation factor to node i,Ng represents the total number of the unit,Nl represents the total number of the load,and represents the frequency deviation incurred by the unbalanced power.

The unbalanced power will be distributed to each node with load and generator according to the static frequency characteristic.Due to the uncertainty of the power frequency characteristic,the static frequency characteristic coefficients are randomly distributed,and the randomly changed power of the generator is:

The active power variation of the load is:

The injection power variation of node i will be:

The updated injection power of node i will be:

Equation(10)can be simplified as:

In the equation:

If node i is not engaged in the primary frequency modulation,then

3.2 Cumulant of node input variable

In the conventional cumulant method,the corresponding cumulant analytical expression is determined by analyzing the mathematical expression of the input variable distribution function for the overall purpose of calculating the probabilistic power flow [26].This method requires the distribution type or the mathematical analytical expression of the input variable to be known.Here,the active power input is taken as a variable related to the random variable,unbalanced power distribution coefficient which is determined using equation(13).However even if the probability distribution of the unit regulation powers and of the unit and the load are known,the probability distribution of cannot be obtained directly,and the probability distribution of the input variable cannot be acquired either.Therefore,the cumulant cannot be calculated by the conventional mathematical expression deduction method of the input variable.Nevertheless,the method proposed in [18]can be used to obtain the cumulant of the input variable based on Monte Carlo sampling and to calculate the cumulant in this paper.

When the probability distribution functions of KGi and KLi are known,sampling by Monte Carlo simulation according to their distribution functions can be applied to acquire m groups of samples of and the origin moment of various orders of can be calculated based on the samples:

Then the cumulant of various orders of can be calculated by the relation between the cumulant and origin moment.

According to the homogeneity principle of cumulants [26],the kth order cumulant a times the size of the random variable is equal to ak times of the kth order of the variable.The cumulant of node input active power variation can be expressed as:

where represents the cumulant of various orders of represents the vth power of

If the probability distribution of the unit regulation powers KGi and KLi of the unit and the load cannot be acquired,the corresponding samples can be obtained from the stored historical data; then the cumulant of the node active power can be calculated.In practical electric power systems,the static frequency response characteristic coefficient can be acquired by processing historical disturbance data,which may provide a large amount of historical sampling values.Then the abovementioned method can be applied to determine out the cumulant.

3.3 Branch power probability model

From the linear deduction of the probabilistic power flow model described in Section 2,the branch power flow can be determined using the Quasi-steady state sensitivity method [27].

Set the selected node N as a reference node and form the augmentation sensitivity matrix,which adds the reference node in the conventional sensitivity matrix where the sensitivity element value obtained by the augmentation of the reference node is zero:

The quasi-stable-stable active power sensitivity is expressed as:

where represents the quasi-stable-stable active power sensitivity,and F is distribution coefficient matrix,which is defined as follows:

where can be obtained by substituting the average value into equation(13).

The branch power flow variation is calculated using the following equation:

where ΔPis the node injection variation vector associated with the unit and load disturbance.

According to the homogeneity and the additivity of the cumulant,cumulants of various orders of the branch power flow variation can be obtained by the cumulant of various orders of the active power variation of each node described in the previous section,which is expressed as follows:

where represent the values.

3.4 Branch power probability distribution

Through the above cumulant method,the expectation,variance,and finite order moment of the output random variables can be easily obtained.Nevertheless,it is also desirable to obtain the distribution function characteristics of the output random variables in the operation system.Therefore,the branch power probability distribution is calculated using only the Gram-Charlier expansion series,which has been described in [26].In addition,the 7 order Gram-Charlier expansion model is adopted.

4 Calculation process

A flow chart of probabilistic power flow considering the uncertainty of the static frequency characteristic is shown in Fig.2.

Fig.2 Flow chart of PLF considering the uncertainty of static frequency characteristic

5 Calculation example analysis

The presented approach is tested on the improved IEEE-30-buses system,which assumes that all generators have output,and the rated active power is the upper limit of the output.The unit key parameters are presented in Table 1.

Table 1 Data of the improved IEEE-30-BUSES unit

Ps Qs PGN 1 0.987 0.000 2.00 2 0.800 0.500 0.80 5 0.500 0.370 0.50 8 0.200 0.373 0.35 11 0.200 0.162 0.30 13 0.200 0.106 0.40 N

Suppose that the system suffers generator tripping fault and loses the generator at node 8,ignoring the system network loss caused by power transfer,then the unbalanced active power P∑Δ =-0.147.With the primary frequency modulation of the system,the unbalanced active power is reallocated by the load and adjustable unit respond according to the static frequency characteristic.According to the unit output active power,units 2 and 5 are limited to adjustment,and only 11 and 13 can share the unbalanced power of the slack bus.Assume that the frequency characteristic coefficient of the load and the generator are subject to normal distribution,KL ～ N(1.5,0.075),and the value range is(1,3); all units are steam turbo generators,KG ～N(25.0,1.25),and the value range is(0,50).Simple random sampling is run 10,000 times using the Monte Carlo simulation method.The frequency deviation and branch power flow change under different unit response conditions are compared and analyzed.

The impact of different frequency modulation responses on the power grid is difficult to assess through only the analysis of the uncertainty of the system’s frequency modulation characteristics.Therefore,contrastive analysis is conducted according to the following three conditions to better reflect the effect:

Case1:11# and 13# units respond;

Case2:11# unit fails to respond;

Case3:13# unit fails to respond.

Fig.3 Comparison for density of frequency deviation probability under the response of different units

If a unit fails to respond,the total unit power regulation of the system decreases,and the frequency deviation increases.Therefore,the frequency deviation is relatively small when all the units respond.The rated capacity and adjustable margin of 13# unit are larger than that of 11# unit.Therefore,the unit power regulation of 13# unit is larger than that of 11# unit under the same difference coefficient.When 13# unit does not respond,the total unit power regulation of the system is small,and the frequency deviation is larger under the same unbalanced power.

In order to better reflect the relationship between the change amount of the branch power flow and the initial value under different unit response conditions,the change amounts are shown with the relative change of branch power flow under different unit response conditions in Tables 2,3,and 4.Branches 7-5,25-24,8-6,6-4,and 2-1 show larger relative change of power flow.Branches 7-5 and 25-24 show small ground state flow value and active flow change after the disturbance,whereas branches 8-6,6-4 and 2-1 show larger ground state flow and active flow change,which deserves close attention.The flow changes of branches 2-1,6-4,and 8-6 are selected for digraph analysis of probability distribution.

Table 2 Ranking of branch power flow change for the response of all units

Branch Initial value Mean value of change Relative change 7-5 -0.0026 2.7415E-02 1054.41%25-24 -0.0021 7.1181E-03 338.96%8-6 -0.1192 1.6956E-01 142.25%6-4 -0.3479 9.0660E-02 26.06%2-1 -0.579 1.1960E-01 20.66%24-23 -0.0241 4.9065E-03 20.36%28-6 -0.1472 2.2915E-02 15.57%4-3 -0.3696 5.1141E-02 13.84%6-2 -0.3849 5.3197E-02 13.82%3-1 -0.3954 5.1774E-02 13.09%17-16 -0.0405 5.2222E-03 12.89%

Table 3 Ranking of branch power flow change for the response of only 13# unit

Branch Initial value Mean value of change Relative change 7-5 -0.0026 2.9368E-02 1129.54%25-24 -0.0021 6.4871E-03 308.91%8-6 -0.1192 1.6964E-01 142.32%6-4 -0.3479 9.6361E-02 27.70%24-23 -0.0241 5.6638E-03 23.50%2-1 -0.579 1.2871E-01 22.23%17-16 -0.0405 7.3256E-03 18.09%28-6 -0.1472 2.3402E-02 15.90%4-3 -0.3696 5.5613E-02 15.05%6-2 -0.3849 5.7262E-02 14.88%3-1 -0.3954 5.6293E-02 14.24%

Table 4 Ranking of branch power flow change for the response of only 11# unit

Branch Initial value Mean value of change Relative change 7-5 -0.0026 2.9557E-02 1136.80%25-24 -0.0021 5.9804E-03 284.78%

Continue

Branch Initial value Mean value of change Relative change 8-6 -0.1192 1.6972E-01 142.38%6-4 -0.3479 8.9462E-02 25.71%2-1 -0.579 1.3134E-01 22.68%28-6 -0.1472 2.3803E-02 16.17%4-3 -0.3696 5.7843E-02 15.65%6-2 -0.3849 5.7637E-02 14.97%3-1 -0.3954 5.8546E-02 14.81%4-2 -0.3089 3.7258E-02 12.06%24-23 -0.0241 2.7442E-03 11.39%

Fig.4 Comparison of the cumulative distribution of the branch power flow under different unit response

Table 5 Comparison of the variance of the distribution of the branch power flow change under different unit responses

ΔPL Distribution Variance All units respond 11# unit fails to respond 13# unit fails to respond Branch 2-1 1.5018E-06 1.18374E-06 9.56339E-07 Branch 6-4 1.3006E-07 4.4479E-09 2.42843E-07 Branch 8-6 2.4072E-09 2.78316E-09 2.08249E-09

According to Fig.3 and Table 5,the effects of different response conditions of the unit on the distribution of branch power flow change are different.Regardless of the response of the unit,the flow change of branch 8-6 is the largest,with more concentrated distribution.Furthermore,the distribution interval lengths of its flow change are the least.These results show that its flow change is not affected by the differences in the response of the units.In contrast,the flow change distribution of branch 2-1 is relatively dispersed,and the distribution interval length of its flow change is always the largest,which shows that its flow change distribution is heavily affected by the differences in unit response.In addition,the flow changes of branches 26-25,29-27,30-27 and 30-29 are 0 or close to 0,i.e.the flow is not changed and remains unaffected by the difference in unit response.

In conclusion,the frequency response characteristic of the generator set significantly affects the flow and frequency deviation.By selecting the branch that demands the most attention,we found that the uncertainty of the frequency response characteristics under different unit response conditions leads to different power flow distributions.Power flow analysis was performed fully considering the uncertainty of the static frequency characteristic of the system,which can provide more comprehensive reference information to the dispatcher for scheduling decisions.

To verify the effectiveness of the proposed method,it was compared with Monte Carlo simulation.Repeated deterministic power flow calculation was conducted in Monte Carlo Simulation,with 10,000 sampling iterations and case 1 was selected,where 11# and 13# units responded.

To further compare the performance of the proposed method in finding the weak link of the system,the branches are ranked based on the active power flow change of the branch.Furthermore,the results are compared with the ranking results obtained of Monte Carlo simulation.Table 6 shows the ranking result comparison of the top branches with larger relative change of active power flow.The comparison shows that although the ranking results of the two methods are not the same,the top branches with larger relative change include the weakest link.

Table 6 Ranking comparison of branch power flow change

Proposed method Monte carlo simulation Branch ranking Mean value of change Relative change Branch ranking Mean value of change Relative change 7-5 2.0359E-02 783.04% 7-5 2.7415E-02 1054.41%8-6 1.8727E-01 157.11% 25-24 7.1181E-03 338.96%25-24 1.5919E-03 75.80% 8-6 1.6956E-01 142.25%6-4 1.0508E-01 30.20% 6-4 9.0660E-02 26.06%2-1 1.3136E-01 22.69% 2-1 1.1960E-01 20.66%28-6 2.6201E-02 17.80% 24-23 4.9065E-03 20.36%24-23 4.2531E-03 17.65% 28-26 2.2915E-02 15.57%4-3 6.4226E-02 17.38% 4-3 5.1141E-02 13.84%6-2 6.4799E-02 16.39% 6-2 5.3197E-02 13.82%3-1 6.1539E-02 15.99% 3-1 5.1774E-02 13.09%

Table 7 Comparison of calculation time

Method Calculation time(s)Method in this paper 0.416 Monte carlo simulation 97.513

6 Conclusions

It is very difficult to assess the system state after disturbance because of uncertainties in the frequency modulation performance of the unit and load.This paper presented a probability power flow method,which considers the uncertainty of the static frequency characteristics of the system after the occurrence of large disturbance.The Monte Carlo sampling method and the cumulant method were combined for analysis and calculation to obtain the system frequency deviation and the probability distribution of the branch power flow variation.The main aspects can be summarized as follows:

(1)The frequency response characteristics of the generator unit significantly influence frequency deviation.However,in the existing probabilistic power flow considering the uncertainty of new energy,the frequency deviation is generally not analyzed and the frequency is determined to be relatively stable.

(2)The uncertainty of the frequency response characteristics under different unit response conditions lead to different power flow distributions.Power flow analysis was performed by fully considering the uncertainty of the system frequency modulation characteristics,and the obtained result was relatively consistent with the actual power grid operation characteristics.The results can provide reference for power system planning and safe operation analysis,enriching the application scenarios of the existing probabilistic power flow.

(3)The research of probabilistic power flow is a systematic subject.This paper presents the probabilistic power flow method based on the uncertainties of load and unit frequency response.The author considers the frequency response characteristics of load and unit in the future,and further improves the practicability of probabilistic power flow.

Acknowledgements

This Work was Supported by the State Grid Scientific and Technological Project(Title:Research on Control Strategy with Fast Demand Response to Severe Power Shortage,SGJS0000DKJS1700263).