Evaluation of Renewable Energy Hosting Capacity of Grid with Multi-Infeed HVDC

Considering the impact of large-scale renewable energy access to a multi-infeed HVDC receiving-end system on its voltage stability and multi-infeed short circuit ratio, a renewable energy acceptance evaluation method based on an improved holomorphic embedding method is proposed. Firstly, to construct a model containing only an AC system, the DC inverter is simplified into an AC source model; secondly, the sigma index is used to judge the static voltage stability of the system bus and determine the access location of renewable energy, and sensitivity index is used to determine the scheme of replacing conventional units with renewable energy; then, with the voltage crossing the set range as the constraint condition, the analytical expression of each bus voltage obtained by the holomorphic embedding method is used to solve the maximum renewable energy capacity of the system; finally, under the maximum hosting scenario of renewable energy, verify the multi-infeed short circuit ratio at each DC drop point. The effectiveness of the proposed method is verified on an improved IEEE39-bus system.


Introduction
As a relatively mature green power technology, renewable energy power generation plays an important role in the process of carbon neutrality in the world.However, with the gradual improvement of the DC transmission system, many regions show a clear trend of receiving DC.Large-scale renewable energy grid connection will hurt the stable operation of the AC/DC interconnection system.Among them, the voltage limit of the renewable energy grid connection point and the change of the AC system's support capacity for DC after the renewable energy is connected are issues that scholars are more concerned about.How to access the maximum capacity of renewable energy on the premise of stable operation of multi-in-feed HVDC systems has become an urgent problem to be solved.Therefore, building a renewable energy hosting capacity evaluation method suitable for multi-in-feed HVDC systems is of great significance.
At present, scholars have made many research achievements on the renewable energy hosting capacity of the communication system [1][2][3][4][5][6] .Wang [1] reviewed the factors affecting the maximum injected power of wind power, such as the system grid structure, the rotating standby level of conventional units, and the reactive power compensation of wind farms.Sun [2] evaluated the wind power acceptance capacity of the system based on the load characteristics and the peaking capacity.Wang [5] evaluated the maximum wind power acceptance capacity based on the power balance by considering the power source type, the peaking capacity, and the peak-to-valley difference and gave measures to improve the wind power acceptance capacity of the grid.However, there are few studies on the assessment of renewable energy acceptance capacity of AC/DC interconnected systems.
To solve the problem that conventional power flow calculation is strict with iterative algorithm and initial value selection, scholar A. Trias [7] proposed the Holomorphic embedding load flow method (HELM).Since HELM has the advantages of determining and unique initial value, non-iterative solution process, and obvious signal when power flow has no solution, this method is gradually applied to the field of power system static stability analysis [8][9][10][11][12] .Du [9] used the improved holomorphic embedding model to solve the static voltage critical point of the predicted system; Liu and Lai [10][11] proposed that based on the holomorphic embedding sigma trajectory, the order of intersection of the bus sigma trajectory and the instability boundary is taken as the criterion of the static voltage stability of the bus.The renewable energy hosting capacity evaluation method proposed in this paper is based on the holomorphic embedding model.The analytical expression of each bus voltage is obtained by solving the holomorphic embedding model, and the corresponding embedding factor value is solved with the voltage out of limit as the constraint condition.Finally, the maximum renewable energy hosting capacity of the system is obtained through the embedding factor value.

Constructing HELM model
The general expression of the conventional power flow equation of the N-bus power system is as follows: where Y ik is the (i,k) element of the bus admittance matrix, S i and V i are the injection power and voltage of bus i, V k is the bus voltage to be obtained, and the superscript * indicates conjugation.
The voltage and the reactive power output of the generator in the conventional power flow equation are embedded into the variable a, which is converted into the holomorphic functions V(a) and Q g (a) on the complex plane concerning the embedding factor a, as shown in Formula (2): The converted power flow equations are shown in the equation (3): where ΔS i is the scaling direction at PQ bus i, S ai =P gi -P li -jQ gi (a)-jQ li and ΔS ai =ΔP gi -ΔP li -ΔQ l ， ΔP gi , ΔP li, and ΔQ li is the scaling direction of active output of PV bus, the scaling direction of load active power and the scaling direction of load reactive power respectively.The above variables need to be customized to determine the scaling direction of the system.

Model solving method
The solution of the holomorphic embedding model is a recursive process.The recursive equation for the solution of the above model is shown in Formulas (4) ~ (8): In the formula, the conditions n≥1, V i [0], and Q gi [0] can be solved by the traditional holomorphic embedding model, and the analytical expressions of voltage and generator reactive output can be obtained according to the recursive equations of equations ( 4) ~ (8).

Equivalent DC inverter station
On the premise of only static stability calculation, Zhao [13] proved that a DC inverter station can be equivalent to an AC source model with impedance, and its equivalent topology is shown in Figure 1: The expressions of equivalent AC source potential E eq and equivalent impedance Z eq are, respectively: where U dc is DC voltage; n is the number of multiple inverter bridges and n=m/6 for m pulse inverter; k is the transformation ratio of the converter; β Is the arc extinguishing angle of the inverter; X is commutation reactance; θ h is the phase angle at the DC access point; δ 1 =-cos -1 (vcosβ) and v is the fundamental current factor.After the DC inverter station is equivalent to the AC source, the renewable energy hosting capacity is calculated, which avoids the simultaneous occurrence of AC and DC systems in the topology and can effectively reduce computational complexity.

Determine the access location of renewable energy
In this paper, the static voltage stability of each bus of the system is solved by the sigma index based on the holomorphic embedding method to determine the access location of renewable energy.
First, write the current balance equation for the two buses system as shown in Figure 2: Define sigma index σ to be equal to: The equivalent equation of the two buses circuit is obtained as follows: where U k is the normalized voltage and By decomposing the real part and the imaginary part, Equation ( 12) can be rewritten as: Equation ( 13) is a quadratic equation, which defines a parabolic boundary, as shown in Figure 3.Its general solution is: If the solution of the quadratic equation exists, the following conditions need to be satisfied: At this time, the sigma index meeting the above conditions is inside the boundary, indicating that the voltage is stable, as shown in the yellow area of Figure 3; If the conditions are not met, it will cross the above boundary, indicating voltage instability, as shown in the gray area of Figure 3.
Equation ( 12) can be used to calculate the normalized voltage U k and σ k to obtain the following equation after embedding a: Similarly, the sigma function can also be expressed in the form of power series: After obtaining the functional expression of the sigma index concerning the embedding factor a, the sigma trajectories of each bus can be drawn on the plane as shown in Figure 3, and the a k corresponding to the intersection of the sigma trajectories of each bus and the instability boundary can be recorded.The a k corresponding to the intersection can quantify the static voltage stability margin of the bus.The larger the a k , the more stable the bus is.To accept more renewable energy, renewable energy units should be connected at the location with strong system stability.

The calculation process of renewable energy hosting capacity
The calculation steps of renewable energy hosting capacity based on the holomorphic embedding method proposed in this paper are as follows: 1) Use the method described in Section 3.1 to simplify all DC inverter stations of the receiving system; 2) Conduct static voltage stability analysis on each PQ bus of the simplified system by using the sigma index described in Section 3.2, and determine the access position of the renewable energy unit by comprehensively considering the bus stability margin, the initial bus voltage, and the electrical distance from the DC drop point; 3) The "access and replace" balance way is used, which means the same capacity of conventional units needs to be replaced after the access to renewable energy units, considering that the reduction of conventional unit capacity will affect the multi-infeed short circuit ratio (MISCR), so the "unit replacement capacity-MISCR" sensitivity analysis is performed, and the small sensitivity units are preferentially selected for replacement; 4) To determine the renewable energy access point, power factor, and the configuration after the replacement scheme of the unit to scale the fully holomorphic embedding model parameters directionally, the model is solved to obtain the analytical voltage function of each bus and further to counter the corresponding embedding factor a when the voltage exceeds the limit; 5) Calculate the renewable energy access capacity at this point and check that MISCR is greater than 3 at the individual DC drop points; 6) If the MISCR is less than 3, the conventional set will need to reduce the renewable energy access capacity and restore the corresponding capacity until each DC drop point MISCR is met greater than 3; 7) If MISCR exceeds 3, the current renewable energy access level is the system's maximum hosting capacity.

Computational simulation
The IEEE39-bus standard model was refined by adding two 500 MW direct currents to 33 and 35 buses of the system while replacing the regular unit at both buses.The improved model is shown in Figure 4: Bring the DC operation parameters into Equation ( 9) to obtain: the equivalent AC voltage source amplitude |E eq |=352 kV, and the equivalent impedance Z eq =0.0907-0.0524j.

Calculate the location of the renewable energy grid connection
The static voltage stability of PQ buses of the simplified AC system is sorted by using the sigma index described in Section 3.2.The static voltage stability margin and voltage amplitude of some buses are shown in Figure 5.
Considering the factors such as the bus voltage vulnerability, the initial voltage amplitude, and the distance from the DC drop point, buses 6, 10, and 23 are selected as the access points of the renewable energy units, and the scaling directions of the three renewable energy units are set ΔS=-(100+48.4j)MVA, the leading power factor of renewable energy units is 0.9.

Determine unit replacement scheme
In the improved IEEE39-bus model, the MISCR at 33 and 35 buses are 3.87 and 3.89, respectively.Since the MISCR is closely related to the short-circuit capacity of conventional units, the impact of unit capacity replacement on MISCR at the DC feed point is an important basis for formulating a unit replacement scheme.The sensitivity analysis of "unit replacement capacity-MISCR" is carried out.The specific scheme is to connect 100 MW renewable energy at any point, reduce the 100 MW active output of 7 conventional units in turn and adjust their capacity to obtain the impact level of each unit's capacity reduction on the DC feed point.The results are shown in Table 2 It can be seen from Table 2 that reducing the unit capacity near the DC feed point will have a great impact on the MISCR.
Since unit 31 is a balanced unit, it is not included in the replacement scope.The unit reduction scheme can be divided into several stages:  Stage 1: Select the three units with the least influence on the MISCR as 39, 38, and 30, and the scaling direction of active power output is ΔP=-100 MW, if there is no bus voltage exceeding the limit before the output of unit 30 drops to 0 (a 1 ≤2.5), the next stage will be entered;  Stage 2: units 39 and 38 continue to reduce output, and 32 units begin to reduce output.The scaling direction of active power output is ΔP=-100 MW, if there is no bus voltage exceeding the limit before the output unit 38 drops to 0 (a 2 ≤5.8), the next stage will be entered;  Stage 3: units 39 and 32 continue to reduce output, and 37 units begin to reduce output.The scaling direction of active power output is ΔP=-100 MW, if there is no bus voltage exceeding the limit before the output of unit 32 drops to 0 (a 3 ≤0.7), the next stage will be entered;  Stage 4: units 39 and 37 continue to reduce output, and 34 units begin to reduce output.The scaling direction of active power output is ΔP=-100 MW.If there is no bus voltage exceeding the limit before the output of unit 39 drops to 0(a 4 ≤1), the next stage will be entered.

Calculate renewable energy hosting capacity
Solve the Holomorphic embedding model scaled by the direction, obtain the analytical expression of the voltage of each bus concerning the embedding factor a, and the bus voltage range is set to be 0.95~1.05(p.u.), and the corresponding embedding factor a when the voltage exceeds the limit is inversely solved.Therefore, it is only necessary to satisfy any one of the equations ( 18 The simulation is performed according to the above steps.When the simulation proceeds to the third stage and a = 0.67, bus 10 first reaches the set voltage upper limit value of 1.05.Table 3 shows the renewable energy hosting capacity of each stage.The total load of the improved IEEE39-bus system is S LD =(6254.23+1487.10j)MVA.The ratio of the installed capacity of renewable energy connected to the power grid to the total load of the system is used to characterize the renewable energy penetration rate.The maximum renewable energy penetration rate of the system is 46.51% under the constraint of voltage out of the limit.

Verify the MISCR at the DC drop point
The capacity replacement of conventional generating units directly affects the short-circuit capacity of the units.The renewable energy units connected to the system in the form of load buses will affect the system impedance and thus change the multi-infeed interaction factor.Therefore, in the process of increasing the access capacity of renewable energy units, the multi-feed short-circuit ratio of the DC feed point is also changing.To prevent the support ability of the AC system to the DC system from decreasing to an unacceptable level, it is necessary to verify the multi-feed short-circuit ratio.
The MISCR at bus 33 is reduced from 3.87 to 3.60 and at bus 35 from 3.89 to 3.51 with the renewable energy connection.The MISCR at the two DC drop points meets the requirement of greater than 3. Therefore, it is determined that the maximum penetration rate of renewable energy of the DC receivingend system is 46.51%.

Conclusion
Evaluating the renewable energy hosting capacity of AC/DC systems has a certain guiding significance for power grid planning.This paper presents a renewable energy acceptance evaluation method for a multi-in-feed HVDC system based on the holomorphic embedding method.The main work is summarized as follows: 1) The improved holomorphic embedding model is constructed, and the solution process is given; 2) On the premise of only static stability calculation, the DC inverter station is simplified as an AC source model with impedance, and the model with only an AC system is constructed, which reduces the complexity of subsequent calculation; 3) The access location of the renewable energy field is determined by using the sigma index based on holomorphic embedding, and the unit replacement scheme is determined by using the sensitivity analysis of "unit replacement capacity-MISCR"; 4) The voltage exceeding the limit is set as the constraint condition, and the MISCR is used as the verification condition.The holomorphic embedding model is used to solve the maximum renewable energy hosting capacity of the system, and the calculation results of the MISCR are used for verification and correction.

Figure 1 Figure 2
Figure 1 Equivalent diagram of DC inverter station

Figure 3
Figure 3 Schematic diagram of system instability boundary

Table 1 .
DC operation parameters table 4.1.Calculation of simplified DC model parametersTo simplify the calculation, the DC inverse station is fed into a simplified impedance-charged AC voltage source model as described in Section 3.1.TheDC operation parameters are shown in Table1:

Table 3 .
Hosting capacity calculation table