Multi-objective partition planning for multi-infeed HVDC system

0 Introduction

The advantages of the high-voltage direct current(HVDC) technique in bulk power supply promote its application in cross-region transmission,particularly in the optimal allocation of renewable energy [1-3].The close proximity of multiple HVDCs fed into the same synchronous grid tends to be inevitable in the load centers,e.g.,those in Eastern China [4-5].Concerning the linecommutated converter (LCC),adequate voltage support is crucial [6-7].Additionally,regarding supply sufficiency,the different infeed power properties due to the intermittency of renewable energy transmitted by HVDCs should be considered.Typically,applying a grid partition in the stage of power planning is effective for identifying the principle to operate the power grid [8-10].

The reliability of LCC-HVDC largely depends on the reactive power supply [11-12].However,the attention has only been paid to the capacity of supplementary reactive power in multi-infeed HVDC system previously.For example,a large capacity of synchronous compensators,static var compensators,etc.is usually installed exactly in the converter stations [13-14].Some LCC-HVDCs are planned to be revamped to voltage source converter (VSC)-based HVDCs.The efficiency and the effects of reactive power support are also important.The reactive power resources with strong mutual effects should be mounted within the same partition to achieve good performance.On the other hand,it is better to make the power supplies within a partition complement each other,which typically enhances the power sufficiency and reliability with high penetration of renewable energy.

For a long time,research on the grid planning and control in multi-infeed HVDC systems mainly focused on improving the transient responses and the recovery performance of HVDCs subjected to AC faults by supplying more reactive power [15-17].Few studies focused on the efficiency of voltage control and the energy complementarity between multiple HVDCs,although for the traditional AC system,relevant research has been performed.For realizing the reasonable coordination and distribution of the reactive power flow,the method of secondary voltage control (SVC) was proposed in [18].To maintain the voltage of the pilot buses and thus the entire voltage quality,SVC requires that the pilot buses be voltage representatives and that the coupling of nodes within a control area be intensive.For handling the situation where weak coupling between different areas cannot be satisfied,coordinated SVC (CSVC) was developed in [19]and [20].This thought makes it is possible to enhance the ability of multiple HVDCs to withstand a disturbance.With the inverter stations regarded as dynamic reactive power sources,the voltage stability in the receiving grid was improved in [21].With regard to the complementary coordination of various energy types,an evaluation index was established on the basis of the hybrid wind-solar-hydro power generation system in [22].A case study indicated that the index can track the system load well and have strong intraday complementarity.The feasibility of the coordinated control in an intermittent power-plant cluster was analyzed in [23].The objective of the aforementioned studies was to optimize the performance of the whole system by dividing it into partitions and applying the coordination strategies to them,inspiring multi-objective coordination and optimization in multi-infeed HVDC systems.

In the present study,the idea of power-grid partitioning is combined with the coordinated optimization of the multiinfeed HVDC system.A multi-HVDC partition method is proposed.Considering the multiple objectives,including the voltage support,energy complementation,and avoidance of HVDCs with the relevant rectifier and the inverter stations located in the same sending and receiving areas,the corresponding indices are quantified in this paper.Thus,the partition model is constructed by incorporating the graph theory with the proposed indices.The HVDC partition problem is transformed into a graph-cut problem,and a spectral clustering algorithm is used to solve it.Finally,the HVDC partition scheme is validated by taking an actual regional grid in Eastern China as an example.

1 Proposed Indices for Partition Method for Multi-Feed HVDC System

Cluster analysis has been widely used in the study of power-grid partitioning in recent years [24-26].HVDC partitioning involves dividing multiple HVDCs into different partitions according to the specific relevance between the HVDCs.Quantitative evaluation of the relevance is the preliminary basis.This section presents four indices that are used in the partition method.

1.1 Index of voltage coordination

To ensure the reliability of power supply delivered by HVDCs,the requirements for the voltage quality in converter stations are strict.Thus,the HVDC converter station typically has a high capacity of reactive power compensation.The voltages of the converter buses in different HVDCs affect each other.The converter buses are regarded as the pilot buses in the SVC/CSVC method.Therefore,the HVDCs with strong coupling should be assigned to the same partition.

The node voltage equation I = YU reflects the connection of the nodes in the system [27].By inversing Y,the node impedance matrix Z = Y-1 is obtained.The mutual impedance Zi,j reflects the degree of electrical coupling between nodes i and j.The mutual impedance Zi,j is thus used as an index to describe the electrical coupling between different HVDC converter buses in this paper.The index is defined as follows:

where Zi,j represents the mutual impedance of converter buses i and j.

A larger CPi,j corresponds to stronger coupling between converters i and j.In accordance with SVC/CSVC,the voltage supports have better effects on the buses with stronger coupling; thus,converters with a large CPi,j are expected to be in the same partition.The value ranges of different indices are normalized as follows:

where CP represents the aggregation of CPi ,j(i ≠ j ).

1.2 Index of energy complementarity

Multiple HVDCs can be regarded as power sources for the grid that they are fed into.Considering the intermittency of renewable energy,it is better to mix various types of power infeed in a partition for ensuring the supply sufficiency to the greatest extent possible.According to this idea,the index CE is derived:

where σ* represents the standard deviation normalized in a manner similar to (2).Pdi,m represents the capacity of the mth power type in the ith HVDC.CEi,j = CEj,i = 0 when i = j or the number of power types included in HVDCs i and j is 1.Only different power types are considered to be complementary in this paper.A smaller capacity difference between any two different power types indicates better balance,better complementary effects,and a larger CE.Thus,the grid partition should pursue larger CE.

1.3 Index of reactive power support from VSCHVDC

In some practical projects,the infeed LCC-HVDCs are replaced with VSC-HVDCs owing to their advantages,e.g.,their flexibility and enhancement of the reactive power supports [28-29].To make full use of VSC-HVDCs,it is reasonable to cluster the closely related VSC-HVDCs and LCC-HVDCs into the same partition.The related index is defined as follows:

where Si represents the VSC-HVDC capacity,and represents the voltage coupling between converters i and j.

A larger Qi,j represents a better support effect between the ith VSC and the jth LCC.

1.4 Index representing risks in case of same sending and receiving ends of HVDCs

When the HVDCs located in the same receiving area also come from the same sending area,the possible fault usually incurs more risks.To reduce such risks,it is a good choice to cluster the HVDCs with different sending area in a partition.The index representing this risk is defined as (5).

where i and j correspond to the ith and jth HVDCs,respectively.

2 Partition approach based on spectral clustering

Spectral clustering [30-32]is one of the most popular clustering algorithms.It is based on spectral graph theory and regards each sample as a vertex in the graph.The edge between vertexes is weighted by the similarity between samples.The clustering problem is equivalent to the segmentation problem of the graph,which has intuitive physical meaning.In the clustering results,the degree of similarity between the samples in the same partition is greater than that between different partitions.In this study,each HVDC is treated as a vertex (V) in the graph(G),and the edge (E) between vertexes is weighted by the correlation between HVDCs.Therefore,HVDC partitioning is converted into the segmentation of the graph G.The implementation steps are described as follows.

2.1 Calculation of similarity matrix

According to the relevance indices mentioned in Section 2,the HVDCs with a large index value are clustered into the same partition.By synthesizing various indices,the similarity matrix can be derived,as indicated by (6).To make the contribution of each index to the correlation evaluation of multi-infeed HVDC system the same,the weight coefficient is defined,as indicated by (7).

2.2 HVDC partitioning procedure

• Construct the Laplacian matrix where

• Calculate the eigenvalues of the matrix Lsvm and sort them as η1 ≥ η2 ≥ …≥ηn.

• Select the variable k to be the clustering number,identify the eigenvectors {u 1 ,u2,… ,uk }corresponding to the first k eigenvalues of the matrixand construct the matrix

• Unitize each row of U to obtain the matrix T,where

• Regard each row of matrix T as coordinate of a vertex related to an HVDC system in the space Rk.Then,the clustering result is obtained by applying the k-means algorithm,and the vertex in each cluster represents an HVDC system with the corresponding serial number.

3 Case study

In this section,a practical power grid in China is taken as an example to validate the proposed partition planning method.Twelve HVDC systems,including two VSCHVDCs,are fed in this grid.The relevant information is presented in Fig.1 and Table1.

Fig.1 Diagram of the multi-infeed HVDC power grid

Table1 Information of HVDC systems

HVDC Energy composition transmitted by HVDC (MW) Converter type Sending area Thermal power Hydropower New energy XT 7300 0 2700 LCC A1 JN 8000 0 0 VSC A2 JS 0 7200 0 LCC A3 LZ 0 3000 0 LCC A4 ZW 8000 0 4000 LCC A5 XS 0 6400 0 LCC A3 JF 0 3000 0 LCC A4 SH 0 3000 0 LCC A4 GN 0 1200 0 LCC A4 YZ 0 8000 0 LCC A3 LS 4000 0 4000 VSC A5 BZ 0 8000 0 LCC A3

By inverting the nodal admittance matrix of the network,the mutual impedance matrix of each HVDC converter bus can be derived,as shown in the Appendix.Other indices are calculated using (3)-(5) according to the information shown in Table1.Then,the similarity matrix of multiple HVDCs is derived by weighting each index with (6) and (7),as shown in Table2.

Table2 Similarity matrix of multiple HVDC systems

XT JN JS LZ ZW XS JF SH GN YZ LS BZ XT—0.610.150.140.200.160.150.160.140.130.230.13 JN0.61—0.190.250.270.180.240.250.170.120.200.10 JS0.150.19—0.150.140.020.100.090.090.020.280.01 LZ0.140.250.15—0.140.070.020.010.010.110.290.07 ZW0.200.270.140.14—0.160.170.150.140.150.240.14 XS0.160.180.020.070.16—0.280.210.320.090.420.02 JF0.150.240.100.020.170.28—0.150.360.120.480.09

According to the similarity matrix W,the Laplace matrix Lsym is constructed,and its eigenvalues are sorted as follows:{ }η ={0,0.8045,0.8730,1.0070,0.0317,1.0560,1.0860,1.1001,1.1674,1.2048,1.2817,1.3877}.The number of clusters is initially determined to be four,and the eigenvectors corresponding to the first four eigenvalues are used in the spectral clustering algorithm,yielding four HVDC partitions,as shown in Fig.2 and Table3.The variable k represents the number of the clusters.

continue

XT JN JS LZ ZW XS JF SH GN YZ LS BZ SH0.160.250.090.010.150.210.15—0.120.100.390.08 GN0.140.170.090.010.140.320.360.12—0.120.410.08 YZ0.130.120.020.110.150.090.120.100.12—0.560.05 LS0.230.200.280.290.240.420.480.390.410.56—0.65 BZ 0.13 0.10 0.01 0.07 0.14 0.02 0.09 0.08 0.08 0.05 0.65 —

Fig.2 Schematic of HVDC partitioning results

Table3 HVDC partitioning results (k = 4)

Partition HVDCs involved HVDC partition 1 XT,JN,ZW,SH HVDC partition 2 JS,LZ HVDC partition 3 XS,JF,GN HVDC partition 4 YZ,LS,BZ

Four types of considerations are obvious reflected in HVDC partitions 1 and 4.According to the similarity matrix,the degree of similarity of HVDCs within the same partition is greater than that between different partitions.The HVDCs in partitions 1 and 4 have diverse powerinfeed types,sending areas,and converter types.In contrast,the power types and converter types in HVDC partitions 2 and 3 are identical,but the coupling relationship within each partition is significantly stronger than that in HVDC partitions 1 and 4 (as shown in Table4),and such a relationship is the main property reflected in the synthesis of four factors.The coupling degree between the HVDC SH and the HVDCs in HVDC partition 3 is relatively high.The reason why they are not grouped into the same partition is that the sending areas are identical in HVDC partition 3.The proposed strategy is an optimization approach based on an existing grid and the four indices presented in the second section.The goal of HVDC partitioning is to provide a better coordination scheme for optimizing the management and control of HVDCs rather than changing the network.

However,suggestions for future planning improvement can be provided according to the HVDC partitioning results.For instance,because VSC technology significantly improves the active and reactive power support,the LCC can be replaced with a VSC.Therefore,the converter types in partition 3 become diverse.Additionally,the transmission corridors and lines can be reused.In contrast,the single power type (hydropower) in HVDC partition 3 provides a higher potential of renewable energy consumption.Increasing the proportion of renewable energy in the process of energystructure adjustment is a promising option in the future.

Table4 Average values of the coupling strengths

Within the same partition Between different partitions Partition Average value Partition pairs Average value HVDC partition 1 0.1033 HVDC partitions 1 and 2 0.1500 HVDC partition 2 0.7833 HVDC partitions 1 and 3 0.0538 HVDC partition 3 0.2700 HVDC partitions 1 and 4 0.0500 HVDC partition 4 0.1667 HVDC partitions 2 and 3 0.0650—— HVDC partitions 2 and 4 0.1222—— HVDC partitions 3 and 4 0.0517

The results obtained when the number of clusters is increased to five are presented in Fig.3 and Table5.The HVDC SH is grouped into a separate partition.This is mainly because the coupling relationship between the SH and the HVDCs in partitions 1,2,and 5 is relatively weak.If the LCC in HVDC partition 3 is replaced with a VSC,the SH can be incorporated into HVDC partition 3.

Fig.3 Schematic of HVDC partitioning results

Table5 HVDC partitioning results (k = 5)

Partitions HVDCs involved HVDC partition 1 XT,JN,ZW HVDC partition 2 JS,LZ HVDC partition 3 XS,JF,GN HVDC partition 4 SH HVDC partition 5 YZ,LS,BZ

To validate the advantages of partitioning,the effects are simulated in PSD-BPA according to the HVDC partitioning results with four clusters.Then,a three-phase short-circuit fault is assumed to occur in the converter buses of XT or XS.The fault occurs at the 20th cycle and lasts for five cycles.The voltages of the converter buses and the extinction angles of the HVDCs are shown in Figs.4-7.

Fig.4 Voltages of the converter buses in partition 1

Fig.5 Extinction angles of HVDCs when a fault occurs in the converter bus of the HVDC XT

Fig.6 Voltages of the converter buses in partition 3

Fig.7 Extinction angles of HVDCs when a fault occurs in the converter bus of the HVDC XS

According to the voltage curves,the voltage recovery time of the XT converter buses was reduced by 0.857 cycles after partitioning.In Fig.6,the voltages of the JF and GN converter buses drop below 0.33 and 0.44 p.u.,respectively,during the fault period,whereas the voltages are maintained above 0.33 and 0.44 p.u.,respectively,if partitioning is performed.This indicates that partitioning the reactive power resources with strong mutual effects in the same partition provides good voltage regulation.As shown in Fig.5,the number of consecutive commutation failures of the HVDC decreases from eight to seven after partitioning.In Fig.7,the number is reduced by three,and the duration of commutation failure of the JF is reduced by 0.428 cycles.It can be concluded that the proposed method significantly improves the security and stability of the multi-infeed HVDC power grid.

4 Conclusions

A partition method is proposed for optimizing the coordinated control and the management of multiple HVDCs located in close proximity.The following conclusions are drawn.First,four indices can be formulated to represent the considered optimization factors.Secondly,the partition planning can be transformed to into a graph-cut problem and solved via the spectral clustering algorithm.Thus,the objectives are implemented in the processing of clustering.

The proposed method in this paper was validated for a specific grid structure in China,and the indices adopted are comprehensive.It is not guaranteed that all the indices are optimal after clustering.However,the results of this study provide constructive suggestions for future planning.

Acknowledgements

This work was supported by the Science and Technology Project of State Grid Corporation of China:“Control Strategy Optimization Technology for Large-Scale Photovoltaic Power Generation on the Sending-end and Receiving-end of DC Power System” (4000-201934198A-0-0-00).

Declaration of Competing Interest

We declare that we have no conflict of interest.

Appendix A

TableA1 Coupling matrix of converter buses

XT JN JS LZ ZW XS JF SH GN YZ LS BZ XT—0.240.060.020.050.040.060.070.040.010.000.00 JN0.24—0.040.060.090.030.050.060.030.020.010.00 JS0.060.04—0.270.060.070.120.080.080.050.020.02 LZ0.020.060.27—0.080.040.050.030.030.150.030.04 ZW0.050.090.060.08—0.100.160.110.110.120.100.08 XS0.040.030.070.040.10—0.630.440.720.260.090.06 JF0.060.050.120.050.160.63—0.411.000.180.120.09 SH0.070.060.080.030.110.440.41—0.330.130.070.06 GN0.040.030.080.030.110.721.000.33—0.160.080.06 YZ0.010.020.050.150.120.260.180.130.16—0.160.13 LS0.000.010.020.030.100.090.120.070.080.16—0.21 BZ 0.00 0.00 0.02 0.04 0.08 0.06 0.09 0.06 0.06 0.13 0.21 —