Two-stage ADMM-based distributed optimal reactive power control method for wind farms considering wake effects

Zhenming Li,Zhao Xu,Yawen Xie,Donglian Qi,Jianliang Zhang

1.Department of Electrical Engineering,The Hong Kong Polytechnic University,Kowloon,HKSAR,P.R.China

2.College of Electrical Electronic Engineering,Zhejiang University,Hangzhou,P.R.China

3.Quzhou Power Supply Company,Zhejiang Electric Power Company of State Grid,Quzhou,P.R.China

Abstract: Since the connection of small-scale wind farms to distribution networks,power grid voltage stability has been reduced with increasing wind penetration in recent years,owing to the variable reactive power consumption of wind generators.In this study,a two-stage reactive power optimization method based on the alternating direction method of multipliers (ADMM)algorithm is proposed for achieving optimal reactive power dispatch in wind farm-integrated distribution systems.Unlike existing optimal reactive power control methods,the proposed method enables distributed reactive power flow optimization with a two-stage optimization structure.Furthermore,under the partition concept,the consensus protocol is not needed to solve the optimization problems.In this method,the influence of the wake effect of each wind turbine is also considered in the control design.Simulation results for a mid-voltage distribution system based on MATLAB verified the effectiveness of the proposed method.

Keywords: Two-stage optimization,Reactive power optimization,Grid-connected wind farms,Alternating direction method of multipliers (ADMM).

0 Introduction

There is a prevailing trend to integrate various renewable energy generation sources into modern power systems featuring sustainable development and decarbonization.Among all others,wind energy distinguishes itself with its advantages of considerable power capacity,high efficiency,and wide availability.Particularly,the doubly fed induction generator (DFIG)has attracted significant attention owing to its low cost and flexible controllability.Specifically,the active and reactive power of the DFIG can be decoupled and controlled independently,which facilitates its manipulation of reactive power generation and voltage support to the grid.

The use of voltage and reactive power control systems is mandated in modern wind farms (WFs).They are used to regulate the voltage at the point of connection (POC)within a certain range to counteract the destabilization effects of intermittent wind power generation.Meanwhile,the terminal voltage of wind turbines (WTs)must be maintained within a feasible range to prevent trips of WTs caused by severe faults [1].As wind generation exhibits a strong potential for grid voltage stabilization,several studies have reported that wind generation is a feasible dispatch resource to control and optimize the reactive grid power flow [2].

Centralized control algorithms such as secondary proportional integral (PI)control,model predictive control,and artificial intelligence algorithms have been applied for reactive power control of WFs and WTs.As reported in[3,4],the secondary PI control can be applied to operate the grid-tied WT reactive power for eliminating the grid voltage fluctuation.A direct voltage control method,which can keep the terminal voltage within an acceptable range,is proposed in [5] to protect the associated grid appliances.Based on the Adaptive Nondominated Sorting Algorithm III and the Latin Hypercube Sampling Algorithm,a robust dynamic reactive power optimization method is reported in[6] for the reactive power control of wind power generation.In [7],the particle swarm optimization (PSO)algorithm is used to find the optimal reactive power output of WFs to minimize the distribution loss.Reference [8] formulates an optimization problem to maximize the renewable energy harvesting within the operational and environmental limitations for a wind-photovoltaic hybrid renewable energy system.However,centralized control schemes have drawbacks such as (i)high communication cost,(ii)heavy computational load [9,10],(iii)slow computing speed,and(iv)stringent hardware requirements in large-scale WFs.

Among distributed optimization control methods [11-14],the ADMM algorithm,which is applied to solve largescale convex optimization problems,has been reported to optimize the reactive power of the grid in a distributed manner [15].Similar research has been reported in [16],where the coordination among reactive power sources of multiple WFs in a voltage-source-converter high-voltage DC system is achieved.To minimize the losses of the generators,converters,filters,and the grid,a distributed optimal reactive power control scheme based on ADMM is proposed in [17].This optimization scheme only demands information exchange between adjacent controllers,which reduces the communication infrastructure.In addition,to achieve the optimal coordination among grid,WFs,and WTs subjected to multiple constraints and optimization objectives,a hierarchical optimization structure has also been reported [18-21].By combining the two-layer optimization structure and the ADMM algorithm,the WF reactive power reference is optimized in the first layer optimization structure according to the dispatch command from the transmission system operators (TSO).Then,the reactive power outputs of WTs in the WF are optimized in the second layer [22-24].

In the practical ADMM algorithm application,the global variable change is usually taken as the convergence criterion.In recent studies,the utilization rate of WFs or WTs is considered as the global variable and the consensus of utilization ratio is modeled as the convergence criterion[17,22].However,the consensus of utilization ratio is an ideal assumption.The idea of area partition is used to manage this problem.It can be considered as a centralized supportive distributed algorithm,which has been discussed in several articles [25-27].In the partitioning algorithm,only the coupled region information is exchanged between adjacent regions.Compared with the aforementioned consensus-based ADMM algorithm,the partition-based ADMM algorithm is more flexible in global variable selection.The idea of partition has been reported in [26,27] to determine whether the algorithm will converge by comparing the change of state variables in the coupled region.

When considering the control and optimization algorithm design,wake effect is an important issue for large-scale WFs [28-32].However,fewer studies have discussed voltage optimization considering the wake effect,which is one of our study’s contributions.

Based on this philosophy,a two-stage distributed reactive power output optimization control method is proposed in this study.The key contributions can be summarized as follows:

i)A two-stage optimization structure is established.The upper layer is used to optimize the reactive power in the distribution network with the objective of tracking TSO reactive power dispatch commands.The lower layer is applied to optimize the reactive power among the WFs and track reactive power commands generated by the distribution network layer.

ii)Based on the idea of partition,the coordination of distributed reactive power flows is simultaneously optimized in both the grid and WF layers.

iii)In the WF layer,the wake effect is modeled with respect to different weighting factors,which is more consistent with the practical situation.

The remainder of this article is organized as follows.The proposed two-stage distributed voltage optimization control scheme is discussed in Section 1.Then,the loss calculation of the grid and WF is elaborated in Section 2.Details about ADMM and the proposed optimization method are presented in Section 3.Simulation verifications are provided in Section 4.Finally,the conclusions and potential future work are summarized in Section 5.

1 Modeling of the proposed two-stage distributed voltage optimization control scheme

1.1 Structure of the grid-connected WFs

Fig.1 shows the typical WF structure connected to the distribution network through transformers.It is supposed that several WFs are connected to the distribution network through collector buses,and each collector bus is connected to several feeders,to which several WTs are connected.

Fig.1 Example of the WF structure connected to the distribution network

1.2 Structure of the double-fed induction WT

Fig.2 shows the structure of the double-fed induction WT when connected to a feeder.The WT blades are connected to the generator through a gearbox.The WT stator is directly connected with the feeder to realize the active and reactive power exchange with the feeder.Additionally,the WT rotor is connected to the feeder through a rotor-side converter (RSC)and a grid-side converter (GSC),where active power,reactive power,and frequency are controlled by AC excitation.With the electronic converters,the active and reactive power produced by WTs can be independently controlled.

Fig.2 Structure of the double-fed induction wind turbine [33]

1.3 Structure of distributed voltage optimization control

Fig.3 shows an example of the proposed structure of the distributed voltage optimization control method.It is composed of two calculation stages:

Fig.3 Example of the proposed structure of the distributed voltage optimization control method

First stage: Calculation ofin the distribution network layer.The reactive power referenceis decided for several areas of the distribution network,with an objective to minimize the total power losses in the distribution network.To solve the optimization problem in a distributed manner,the distribution network is separated into several areas,and each area is equipped with a controller to ensure that it can operate only with the local measurements and data from the neighboring area.

Second stage: Calculation ofin the WF layer.The reactive power referenceof each WT is decided by WT controllers according to thereceived from the first layer.The WTs in the WF operate in a distributed manner to minimize the total power losses inside the WF.To make full use of the reactive power potential under the premise of considering the active power output,wake effects will be considered while tracking thecommand.

2 Loss calculation of the grid and the WF

Loss calculation is presented in this section,including the loss calculation of the distribution network and the WFs.

2.1 Loss calculation of the grid

As for the voltage optimization control of the distribution network,it is a primary condition that the voltage of each node in the system runs within the allowable range.Furthermore,economic operation is also an important factor to consider.In this study,active power loss is proposed as an indicator to reflect network loss:

whereIijis the current onLineij,Rijis the resistance ofLineij,Viis the voltage ofNodei,Pijis the active power at the head ofLineij,Qijis the reactive power at the head ofLineij,andi,jis the node number of the distribution network withNnodes in total,i,j∈{1,2,…N}.

Based on Kirchhoff's law of voltage and current,we can obtain the power flow as follows:

whereXijis the reactance ofLineij;VNis the rated voltage of the network (supposing thatNode1is the generator node);∑Pjkand ∑Qjkare the sums of the active and reactive power fromNodejtoNodek,respectively;andare the active and reactive power of nodal nodes,respectively;andare the active and reactive power produced byNodej,respectively,and are defined as

Furthermore,the reactive power that a WF can produce is limited by its capacity and active power,that is,

whereis the maximum reactive power that a WF connects toNodejcan produce,andis the apparent power of the WF.

According to the operational requirements of the distribution network,the voltage is supposed to satisfy,

2.2 Loss calculation of the WF

Because the WF model is also a radial network [12],the WF connection mode can be determined by referring to the distribution network model.The WF loss is primarily composed of loss from collector bus to slack bus,feeder loss,and component loss of WTs.Reference [17] provides the calculation model of the loss of each part,which is referenced in this study.

The collector bus losses can be calculated as,

whereRcolis the resistance between the collector and slack buses.PcolandQcolare active and reactive power from the slack bus to the collector bus,respectively.Vcolis the voltage of the collector bus.

The feeder loss is modeled as,

whereRi,jis the resistance between nodeiand nodej,Pi,jandQi,jare active and reactive power from nodeito nodej,respectively.Viis the voltage of nodei.

The component loss of a WT is modeled as,

whereis defined as copper losses in the stator and rotor,is the los ses of converters (rotor- and grid-side converters),andis the losses of the filters.

Based on the calculation models in [17],andcan be calculated as,

whereR=diag(Rs,Rs,Rr,Rr),RsandXsare the equivalent resistance and reactance of the stator,respectively.RrandXrare the equivalent resistance and reactance of the rotor,respectively.Other parameters can be calculated according to the following formula,

3 Distributed voltage optimization control scheme

The distributed voltage optimization control scheme is realized based on ADMM and partitioning.The interactive information only includes the state of the coupling branch,which reduces the demand for data transmission.A parallel computing architecture can be adopted,and when the number of nodes increases,it has good scalability.

3.1 Concept of ADMM-based optimization method

The ADMM is an algorithm that is intended to blend the dual ascent decomposability with the superior convergence properties of the method of multipliers [15].The algorithm solves problems in the form:

with variablesx∈ℜnandz∈ℜm,whereA∈ℜp×n,B∈ℜp×m,andc∈ℜp.f,andgare convex.

As in the method of multipliers,the augmented Lagrangian is formed as follows:

ADMM consists of the iterations,

3.2 Concept of radial network partition

According to the optimization theory,the local optimal solution of a convex optimization problem is equal to the global optimal solution [27],which makes it possible to find the global optimal solution by using a partitioned distributed algorithm to solve the optimization problem.In the voltage control field,partitions are usually decided by the system operators themselves.How to partition efficiently or achieve better performance is not the key issue to be discussed in this paper.

For a distribution network or wind farm with a radial form,G=(N,E)is represented as a radial network,whereNis the set of network nodes,andEis the set of network lines.ei,j∈Erepresents a network branch,wherei,j∈N.For the problem considered in this study,it is assumed that the distribution network and the wind farm are divided intonarearegions,respectively.The node set in theith partition is represented asNi,and the branch set in theith partition is represented asEi,wherei={1,2,…,narea}.The set of branches in the coupling region is denoised toXi,j,wherei,j= {1,2,…,narea} andi≠j.For example,as Fig.4 shows,

Fig.4 Example of a radial network partition

3.3 ADMM-based grid optimization scheme

To make the partition problem equivalent to the original problem,the state variables of the coupling branch in each adjacent region must be equal.Therefore,the grid optimization objective can be decomposed into the distributed coordination calculation of sub-optimization problems in each region.The distributed reactive power optimization control model based on partition coordination is as follows:

whereirepresents theith partition in the network,i,j= {1,2,…,narea}.xgrid,irepresents the set of all state variables in theith region,wheremandnrepresent the number of grid nodes,m,n∈Ni.Pi,mnandQi,mnare the active and reactive power from nodemto nodejin theith region,respectively.Ui,mis the voltage of nodem,andis the reactive power produced by the WF connected to nodem.

The constraints of this optimization problem are primarily considered from the power flow constraints,the safe operating condition of the power grid,the reactive power capacity constraints of the WFs,and ADMM algorithm coupling constraints.That is,the constraints with the consideration of area partition are as follows:

whereiis theith region in the grid,i={1,2,…,narea},andm,n,andkare themth,nth,andkth nodes in the grid,respectively;m,n,k={1,2,…,N}.The optimization problem of the entire system is decomposed into a coordinated calculation of sub-optimization problems ofnarearegions.

To solve the above optimization problem by the ADMM method,the sub-objective function can be transformed into the sub-augmented Lagrangian function as shown below,

whereρgridis the penalty parameter of ADMM algorithm,andygrid,iis the global variable of the suboptimization problem,whereygrid,i=(xgrid,i+xgrid,j)/2.λgrid,iis a set of state variables inXi,j,w here

Such that the optimization problem is defined as,

with the constraints of (19)-(25).

The specific procedure of the proposed algorithm is illustrated in Table 1.

Table 1 Procedure of ADMM-based optimization algorithm of the grid

continue

3.4 ADMM-based optimization scheme of the WF

Similar to the grid optimization problem,the WF optimization problem can also be decomposed into distributed coordination calculation of sub-optimization problems in each region.The distributed reactive power optimization control model based on partition coordination is as follows:

whereiis theith region,i,j= {1,2,…,nareaWF},andxWF,iis the set of all state variables in theith region;xWF,i={Qg,i,j,Ui,j},wherejis thejth WT in theith region.is the total loss of WTs and feeders in theith region,where

Owing to the existence of the wake effect [34],the WTs in the front receive more wind energy and produce more active power,thus having less reactive power potential.Similarly,the back WTs have greater reactive power potential.Therefore,ωi,jis used to describe the output under the wake effect,whereωi,jis represented as the weight of reactive power output of thejth WT in theith region.It is supposed that the active power output of the WT is greater because the weight of the WT is less.

Constraints of the optimization problem are chiefly considered from power flow,reactive power capacity,and ADMM algorithm coupling constraints,that is,the constraints are as follows:

To solve the above optimization problem by the ADMM method,the sub-objective function can be transformed into the sub-augmented Lagrangian function as shown below,

The definition ofρWF,yWF,i,andλWF,iis similar to the definition of the grid optimization problem.

Such that the optimization problem can be transformed into,

and the constraints are as (29)-(35)show.

The specific procedure of the proposed algorithm is illustrated in Table 2.

Table 2 Procedure of ADMM-based optimization algorithm of the WF

4 Simulation results

4.1 Case study setup

A distribution network (IEEE 33 bus system)with five WFs,with each WF having two feeders and 2 × 2 MW DFIG-based WTs connected to each feeder is used for validating the performance of the proposed strategies.The parameters of the DFIG-based WF can be found in [17],and the other parameters that are used in simulations are listed in Table 3.

Table 3 Parameters used in the simulation

Table 4 Control strategies adopted for comparison

4.2 Control strategies

Because the consensus ADMM algorithm is one of the key optimization methods based on ADMM adopted among recent articles,this paper primarily adopts three control strategies to conduct comparative simulations,as shown in the table below.Strategy A is the proposed twostage optimization algorithm,and Strategies B and C are consensus control of the grid and WFs,respectively,which are dispatched with the reactive power commands according to the consensus protocol of the utilization ratio among the WFs in the grid and WTs in the WF,respectively.

4.3 Simulation results

A)Control performance of the proposed method

Assume that the coupling region of the grid and the WFs are as shown in Figs.5 and 6,respectively.Considering the reactive power scheduling command issued by the TSO to be 1.55 p.u.,the reactive power output assigned to each WF is shown in Table 5,and the outputs of each WT in the WFs connected to each node are shown in Table 6.

Fig.5 Grid structure with partition

Fig.6 WF structure with partition

Table 5 Reactive power output assigned to each WF

Table 6 Reactive power output assigned to each WT in the WF

Distributed optimization is an iterative process.During each iteration,neighboring regions exchange coupling branch states.Communication times are positively correlated with optimization problem solving times.In the proposed algorithm,when the convergence conditions are set asε=10-4,both grid and wind farm sides can achieve convergence within 70 iterations under a parallel computing structure,which meets the reactive power dispatch requirements.

B)Comparison with other strategies

As mentioned in Subsection 4.2,the comparative results are as shown in Table 7.

Table 7 Comparative results between four strategies

As observed from the above results,compared with the consensus ADMM based optimization,the total loss of the proposed optimization method is smaller under the same objective function.Thus,the effectiveness of the proposed method is verified.

5 Conclusions and discussions

Herein,we proposed a two-stage distributed reactive power output optimization control method considering wake effects for WFs and demonstrated the effectiveness and rationality of the proposed strategies.With the proposed method,each area only exchanges information of the coupled region with its neighbors,which effectively protects information privacy.The proposed optimization framework simultaneously achieves the goals of tracking TSO control commands,optimizing reactive power output of WFs at the distribution network level,and optimizing fan output at the WF level.The performance and convergence of the proposed method were verified via simulations considering the IEEE 33-bus and 4 × 2 MW WF systems.

However,this study only discussed the response of the distribution network and WF sides to the TSO dispatching commands.The dynamic response of the proposed method under variable wind speed conditions and the autonomous response under voltage fluctuation conditions should be considered in future studies.

Acknowledgements

The authors would like to thank the support of The National Key Research and Development Program of China (Basic Research Class)(No.2017YFB0903000),and the National Natural Science Foundation of China (No.U1909201).

Declaration of Competing Interest

We declare that we have no conflict of interest.