Voltage Stability Bifurcation Analysis for AC/DC Systems with VSC-HVDC

Abstract

A voltage stability bifurcation analysis approach for modeling AC/DC systems with VSC-HVDC is presented. The steady power model and control modes of VSC-HVDC are briefly presented firstly. Based on the steady model of VSC-HVDC, a new improved sequential iterative power flow algorithm is proposed. Then, by use of continuation power flow algorithm with the new sequential method, the voltage stability bifurcation of the system is discussed. The trace of the P-V curves and the computation of the saddle node bifurcation point of the system can be obtained. At last, the modified IEEE test systems are adopted to illustrate the effectiveness of the proposed method.

1. Introduction

As one of the key technologies of large scale access of distributed energy resources, the HVDC transmission system has great potential for further development [1, 2]. Therefore, in the past decades, the problem associated with HVDC converters connected to weak AC networks has become an important research field. The one of particular interest, and highest in consequences, is the AC voltage stability at the HVDC terminals of the AC/DC systems [3, 4].

Voltage source converter-based HVDC (VSC-HVDC) is a new generation technology of HVDC, which overcomes some of the disadvantages of the traditional thyristor-based HVDC system, with a very broad application prospect. Compared to the conventional HVDC systems, the prominent features of the VSC-HVDC system are its potential to be connected to weak AC systems, independent control of active and reactive power exchange, and so on [4–6]. Due to those characteristics, many researches have been done for the exploitation of VSC-HVDC to enhance system stability of AC/DC systems, that is, the improvement of transient stability [7, 8], the power oscillations damping [9, 10], the improvement of stability and power quality for wind farm based on VSC-HVDC grid-connected [11, 12], the stability analysis of multi-infeed DC systems with VSC-HVDC [13], and the keeping voltage stable [3, 14–17].

In [14, 15], the voltage stability analyses of AC/DC systems with VSC-HVDC were mainly based on simulation software, and the analysis based on power flow calculation was slightly inadequate. The power flow calculation of AC/DC systems is the premise and foundation of static security analysis, transient stability, voltage stability, small signal stability analysis, and so on [18–21]. At present, there are two main types of power flow algorithms for AC/DC systems, sequential iterative method [21, 22] and integrated iterative method [23, 24]. The computational practice indicates that the convergence of integrated method is good, but the inheritance of the program is relatively poor, and the writing of program code needs huge work. The sequential iterative method has better program inheritance for pure AC program, but its convergence is not good. In view of these shortcomings, a modified sequential iterative power flow algorithm is proposed in this paper.

In [16], the continuation power flow (CPF) algorithm was presented to solve available transfer capability problem, but the saddle node bifurcation point was not discussed. In [17], the PV and QV curves were used to investigate voltage stability of a weak two area AC network. Though many methods for voltage stability analysis of AC/DC systems with VSC-HVDC have been proposed in different ways, few papers carefully consider the influence induced by different control modes of VSC-HVDC, which actually plays an important role on the voltage stability of the AC/DC systems. Motivated by the previous discussions, our main aim in this paper is to investigate the problem of power flow and voltage stability bifurcation for AC/DC systems with VSC-HVDC.

The rest of the paper is organized as follows. In Section 2, the steady model and control modes of AC/DC systems with VSC-HVDC are presented. In Section 3, the improved sequential iterative algorithm, the parameterization power flow and converter equations of VSC-HVDC, and the CPF strategy are presented. In Section 4, the model and method are applied to the modified IEEE 14- and IEEE 118-bus test systems with VSC-HVDC. Finally, in Section 5, the paper is completed with a conclusion.

2. Steady Model of VSC-HVDC

2.1. Power Flow Equations of VSC-HVDC

Figure 1 shows the single-line representation of two-terminal VSC-HVDC system. In Figure 1, 𝑖 is the number of VSC converters. 𝐈_𝑖 is the current flowing through transformer. 𝐔_s𝑖=𝑈_s𝑖∠𝜃_s𝑖 is the AC side voltage vector of VSC converter. 𝐔_c𝑖=𝑈_c𝑖∠𝜃_c𝑖 is the output voltage vector of VSC converter. 𝑅_𝑖 is the equivalent resistance of internal loss and converter transformer loss of VSC. 𝑗𝑋_𝑙𝑖 is the impedance of converter transformer. 𝑃_s𝑖 and 𝑄_s𝑖 are the power of AC system injected into converter transformer. 𝑗𝑋_𝑓 is the impedance of AC filter. 𝐈_d and 𝐔_d are the DC current vector and DC voltage vector, respectively. 𝑃_c𝑖 and 𝑄_c𝑖 are the power flowing through converter bridge. 𝑃_d𝑖 is the DC power. From Figure 1, 𝐈_𝑖 can be expressed as follows: 𝐈_𝑖=𝐔_s𝑖−𝐔_c𝑖𝑅_𝑖+𝑗𝑋_𝑙𝑖. (1) The complex power 𝑆^∼_s𝑖 of AC system is given by 𝑆^∼_s𝑖=𝑃_s𝑖+𝑗𝑄_s𝑖=𝐔_s𝑖𝐈^∗_𝑖. (2) To facilitate discussion, assume that 𝛿_𝑖=𝜃_s𝑖−𝜃_c𝑖, |𝑌_𝑖|=1/𝑅²_𝑖+𝑋²_𝑙𝑖, 𝛼_𝑖=arccot(𝑋_𝑙𝑖/𝑅_𝑖), and substituting (1) into (2) yields 𝑃_𝑠𝑖=−𝑌_𝑖𝑈_s𝑖𝑈_c𝑖cos𝛿_𝑖+𝛼_𝑖+𝑌_𝑖𝑈²_s𝑖cos𝛼_𝑖,𝑄_𝑠𝑖=−𝑌_𝑖𝑈_s𝑖𝑈_c𝑖sin𝛿_𝑖+𝛼_𝑖+𝑌_𝑖𝑈²_s𝑖sin𝛼_𝑖+𝑈²_s𝑖𝑋_f𝑖. (3) Similarly, one has 𝑃_c𝑖=𝑌_𝑖𝑈_s𝑖𝑈_c𝑖cos𝛿_𝑖−𝛼_𝑖−𝑌_𝑖𝑈²_c𝑖cos𝛼_𝑖,𝑄_c𝑖=−𝑌_𝑖𝑈_s𝑖𝑈_c𝑖sin𝛿_𝑖−𝛼_𝑖−𝑌_𝑖𝑈²_c𝑖sin𝛼_𝑖. (4) Since the loss of converter bridge-arm is equivalent by 𝑅_𝑖, 𝑃_d𝑖 is qual to 𝑃_c𝑖, and thus 𝑃_d𝑖=𝑈_d𝑖𝐼_d𝑖=𝑌_𝑖𝑈_s𝑖𝑈_c𝑖cos𝛿_𝑖−𝛼_𝑖−𝑌_𝑖𝑈²_c𝑖cos𝛼_𝑖. (5) And 𝑈_c𝑖 can be described as 𝑈_c𝑖=√6𝑀_𝑖𝑈_d𝑖4, (6) where 𝑀 (0<𝑀<1) is defined as the PWM's amplitude modulation index.

Figure 1 

Simplified circuit diagram of single-phase VSC-HVDC.

The steady model of VSC-HVDC is given by (1)–(6) in the per-unit system (p.u.).

2.2. Steady-State Control Modes of VSC-HVDC

Several regular control modes for each VSC converter are chiefly as follows: (1) constant DC voltage control, constant AC reactive power control; (2) constant DC voltage control, constant AC voltage control; (3) constant AC active power control, constant AC reactive power control; (4) constant AC active power control, constant AC voltage control.

3. Voltage Stability Model for AC/DC Systems with VSC-HVDC

3.1. The Modified Power Flow Algorithm Based on Sequential Iteration Method

For the previous model of AC/DC system with VSC-HVDC, the simplified power flow equations are given by 𝐟_ac𝐟=0,_ac-dc𝐟=0,_dc=0, (7) where 𝐟_ac=[Δ𝑃_a1,Δ𝑄_a1,…,Δ𝑃_a𝑛AC,Δ𝑄_a𝑛AC]^T, 𝐟_ac-dc=[Δ𝑃_t1,Δ𝑄_t1,…,Δ𝑃_t𝑛VSC,Δ𝑄_t𝑛VSC]^T, 𝐟_dc=[Δ𝑑₁₁,Δ𝑑₁₂,Δ𝑑₁₃,Δ𝑑₁₄,…,Δ𝑑_𝑛VSC1,Δ𝑑_𝑛VSC2,Δ𝑑_𝑛VSC3,Δ𝑑_𝑛VSC4]^T.

To expand the Taylor series of (7), the second-order item and higher-order terms are omitted, and the modified equation based on Newton-Raphson is given by 𝐟_N=−𝐉_NΔ𝐱_N, (8) where 𝐟_N=[𝐟^T_ac,𝐟^T_ac-dc,𝐟^T_dc]^T, Δ𝐱_N=[Δ𝑥^T_ac1,Δ𝑥^T_ac2,Δ𝑥^T_ac-dc,Δ𝑥^T_dc]^T, Δ𝐱_ac1=[Δ𝑈_a1,Δ𝜃_a1,…,Δ𝑈_a𝑛AC,Δ𝜃_a𝑛AC]^T, Δ𝐱_ac2=[Δ𝑈_t1,Δ𝜃_t1,…,Δ𝑈_t𝑛VSC,Δ𝜃_t𝑛VSC]^T,Δ𝐱_ac-dc=[Δ𝑃_t1,Δ𝑄_t1,…,Δ𝑃_t𝑛VSC,Δ𝑄_t𝑛VSC]^T, Δ𝐱_dc=[Δ𝑈_d1,Δ𝐼_d1,Δ𝛿₁,Δ𝑀₁,…,Δ𝑈_d𝑛VSC,Δ𝐼_d𝑛VSC,Δ𝛿_𝑛VSC,Δ𝑀_𝑛VSC]^T.

And 𝐉_N is given by 𝐉_N𝐱_ac1,𝐱_ac2,𝐱_ac-dc,𝐱_dc=𝜕𝐟_ac𝜕𝐱_ac1𝜕𝐟_ac𝜕𝐱_ac2𝜕𝐟_ac𝜕𝐱_ac-dc𝜕𝐟_ac𝜕𝐱_dc𝜕𝐟_ac-dc𝜕𝐱_ac1𝜕𝐟_ac-dc𝜕𝐱_ac2𝜕𝐟_ac-dc𝜕𝐱_ac-dc𝜕𝐟_ac-dc𝜕𝐱_dc𝜕𝐟_dc𝜕𝐱_ac1𝜕𝐟_dc𝜕𝐱_ac2𝜕𝐟_dc𝜕𝐱_ac-dc𝜕𝐟_dc𝜕𝐱_dc=𝐉_a-a1𝐉_a-a2𝐉𝟎𝟎_ad-a1𝐉_ad-a2𝐉_ad-ad𝟎𝐉_d-a1𝐉_d-a2𝐉_d-ad𝐉_d-d. (9) The power flow equation of VSC-HVDC is given as follows: 𝐟₁𝐟₂𝐉=−¹¹𝐉₁₂𝐉₂₁𝐉₂₂Δ𝐱₁Δ𝐱₂, (10) where 𝐟₁=𝐟_ac, 𝐟₂=[𝐟^T_ac-dc,𝐟^T_dc]^T, Δ𝐱₁=Δ𝐱_ac1, Δ𝐱₂=[Δ𝐱^T_ac2,Δ𝐱^T_ac-dc,Δ𝐱^T_dc]^T, 𝐉₁₁=𝐉_a-a1, 𝐉₂₁=^𝐉ad-a1_𝐉d-a1, 𝐉₁₂=[𝐉_a-a2,𝟎,𝟎], 𝐉₂₂=^{𝐉ad-a2𝐉ad-ad𝟎}_{𝐉d-a2𝐉d-ad𝐉d-d}.

The number of the power flow equations for (10) is 2(𝑛−1)+4𝑛_VSC, and the variables number is 2(𝑛−1)+6𝑛_VSC. The 2𝑛_VSC variables can be eliminated by control modes of VSC-HVDC. So, (10) has solutions. The dimensions of 𝐟₁ are 2(𝑛_AC−1). The dimensions of Δ𝐱₁ are 2(𝑛_AC−1). And the inverse matrix of 𝐉₁₁ is exists. The dimensions of 𝐟₂ are 6𝑛_VSC. The dimensions of Δ𝐱₂ are 8𝑛_VSC, and the 2𝑛_VSC dimensions of Δ𝐱₂ can be eliminated by control modes of VSC-HVDC, and so the inverse matrix of 𝐉₂₂ exists.

To expand (10), −𝐉₁₁Δ𝐱₁+𝐉₁₂Δ𝐱₂=𝐟₁,−𝐉₂₁Δ𝐱₁+𝐉₂₂Δ𝐱₂=𝐟₂. (11)

Then the new modified sequential iterative form is given by Δ𝐱₁=−𝐉₁₁−𝐉₁₂𝐉₂₂⁻¹𝐉₂₁⁻¹𝐟₁−𝐉₁₂𝐉₂₂⁻¹𝐟₂,Δ𝐱₂=−𝐉₂₂−𝐉₂₁𝐉₁₁⁻¹𝐉₁₂⁻¹𝐟₂−𝐉₂₁𝐉₁₁⁻¹𝐟₁. (12)

It can be seen from the previous derivation process that the modified algorithm does not make any hypothesis. The mutual influence between the AC system and DC system is fully considered in the iteration solution procedure. By means of the previous method, the problem of AC variables coupling DC variables is solved strictly in the mathematics expression. The matrix 𝐉₁₁ and 𝐉₁₂ can be obtained by the program of pure AC system.

3.2. Parameter-Dependent Power Flow and Converter Equations of VSC-HVDC

According to connected or not connected with a converter transformer, the buses of AC/DC systems with VSC-HVDC are divided into two kinds, DC bus and pure AC bus [23]. The bus connected to primary side of a converter transformer is considered as a DC bus. The bus not connected to a converter transformer is considered as a pure AC bus. 𝑛 is the total bus number of the system. 𝑛_VSC is the number of VSC converters and also is the number of DC buses. So, the number of pure AC buses is 𝑛_AC=𝑛−𝑛_VSC.

Considering the load changes in several areas or a particular area (at a bus and/or at a group of buses) of the AC/DC systems with VSC-HVDC, the power flow equations for a pure AC bus are given by Δ𝑃_a𝑖=𝑃_a𝑖−𝑈_{a𝑖𝑗∈𝑖}𝑈_𝑗𝐺_𝑖𝑗cos𝜃_𝑖𝑗+𝐵_𝑖𝑗sin𝜃_𝑖𝑗+𝑃_G𝑖−𝑃_L𝑖𝜆=0,Δ𝑄_a𝑖=𝑄_a𝑖−𝑈_{a𝑖𝑗∈𝑖}𝑈_𝑗𝐺_𝑖𝑗sin𝜃_𝑖𝑗−𝐵_𝑖𝑗cos𝜃_𝑖𝑗+𝑄_G𝑖−𝑄_L𝑖𝜆=0, (13) where the subscript “a” identifies that the bus is a pure AC bus, a=1,2,…,𝑛_AC. The subscript “𝑖” is the number of the buses, 𝑖=1,2,…,𝑛. The subscript “𝑗” identifies that all the buses connected to the bus “𝑖” (expressed in the terms of 𝑗∈𝑖). 𝑈 and 𝜃 are the bus voltage amplitude and phase angle, respectively. 𝐺 and 𝐵 are the real part and imaginary part of nodal admittance matrix, respectively. 𝑃_G𝑖 and 𝑄_G𝑖 are the power of generator. 𝑃_L𝑖 and 𝑄_L𝑖 are the loads at the bus 𝑖. 𝜆∈𝑅 are the parameters such real/reactive power demand at the buses and transmission line parameters.

For a DC bus, the power flow equations are given by Δ𝑃_t𝑖=𝑃_t𝑖−𝑈_{t𝑖𝑗∈𝑖}𝑈_𝑗𝐺_𝑖𝑗cos𝜃_𝑖𝑗+𝐵_𝑖𝑗sin𝜃_𝑖𝑗±𝑃_t𝑖+𝑃_G𝑖−𝑃_L𝑖𝜆=0,Δ𝑄_t𝑖=𝑄_t𝑖−𝑈_{t𝑖𝑗∈𝑖}𝑈_𝑗𝐺_𝑖𝑗sin𝜃_𝑖𝑗−𝐵_𝑖𝑗cos𝜃_𝑖𝑗±𝑄_t𝑖+𝑄_G𝑖−𝑄_L𝑖𝜆=0, (14) where the subscript “t” identifies the bus as a DC bus. “±” signs correspond to the rectifiers and inverters of VSC-HVDC, respectively.

The basic power flow equations for VSC-HVDC converters are given as follows: Δ𝑑_𝑘1=𝑃_t𝑘+√64𝑀_𝑘𝑈_t𝑘𝑈_d𝑘|𝑌|cos𝛿_𝑘+𝛼_𝑘−𝑈²_t𝑘|𝑌|cos𝛼_𝑘=0,Δ𝑑_𝑘2=𝑄_t𝑘+√64𝑀_𝑘𝑈_t𝑘𝑈_d𝑘|𝑌|sin𝛿_𝑘+𝛼_𝑘−𝑈²_t𝑘|𝑌|sin𝛼_𝑘−𝑈²_t𝑘𝑋_f𝑘=0,Δ𝑑_𝑘3=𝑈_t𝑘𝐼_d𝑘−√64𝑀_𝑘𝑈_t𝑘𝑈_d𝑘|𝑌|cos𝛿_𝑘−𝛼_𝑘+38𝑀_𝑘𝑈_d𝑘²|𝑌|cos𝛼_𝑘=0, (15) where the subscript “d” identifies that the variable as the DC side of VSC. “𝑘” is the 𝑘th of VSC connected to DC network, 𝑘=1,2,…,𝑛_VSC.

For the 𝑖th VSC converter, there are four unknown variables in (15); that is, 𝐱_d𝑖=[𝑈_d𝑖,𝐼_d𝑖,𝛿_𝑖,𝑀_𝑖]^T, and one more equation is needed to solve (15); that is, 𝐈_d=𝐆_d𝐔_d, (16) where 𝐆_d is the nodal admittance matrix of DC network.

Moreover, the DC network equation is given by Δ𝑑_𝑘4=±𝐼_d𝑘−^𝑛AC_𝑠=1𝑔_d𝑘𝑠𝑈_d𝑠=0, (17) where 𝑔_d𝑘 is the matrix element of nodal admittance matrix of DC network, 𝑠=1,2,…,𝑛_AC.

In this paper, the concerned parameters are the real and reactive loads changes at the buses that can vary according to the following equations: 𝑃_𝑖=𝑃_𝑖0(),𝑄1+𝜆_𝑖=𝑄_𝑖0(1+𝜆), (18) where 𝑃_𝑖0 and 𝑄_𝑖0 are the initial active and reactive loads, respectively. 𝑃_𝑖 and 𝑄_𝑖 are the active and reactive loads at bus 𝑖, respectively.

For the previously AC/DC systems with VSC-HVDC, the simplified power flow equation with parameter 𝜆-dependent is given by 𝐟(𝐱,𝜆)=0, (19) where 𝐟∈𝑅^{2(𝑛−1)+4𝑛VSC+1}, 𝐱∈𝑅^{2(𝑛−1)+4𝑛VSC+1}. 𝐟 is the balance equation of power flow. 𝐱 is the system state variable such bus voltage magnitude and phase angles, DC system variables. 𝑛₁ and 𝑛₂ are the number of PQ and PV AC buses. The number of power flow equations for (19) is 2(𝑛−1)+4𝑛_VSC+1=2𝑛₁+𝑛₂+4𝑛_VSC+1.

3.3. Continuation Power Flow Algorithm Based on Sequential Iteration Method

CPF is a powerful tool to numerically generate P-V curve to trace power system stationary behavior due to a set of power injection variations [25, 26]. It uses predictor-corrector scheme to find a solution path of a set of power flow equations that have been reformulated to include a load parameter. (𝑥_𝑙,𝜆_𝑙)^T is the initial state of the power flow solution curve, where the subscript “𝑙” is the iteration number of predictor-corrector scheme based on Newton-Raphson algorithm in CPF.

3.3.1. The Predictor Step

The predictor step is a stage in first-order differential form. Once a base solution has been found (𝜆=0), a prediction of the next solution can be made by taking an appropriately sized step in a direction tangent to the solution path. Taking the derivatives of (19) will result in the following total differential form equation: 𝑓^′_𝑥𝑓^′_𝜆𝑑_𝑥𝑑_𝜆=0, (20) where 𝑓^′_𝑥=𝜕𝑓/𝜕𝑥 is the Jacobian matrix of power flow equation with 𝑥. 𝑓^′_𝜆=𝜕𝑓/𝜕𝜆 is the partial derivative of power flow equation with 𝜆. [𝑑_𝑥𝑑_𝜆]^T is the tangent vector, which is the computation target of the predictor step.

Since the insertion of 𝜆 in the power flow equation added an unknown variable, one more equation is needed to solve the previously equation. This is satisfied by setting one of the components of the tangent vector to +1 or −1. This component is referred to as the continuation parameter. Equation (20) now becomes [23] 𝑓^′_𝑥𝑓^′_𝜆𝑒_𝐾𝑑_𝑥𝑑_𝜆=0±1, (21) where 𝑒_𝐾 is a row vector with all elements equal to zero except for the 𝐾th element being equal 1. The dimensions of 𝑒_𝐾 are 1×[2(𝑛−1)+6𝑛_VSC+1]. The introduction of the additional equation makes the Jacobian matrix non-singular at the critical operation point. The modified equation of Newton-Raphson algorithm for (21) is Δ𝐟=−𝐉Δ𝐱, (22) where Δ𝐟=[𝐟^T_ac,𝐟^T_ac-dc,𝐟^T_dc,𝐟^T_𝜆]^T, 𝐟_𝜆=±1, Δ𝐱=[Δ𝑥^T_ac1,Δ𝑥^T_ac2,Δ𝑥^T_ac−dc,Δ𝑥^T_dc,Δ𝑥^T_𝜆]^T, 𝐉 is referred to as the Jocobian matrix of (22), and the dimensions of 𝐉 are [2(𝑛−1)+4𝑛_VSC+1]×[2(𝑛−1)+6𝑛_VSC+1].

The initial values of the variables of VSC-HVDC system for power flow program iteration are given by 𝑈⁽⁰⁾_d𝑘=𝑈_d^ref_𝑘,(),𝑈𝑘∈𝐶𝑉_(0)d𝑘=𝑈^N_d𝑘,(),𝐼𝑘∉𝐶𝑉_(0)d𝑘=𝑃_t𝑘𝑈⁽⁰⁾_d𝑘,𝛿_𝑘⁽⁰⁾𝑃=arctan^t𝑘𝑈²_t𝑘/𝑋_L𝑘+𝑈²_t𝑘/𝑋_f𝑘−𝑄_t𝑘,𝑀_𝑘⁽⁰⁾=2√63𝑃_𝐭𝑘𝑋_L𝑘𝑈_t𝑘𝑈⁽⁰⁾_d𝑘sin𝛿_𝑘⁽⁰⁾, (23) where 𝑘∈𝐶𝑉 identifies that the 𝑘th VSC is constant DC voltage control mode. 𝑘∉𝐶𝑉 identifies that the 𝑘th is not constant DC voltage control mode. Superscript “0” identifies the initial value of the 0th iteration. Superscript “ref” identifies that the variable value is reference value. Superscript “N” identifies rated value.

The 𝑃_t𝑘 is given in estimation by 𝑃_t𝑘=−^𝑛VSC_{𝑠=1,𝑠∉𝐶𝑉}𝑃_t^ref_𝑠. (24)

Based on the previous analysis, the tangent vector [𝑑_𝑥𝑑_𝜆]^T is obtained. The prediction value of the next solution is given by 𝑥^′_𝑙+1𝜆^′_𝑙+1=𝑥_𝑙𝜆_𝑙𝑑+ℎ^𝑥𝑑_𝜆, (25) where [𝑥^′_𝑙+1𝜆^′_𝑙+1]^T is prediction value, which is an approximate solution. ℎ is the step size of the prediction.

3.3.2. The Corrector Step

In the corrector step, the prediction value of [𝑥^′_𝑙+1𝜆^′_𝑙+1]^T is substituted into (19), and its iteration form is reformulated as 𝑓^′_𝑥𝑓^′_𝜆𝑓()010Δ𝑥Δ𝜆=−𝑥,𝜆. (26) The iteration form is now reformulated as 𝑓^′_𝑥𝑓^′_𝜆𝑒_𝐾0𝑓()0Δ𝑥Δ𝜆=−𝑥,𝜆. (27)

4. Case Studies and Validations

Two cases are considered and compared: (1) the system with existing AC transmission line and (2) the system with a new dc transmission line based on VSC-HVDC.

The system parameters of the four different control modes and the VSC converters adopted in the case studies are prespecified as listed in Table 1. The initial DC variable parameters of VSC-HVDC system are shown in Table 2. In Table 2, the 𝑁_bus identifies bus number of VSC-HVDC link connected to AC systems. The 𝑃_L and 𝑄_L are the load power of the bus VSC connected to. The 𝑅_d is the resistance of DC network. 𝑅, 𝑋_𝑙, 𝑃_L, and 𝑄_L are given parameters. 𝑃_sand 𝑄_s are calculated by power flow calculation of the original AC system, which is equal to the branch power of the original AC system.

Table 1 

System parameters of operation and control modes of VSC-HVDC.

Table 2 

Initial DC variable parameters (p.u.) of VSC-HVDC system.

4.1. Modified IEEE 14-Bus Text System

First, the proposed method has been applied to the modified IEEE 14-bus system shown in Figure 2. The AC line parameters of the system are the same as the IEEE 14-bus system. The difference is that a two-terminal VSC-HVDC transmission line is placed at bus 13 and bus 14 to replace the AC transmission line 13-14; that is, the VSC₁ and VSC₂ are connected to AC line of bus 13 and bus 14, respectively.

Figure 2 

The modified IEEE 14-bus system with VSC-HVDC.

This paper chooses the commutation bus of the buses 9, 12, 13, and 14 as the research objects. According to the different control modes of VSC-HVDC showed in Table 1, the detailed analysis of the voltage stability bifurcation for the system can be divided into three cases.

4.1.1. The Operation Mode of “1”

Figure 3 shows the P-V curves and load margins of partial buses (buses 9, 12, 13, and 14) of the original pure IEEE-14 system (“AC” for short). Figure 4 shows the P-V curves and load margins of partial buses (buses 9, 12, 13, and 14) of the modified IEEE-14 system (“AC/DC” for short), in which the AC/DC system operates under the control mode “1” in Table 1 as the load increases. And Table 3 shows the power flow data of the VSC-HVDC operating in mode “1” at initial state and the maximum load state of the AC/DC system. Here, the maximum load state is corresponding to the saddle node bifurcation point of the system.

Table 3 

Power flow data at initial and maximum load states for mode “1.”

Figure 3 

PV curves for IEEE-14 bus system.

Figure 4 

PV curves for modified IEEE-14 bus system in operating mode “1.”

In Figure 3, when 𝜆_AC=5.2120p.u., the saddle node bifurcation point is acquired, and the system is in maximum loading state. In Table 3, the “AC bus” of the lower half part includes the four PV buses (bus 2, bus 3, bus 6, and bus 8) and a slack bus (bus 1) of the modified IEEE-14 test system. The upper row of the twin-row is the initial state data, and the lower row of the twin-row is the maximum load state (corresponding with the 𝜆_AC/DC = 5.6358 p.u. in Figure 4) data of power flow.

It can be seen in Figure 4 and Table 3, in operation mode “1,” that the voltage magnitude decreases as the load increases, and the bus 14 (AC/DC) has the weakest voltage profile; so, it is the critical voltage bus needing reactive power support. When the load margin at 𝜆_AC/DC = 5.6358 p.u., the modified AC/DC systems present a collapse or saddle node bifurcation point, where the system Jacobian matrix becomes singular. And now the voltage drop of bus 14 (AC/DC) is the most obvious: 𝑈₁₄=0.6451<𝑈₁₃=0.8381<𝑈₁₂=0.8816<𝑈₉=0.9083 (the parameters are all the AC/DC systems).

4.1.2. The Operation Modes of “2” and “3”

Figures 5 and 6 show the P-V curves and load margins of partial buses (buses 9, 12, 13, and 14) of the AC/DC system under the control mode of “2” and “3”.

Figure 5 

PV curves for modified IEEE-14 bus system in operating mode “2.”

Figure 6 

PV curves for modified IEEE-14 bus system in operating mode “3.”

As shown in Figures 5 and 6, in operating mode “2” and mode “3,” the voltage magnitudes of bus 14 (AC/DC) and bus 13 (AC/DC) remain almost constant as the load increases, respectively. This is because the VSC₂ in mode “2” and VSC₁ in mode “3” both are in constant AC voltage control mode. Besides, the load margins obtained in mode “2” (𝜆_AC/DC=6.9192 p.u.) and mode “3” (𝜆_AC/DC=5.8980 p.u.) are bigger than those in mode “1,” which demonstrated that the VSC-HVDC system supplies voltage support to the AC bus voltage, thanks to the benefits of the fast and independent reactive power output of VSC-HVDC.

4.1.3. The Operation Mode of “4”

Figure 7 shows the P-V curves and load margins of partial buses (buses 9, 12, 13, and 14) of the AC/DC system under the control mode of “4” in Table 1 with the load varies.

Figure 7 

PV curves for modified IEEE-14 bus system in operating mode “4.”

As shown in Figure 7, the voltage magnitude of bus 13 (AC/DC) and bus 14 (AC/DC) remains almost constant as the load increases. This because to the control modes of VSC₁ and VSC₂ are constant AC voltage. The load margin for this operation mode is 𝜆_AC/DC=7.7192p.u., which is bigger than those in other modes. Therefore, in the case of the VSC-HVDC operation in mode “4,” the AC/DC system has better voltage stabilization. But as has been pointed out in [27], when the AC/DC system is disturbed (such as kinds of faults), in order to maintain AC bus voltage, the VSC converter in mode “4” has to provide large amount of reactive power to the AC/DC system. Consequently, the overload degree of VSC converter is more severe under this operation mode than overload degree of VSC converter under other control modes.

By contrasting the four control modes of VSC converters, the results show that the requirements of essential reactive power for AC system can be supplied by VSC-HVDC system, and the certain voltage support capability to AC bus by VSC-HVDC link is validated. But it should be pointed out that the appropriate control pattern is the basis to exploit the reactive power compensation property of the VSC-HVDC system.

4.2. Modified IEEE 118-Bus Text System

The modified IEEE 118-bus system is analyzed in this section [28]. The relevant part of the network is shown in Figure 8, which shows the locations of the VSC-HVDC link. The VSC-HVDC replaced an existing AC transmission line (75–118), as shown in Figure 8, and the VSC₁ and VSC₂ are connected to AC line of bus 75 and bus 118, respectively.

Figure 8 

Relevant part of the modified IEEE 118-bus system withVSC-HVDC.

The performance comparison of the modified IEEE-118 test system for four different operating modes is shown in Table 4. In Table 4, the CPU time for operation mode of “4” is longer than other operation modes. The reason is that the operation mode of “4” is in constant AC voltage control mode. As the load increases, the VSC-HVDC system supplies voltage support to the AC bus voltage, and as a result, the load margin of the AC/DC system in mode “4” is bigger than other operation modes, and the calculation program for the load margin of the mode “4” needs more iteration; so, the CPU time is longer. Table 4 shows that the model and algorithm presented in this paper have certain flexibility with the increase of the network scale. However, it is to be remarked that the CPU time is long. The reason is that with the embedded VSC-HVDC transmission line, the types and the number of the system variables have greatly increased, that is, 𝑃_c𝑖,𝑄_c𝑖,𝑈_d𝑖,𝐼_d𝑖,𝛿_𝑖,𝑀_𝑖,𝑈_c𝑖,𝜃_c𝑖, and so forth, and the dimensions of the system equations and the Jacobian matrix of the AC/DC system have a higher order than pure AC systems.

Table 4 

Performance comparison of the modified IEEE-118 test system for four different operating modes.

5. Conclusions

In this paper, a new method has been developed to analyze voltage stability for AC/DC systems with VSC-HVDC. The impacts of load variations and different VSC-HVDC control patterns on P-V curves and saddle node bifurcation point of the system were numerically analyzed. The simulation results indicate that the constant AC voltage control of VSC converter is superior to other control modes in voltage stability and also show that VSC-HVDC significantly improved the stability of the system compared to a pure AC line. At last, the importance of suitable control mode for the operating of VSC-HVDC was discussed, and some numerical examples have been included to demonstrate the validity of the obtained results.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (Grant nos. 61104045, 51277052, 51107032, and 51077125) and in part by the Fundamental Research Funds for the Central Universities of China (Grant no. 2012B03514).