^{1}

^{2}

^{3}

^{1}

^{4}

^{5}

^{1}

^{2}

^{3}

^{4}

^{5}

The Vietnamese power system has experienced instabilities due to the effect of increase in peak load demand or contingency grid faults; hence, using flexible alternating-current transmission systems (FACTS) devices is a best choice for improving the stability margins. Among the FACTS devices, the thyristor-controlled series capacitor (TCSC) is a series connected FACTS device widely used in power systems. However, in practice, its influence and ability depend on setting. For solving the problem, this paper proposes a relevant method for optimal setting of a single TCSC for the purpose of damping the power system oscillations. This proposed method is developed from the combination between the energy method and Hankel-norm approximation approach based on the controllability Gramian matrix considering the Lyapunov equation to search for a number of feasible locations on the small-signal stability analysis. The transient stability analysis is used to compare and determine appropriate settings through various simulation cases. The effectiveness of the proposed method is confirmed by the simulation results based on the power system simulation engineering (PSS/E) and MATLAB programs. The obtained results show that the proposed method can apply to immediately solve the difficulties encountering in the Vietnamese power system.

During over several decades, a number of electrical power systems in the world have been faced with the serious power system blackouts, such as Indian on July 30 and 31, 2012 [

The event of the small-signal instability during the blackout WSCC system on Aug. 10, 1996: (a) the observed California Oregon Interconnections power (Dittmer control center); (b) the simulated California Oregon Interconnections power (initial base case).

In recent years, the economic tempo in the Vietnam has been developed rapidly; the total load has increased continuously. Accordingly, the Vietnamese power system has been faced with the serious power system blackouts, such as on Dec. 27, 2006; July 20, 2007; Apr. 09, 2007; and the latest event on May 22, 2013. All of the technical problems that the Vietnamese power system identified after these events had already been reported [

The power system oscillations occur in the power systems because of the contingencies, such as the grid faults and sudden load changes; the dampening of these oscillations is necessary for a secure system operation. If the controlled systems react quickly against faults, the power system stability will enhance significantly. The advanced power electronics has led to a new design called flexible alternating-current transmission systems (FACTS) by Electrical Power Research Institute (EPRI). The FACTS devices make more use of the exiting capacities in the power system and enhance the power system stability. For example, the parameters in the power system are controlled and the load flow is modified to preclude the overload of transmission lines after the grid faults. The FACTS devices are widely used to improve the efficiency of power system operation. However, the benefits derived from FACTS controllers, such as the small-signal stability and transient stability that depend greatly on their optimal placement in the power systems [

The methods for solving location problems can be classified into two categories: (i) analytical techniques and (ii) heuristic optimization approaches [

For the analytical approaches, the modal controllability index has been developed by the authors in [

It is observed that most of existing methods in the previous literatures have been proposed recently for the location of FACTS. These methods have several drawbacks; firstly, the computation of critical modes may be questionable in case of large-scale power system since they may not be unique. Moreover, the computation of them also depends on the local or interarea modes. Secondly, the computation of participation factors is only based on the state variables and neglects the input-output behavior. Thirdly, it just focuses on analyzing the small-scale power systems. Therefore, in order to overcome these drawbacks, this paper is a continuation of [

The main new contributions of this paper are summarized as follows:

To develop a relevant method to determine the optimal location of FACTS on the small-signal stability analysis

To propose an association between proposed method and the Hankel-norm approximation method to limit the time calculation, so that the proposed method can be easily implemented for complexity and large-scale power systems.

The remainder of this paper is organized as follows: Section

The Vietnamese 500 kV power system operated in 1994

The result of load flow calculation on the Vietnamese 500/220 kV power system 2020.

The methodological method for dynamic modeling of general

Next, the linearized model is given as [

It can be identified as

Equation (

The main role of TCSC is to control fast the active power flow, increase the power transfer on transmission line, and enhance the stability of the power system. The basic structure TCSC consists of a fixed series capacitor bank C in parallel with a thyristor-controlled reactor (TCR), as shown in Figure

The TCSC controller: (a) structure; (b) equivalent; (c) reactance versus firing angle characteristic cure.

The variable

The transfer function mode of TCSC controller.

The new equivalent impedance of the line, when placed TCSC, as shown in Figure

The TCSC has been instated on the transmission line between bases

At bus

Similarly, at bus

The linearized model of TCSC is given as follows:

Incorporating (

It can be identified as

Therefore, (

The system often has two properties, controllability and observability, which play an important role regarding the determination of the optimal location of TCSC in the power system. From that, the input and output variables need to be used in order to observe and control the system. Therefore, (

System (

Every actuator in the power systems is the energy that can be limited, such that controllability matrix has been used for the purpose of dealing with the amount of input energy. This input energy is required to reach a given state from the origin. The property of controllability of the system can be described in a quantitative manner by the controllability function. Typically, it is defined for dynamic system as the minimum input energy that necessary drives the system from state

It can be proven that the above system (

For the real dynamic system,

Matrices

Matrices

The system states are the internal variables, which are hard to directly measure but the outputs can be measured easily at the same time. The observability property plays an important role in the analysis of optimal location.

System (

The observability property of the system can be characterized in a quantitative manner by the observability function. It is defined for dynamic system as output energy generated by the state

It can be proven that system (

For the real dynamic system,

Matrices

Matrices

If system (

Formal remarks for controllability and observability are white and black. In reality, some states are very calamitous to control or have little effect on the outputs. The degree of observability and controllability can be evaluated by the sizes of

It is hard to directly calculate the Gramian matrices from expression (

Obviously, expression (

In particular, we can pose the problem seeking the minimum input energy (i.e., the control energy) that must derive system (

Clearly, expression (

In order to compute easily, we focus on analyzing the infinite horizon Gramian. The Gramian matrices are computed by using expression (

Considering system (

The so-called similarity transformation transforms the controllability and observability Gramian matrices in the following way:

However, the result turns out to be the invariant output and input behaviors, that is,

Expression (

in which

After obtaining the new system in a balanced form as shown in (

In this study, the system reduction is performed by eliminating all state variables corresponding to the Hankel singular values that are smaller than 10^{−3}. Therefore, the system can be converted to the following order reduction form:

If the number of inputs is more than that of outputs (the so-called all-pass dilation of the system), then,

According to the conclusion in [

Clearly, expression (

The main objective is to dampen the active power and generator angle oscillations in the power system when the active power perturbation occurs in the transmission line. The proposed algorithm is used to determine the optimal location of TCSC based on the Gramian critical energy, and the flowchart of the algorithm is shown in Figure

Flowchart of the proposed algorithm.

Place the TCSC controller in the line

Run steady state.

Generate the active power perturbation signal in the transmission line

Compute the matrices

Perform the order reduction for system based on the Hankel-norm method.

Estimate the stable condition based on the state matrix

Compute the controllability Gramian matrix of the new system (after performing the order reduction) corresponding to the active power perturbation signal

Iterate the steps from

Compute the energy for each placement corresponding to a set of the active power perturbations

Iterate the steps from

Compare the maximum total energy to evaluate the optimal placement for TCSC controller.

The Vietnamese 500/220 kV transmission system is used in the study to illustrate the effectiveness of the proposed method. This system consists of 29 substations of 500 kV, 162 substations of 220 kV, 16 double-circuit lines and 20 single-circuit lines of 500 kV, 205 double-circuit lines and 67 single-circuit lines of 220 kV, and 179 generator units. The generated total power is about 42,179 MW; the peak load demand is about 40,703 MW. The power system simulation engineering (PSS/E) program is used to analyze the transient and small-signal stability and MATLAB is used to calculate Gramian matrices and perform order reduction. All dynamic models such as generators, excitation systems, transmission lines, and loads are modeled by using PSS/E (from dynamic model library) [

The optimal placement of TCSC in the Vietnamese network is determined based on (i) a combination of the controllability Gramian critical energy with the order reduction method on the small-signal stability analysis applied to search for several feasible locations; (ii) the transient stability analysis performed to compare and determine optimal locations through various simulation cases.

In addition, the existing series compensation ratio on the Vietnamese 500 kV power system is considered to calculate and listed out in Table

The existing compensation ratio on the 500 kV system.

Case number | The line between buses | Compensation ratio (%) | On circuit |
---|---|---|---|

( | NhoQuan-Hatinh (NQ-HT) | 57.7 | 1 |

( | NQ-HT | 54.4 | 2 |

( | VungAng-Danang (VA-DN) | 64 | 1 |

( | VA-DN | 64 | 2 |

( | Danang-Docsoi (DN-DS) | 63 | — |

( | Docsoi-Pleiku (DS-Plei) | 58 | — |

( | Thanhmy-Pleiku (TM-Plei) | 52 | — |

( | Pleiku-Daknong (Plei-DakN) | 76 | — |

( | Daknong-Caubong (DakN-CB) | 61 | — |

( | Pleiku-Dilinh (Plei-DL) | 70 | — |

( | Dilinh-Tandinh (DL-TD) | 59 | — |

( | Pleiku-Caubong (Plei-CB) | 70 | — |

The correctness of the proposed method for the optimal location of TCSC is verified on the small-signal analysis; the dynamic model of TCSC is chosen as shown in Figure

Parameters of the TCSC controller.

Parameter | Value | Parameter | Value |
---|---|---|---|

| 0.1 sec | | 0.75 |

| 0.1 sec | | 1.2 |

| 0.4 sec | | 0.25 |

| 10 sec | | 0.015 sec |

The studied contingency cases of active power perturbation were selected from single-line outage cases on the basis of the real power flow performance index (PI) introduced in [

Observing from (

Studied contingency cases.

Case number | Active power perturbation signal in the line between buses |
---|---|

( | NhoQuan-HoaBinh (NQ-HB) |

( | VietTri-Pitoong (VT-PT) |

( | ThangLong-PhoNoi (TL-PN) |

( | NhoQuan-SonLa (NQ-SL) |

( | Pitoong-HoaBinh (PT-HB) |

( | QuangNinh-HiepHoa (QN-HH) |

( | HiepHoa-VietTri (HH-VT) |

( | HiepHoa-Pitoong (HH-PT) |

( | Pitoong-SonLa (PT-SL) |

( | QuangNinh-ThangLong (QN-TL) |

( | QuangNinh-PhoNoi (QN-PN) |

( | QuangNinh-MongDuong (QN-MD) |

( | ThuongTin-Nho Quan (TT-NQ) |

( | ThuongTin-PhoNoi (TT-PN) |

( | VungAng-HaTinh (VA-HT) |

( | DaNang-ThanhMy (DN-TM) |

( | SongMay-TanUyen (SM-TY) |

( | SongMay-TanDinh (SM-TD) |

( | NhaBe-PhuLam (NB-PL) |

( | TanDinh-CauBong (TD-CB) |

( | PhuLam-MyTho (PL-MT) |

( | PhuLam-DucHoa (PL-DH) |

( | DucHoa-MyTho (DH-MT) |

( | MyTho-DuyenHai (MT-DH) |

( | OMon-MyTho (OM-MT) |

( | VinhTan-SongMay (VT-SM) |

( | PhuMy-NhaBe (PM-NB) |

( | CauBong-PhuLam (CB-PL) |

( | NhaBe-MyTho (NB-MT) |

( | CauBong-DucHoa (CB-DH) |

( | SongMay-PhuMy (SM-PM) |

The Vietnamese 500/220 kV power system created the state matrix (

Once again, the state matrix of the system is reduced the dimension by applying the balanced order reduction method to eliminate the singular values that are smaller than 10^{−3}. Figure

Eigenvalue distribution before and after using balanced reduction.

Figure ^{−∞} to 10^{5} are the same with the same control signal. The bandwidth has a range from 10^{5} to 10^{∞}; the frequency response of the reduced system is flat (a straight line) since its order is smaller than the original system. Therefore, we can conclude that two systems are equivalent in terms of the input-output behaviors.

Frequency response of the original and reduced system.

Figure ^{−3} (choice of this study). The eigenvalues distribution of the system before and after using balanced reduction is plotted in Figure

Distribution of Hankel singular values.

The controllability Gramian energy indices were determined based on the proposed algorithm as shown in Section

Energy values according to several feasible locations of TCSC.

Case number | Contingency cases (from Table | (Trace of ^{5} | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

TCSC is placed on the line between buses | ||||||||||||

None | NQ-HT | DN-VA | PLei-TM | PLei-DS | DN-DS | PLei-DL | PLei-CB | PLei-DakN | DakN-CB | DL-TD | ||

| ||||||||||||

( | NQ-HB | 0.0530 | 0.1354 | 0.1189 | 0.0780 | 0.0516 | 0.0614 | 0.0693 | 0.0780 | 0.0626 | 0.0290 | 0.0381 |

( | VT-PT | 0.0020 | 0.0142 | 0.0132 | 0.0020 | 0.0067 | 0.0080 | 0.0103 | 0.0118 | 0.0089 | 0.0055 | 0.0064 |

( | TL-PN | 0.0012 | 0.0947 | 0.0873 | 0.0374 | 0.0368 | 0.0483 | 0.0506 | 0.0526 | 0.0450 | 0.0215 | 0.0254 |

( | NQ-SL | 0.0242 | 0.0923 | 0.0840 | 0.0520 | 0.0354 | 0.0429 | 0.0458 | 0.0502 | 0.0418 | 0.0185 | 0.0242 |

( | PT-HB | 0.0034 | 0.0606 | 0.0575 | 0.0120 | 0.0240 | 0.0294 | 0.0307 | 0.0326 | 0.0282 | 0.0122 | 0.0156 |

( | QN-HH | 0.0010 | 0.0108 | 0.0099 | 0.0011 | 0.0044 | 0.0043 | 0.0048 | 0.0059 | 0.0047 | 0.0029 | 0.0032 |

( | HH-VT | 0.0012 | 0.0095 | 0.0097 | 0.0025 | 0.0044 | 0.0056 | 0.0061 | 0.0059 | 0.0054 | 0.0030 | 0.0031 |

( | HH-PT | 0.0032 | 0.0129 | 0.0122 | 0.0040 | 0.0061 | 0.0074 | 0.0092 | 0.0101 | 0.0079 | 0.0060 | 0.0055 |

( | PT-SL | 0.0010 | 0.0137 | 0.0117 | 0.0064 | 0.0059 | 0.0031 | 0.0074 | 0.0109 | 0.0071 | 0.0043 | 0.0057 |

( | QN-TL | 0.0029 | 0.0613 | 0.0564 | 0.0135 | 0.0242 | 0.0307 | 0.0337 | 0.0364 | 0.0301 | 0.0156 | 0.0176 |

( | QN-PN | 0.0032 | 0.0957 | 0.0887 | 0.0129 | 0.0372 | 0.0489 | 0.0514 | 0.0530 | 0.0458 | 0.0223 | 0.0253 |

( | QN-MD | 0.0216 | 0.0361 | 0.0343 | 0.0160 | 0.0140 | 0.0197 | 0.0194 | 0.0188 | 0.0171 | 0.0082 | 0.0090 |

( | TT-NQ | 0.0054 | 0.1523 | 0.1368 | 0.0850 | 0.0600 | 0.0766 | 0.0864 | 0.0941 | 0.0763 | 0.0382 | 0.0450 |

( | TT-PN | 0.0152 | 0.1173 | 0.1072 | 0.0630 | 0.0460 | 0.0598 | 0.0646 | 0.0685 | 0.0574 | 0.0279 | 0.0330 |

| | | | | | | | | | | | |

| ||||||||||||

| ||||||||||||

( | VA-HT | 0.0182 | 0.2777 | 0.2887 | 0.1660 | 0.1227 | 0.1616 | 0.1759 | 0.1900 | 0.1546 | 0.0780 | 0.0930 |

( | DN-TM | 0.0287 | 0.2015 | 0.4708 | 0.4982 | 0.5948 | 0.1894 | 0.3267 | 0.3003 | 0.2733 | 0.1642 | 0.1462 |

( | SM-TY | 0.0019 | 0.0037 | 0.0035 | 0.0002 | 0.0015 | 0.0022 | 0.0656 | 0.0125 | 0.0132 | 0.0296 | 0.0603 |

( | SM-TD | 0.0085 | 0.0453 | 0.0646 | 0.0399 | 0.0412 | 0.0416 | 0.3961 | 0.1184 | 0.0281 | 0.1382 | 0.3472 |

( | NP-PL | 0.0043 | 0.0683 | 0.0829 | 0.0492 | 0.0502 | 0.0460 | 0.1330 | 0.3648 | 0.1638 | 0.3823 | 0.0413 |

( | TD-CB | 0.0039 | 0.0268 | 0.0380 | 0.0359 | 0.0248 | 0.0232 | 0.1742 | 0.1055 | 0.5285 | 0.1123 | 0.1103 |

( | PL-MT | 0.0029 | 0.0158 | 0.0253 | 0.0229 | 0.0167 | 0.0188 | 0.0607 | 0.1077 | 0.0451 | 0.1217 | 0.0498 |

( | PL-DH | 0.0013 | 0.0079 | 0.0062 | 0.0013 | 0.0030 | 0.0021 | 0.0381 | 0.0175 | 0.0092 | 0.0255 | 0.0262 |

( | DH-MT | 0.0048 | 0.0180 | 0.0273 | 0.0095 | 0.0177 | 0.0189 | 0.0477 | 0.1083 | 0.0492 | 0.1166 | 0.0413 |

( | MT-DHa | 0.0018 | 0.0077 | 0.0174 | 0.0136 | 0.0117 | 0.0133 | 0.0425 | 0.0612 | 0.0269 | 0.0659 | 0.0312 |

( | OM-MT | 0.0023 | 0.0176 | 0.0256 | 0.0271 | 0.0166 | 0.0177 | 0.0672 | 0.0930 | 0.0419 | 0.1001 | 0.0498 |

( | VT-SM | 0.0082 | 0.0120 | 0.0289 | 0.0124 | 0.0195 | 0.0198 | 0.0969 | 0.0826 | 0.0343 | 0.0803 | 0.0911 |

( | PM-NB | 0.0011 | 0.0370 | 0.0341 | 0.0340 | 0.0184 | 0.0111 | 0.1370 | 0.1166 | 0.0620 | 0.0931 | 0.0798 |

( | CB-PL | 0.0043 | 0.0815 | 0.1142 | 0.0150 | 0.0734 | 0.0774 | 0.0584 | 0.5821 | 0.2359 | 0.6319 | 0.1125 |

( | NB-MT | 0.0034 | 0.0078 | 0.0114 | 0.0100 | 0.0077 | 0.0089 | 0.0774 | 0.0357 | 0.0123 | 0.0391 | 0.0502 |

( | CB-DH | 0.0150 | 0.0346 | 0.0499 | 0.0380 | 0.0330 | 0.0357 | 0.0347 | 0.2696 | 0.1048 | 0.3000 | 0.0600 |

( | SM-PM | 0.0112 | 0.0413 | 0.0424 | 0.0121 | 0.0245 | 0.0202 | 0.2338 | 0.0395 | 0.0303 | 0.0146 | 0.1432 |

| | | | | | | | | | | | |

| ||||||||||||

| | | | | | | | | | | |

To verify the effectiveness of the proposed method, several dynamic cases are analyzed based on the transient stability to compare the suitable location (the line between buses PLei and CB) and other feasible ones (as shown in Table

Scenarios on three-phase fault for test of transient stability.

Case number | Fault is nearly bus | Line outage |
---|---|---|

( | 500 kV DN | 500 kV DN-VA |

( | 500 kV Plei | 500 kV PLei-DL |

( | 500 kV CB | 500 kV TD-CB |

( | 500 kV NQ | 500 kV NQ-TT |

The simulation was done on scenario number 1 based on the relative angle oscillations of generator, supposing that the system has been operating at maximum load in order to compare the difference locations of TCSC (that are on the lines PLei-CB and PLei-TM) and without TCSC. Figure

The relative angle oscillations of generators: (a) Yaly; (b) PhuMy-3; (c) VinhTan; (d) QuangNinh.

The simulation of this case was done on scenario number 2 based on the active power of the line and the relative angle oscillations of generators to compare the suitable location of TCSC and other feasible locations as NQ-HT and DN-VA. In this case, the system has been operating at maximum load. Observing from Figure

The transient response: (a) the relative angle oscillations of generator VinhTan; (b) the active power oscillations in the line CauBong-DucHoa; (c) the active power oscillations in the line Pleilu-ThanhMy.

From Table

The transient response: (a) the relative angle oscillations of generator VinhTan (b); the relative angle oscillations of generator DuyenHai; (c) the active power oscillations in the line HaTinh-VungAng; (d) the active power oscillations in the line Pleiku-ThanhMy; (e) the response of TCSC controller.

Observing from Table

The relative angle oscillations of generator: (a) QuangNinh; (b) VinhTan.

Also, for comparison of the other locations, the simulation for this case was done on scenario number 2, supposing that the system has been operating at minimum load. Figure

The transient response: (a) the relative angle oscillations of generator SonLa; (b) the active power oscillations in the line NhoQuan-SonLa; (c) the active power oscillations in the line HaTinh-VungAng.

In this paper, a relevant stochastic method for the optimal placement of TCSC controller has been presented to enhance the rotor angle stability and dampen the power system oscillations in the multimachine systems. This proposed method is developed from the energy approach based on the controllability and observability Gramian matrices of the linearized multimachine systems. The optimal placement depends on the trace indices of the Gramian matrices that have been calculated on the active power perturbation in the line of the network (for this study, the applied network is the Vietnamese 500/220 kV power system).

The optimal placement for TCSC controller is determined based on the Gramian critical energy values that have been calculated on the small-signal stability analysis. However, the acquired results showed that the power system could operate perfectly under the influence of the transient conditions.

The time-domain simulation results on the transient stability analysis show that the rotor angle oscillations of generator and the power oscillations in the line are significantly dampened when the TCSC is placed in the line between PLeiku and CauBong, which has the maximum total energy value.

The Gramian-based reduction method has been also introduced for the purpose of reducing the calculation time of the Gramian critical energy when dealing with the large-scale power systems.

This paper is an applied and fully constituted version of the paper published in International Transactions on Electrical Energy Systems, 2016, Volume 26, Issue 7, pp. 1493–1510.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors sincerely acknowledge the financial support provided by Duy Tan University, Da Nang, Vietnam, Ton Duc Thang University, Ho Chi Minh, Vietnam, Industrial University of Ho Chi Minh City, Ho Chi Minh, Vietnam, and Quy Nhon University, Binh Dinh, Vietnam, for carrying out this work.

^{∞}-error bounds