^{1}

^{2}

^{2}

^{2}

^{1}

^{1}

A fast method based on the phase trajectory to compute DSR is developed. Firstly, the phase trajectory sensitivity has more linear effect than power angle sensitivity. According to the phase trajectory boundary function, controlling unstable equilibrium generators could be identified. The PDSR is finally obtained by the sensitivity analysis between the phase and generators’ active power. Test results on the New England 10-genrator 39-bus system are presented and prove the effectiveness of this approach.

The dynamic security region (DSR) is defined as the set of input power space before the accident. All the injection points in the set can guarantee the transient stability of the system after a given accident. DSR is related to the network topology and expected accident of the system, but not related with the change of the base point, and can be calculated offline. For online applications, the transient stability could be quickly identified, depending on whether the current injection is within the DSR. At the same time, it is possible to compute the distance from the operating point to each boundary, which represents the security margin of the system in different directions. Compared with traditional methods such as time domain simulation, DSR can provide more comprehensive security margin and auxiliary control decision for system operators.

A large number of studies have shown that [

The methods to calculate PDSR can be divided into two categories including fitting and analytic methods. The fitting method uses a large number of critical injection points calculated by numerical simulation to fit the PDSR boundary expression [

Analytic method is the use of transient stability direct method to quickly calculate the PDSR boundary [

In paper [

In recent years, with the construction and development of the PMU and WAMS in the electrical power system, it has become possible to obtain the generator trajectory information in real time [

In this paper, a method to calculate the PDSR boundary is proposed. It uses the generators’ phase trajectory including the power angle δ and angular velocity ω. Based on the transient stability criterion of the phase trajectory [

The motion equation of single machine infinite system could be expressed as:

where δ is the phase angle of generators; ω is angular velocity deviation from the synchronous electrical angular velocity. M is the inertia time constant; P_{m} is the generators’ mechanical power; P_{e} is the generators, electromagnetic power; D is the generators’ damping coefficient.

In phase trajectory analysis, the phase angle δ is the abscissa and the angular velocity ω is the ordinate. The phase trajectories of the generator are shown in

Ignoring the damping and regardless of regulator and governor role, formulation (1) can be written:

In order to study the relationship between the trend of phase trajectory and the transient stability of the system, the first order derivative D_{1} and the second order derivative D_{2} [

When the second order derivative D_{2} is 0, the formulation needs:

As shown in _{1}. The phase plane is divided into two parts. The left part of the second derivative has D_{2} < 0, and the right part of the second derivative has D_{2} > 0.

In paper [

In Single-machine Infinite System, when P_{m} and P_{e} remian unchanged, for- mulation (6) shows:

When ω decreases monofonicallythe phase boundary function f increases monofonically. The formulation to judge the transient stability could be writed:

The maximum of function

The traditional formulation to judge the transient stability using the phase angle:

where δ_{u} is the unstable equilibrium point (UEP);

When the power system is stable:

When P_{m} increases,

For the sake of convenience, the following P_{m} is denoted as P. From the above section, we know that if the system transient stability, the maximum value of f is the value of the the back point

When the back point closes to the unstable equilibrium point δ_{r} → δ_{u}, we define the deceleration power

The trajectory sensitivity formula could be writen:

When

It can be seen that, compared with the power angle stabilitycriterion, when the generator’s active output changes, the phase trajectory stability criterion _{m} changes, the unstable equilibrium point

For n generators of the power system, the movement process described as:

where

According to the above analysis, the transient stability criterion of multi-ma- chine system based on phase trajectory is shown as follows:

For the multi-machine system phase trajectory analysis, the change of active power output of generator

For a system with n generators, a phase trajectory sensitivity matrix is defined for a certain operating point:

When the active power of generators changes ΔP, the phase trajectory formulation to judge the transient stability:

where

For the multi-machine system, how to correctly identify the cause of system instability, in other words, how to find the cause of the system instability by

The practical dynamic security region is defined [

where ^{ }is the upper limit and lower limit of the active power injection; ^{n}^{−1} is N − 1 dimensional real number space.

In the phase trajectory analysis, the criteria to judge transient stability of the system are as follows:

From the above analysis, combined with the sensitivity matrix S, when the generators’ active power output changes, the formula to judge transient stability system is as follows:

The formula is deformed as follows:

Comparing formula (18), we can see that formula (21) is the effective deformation of the practical dynamic security region. From paper [

The upper and lower limit of generator’ active power:

The constraints of balancing machine’ active power:

Considering the power balance of the system:

Then the bus injection constraint is given:

Substituting Equation (24) into Equation (21):

Defining the sensitivity of the phase trajectory function f_{i} of the generator i to the generator j:

The PDSR boundary dominated by the generator i based on the phase trajectory is:

For the actual system, there are a large number of generators. The security region defined by formula (29) only needs to establish the boundary of the security domain where the critical generator is destabilized. In this case, the critical function f in the plane is only considered to be easy to be destabilized.

When adjusting active power output of the generator i to the upper limit, if the system becomes transient unstable, there is unstable mode of system controlled by generator i. And this controlling PDSR boundary is under the node injection constraint.

The dichotomy is used to search the critical point of the unstable mode. The generator set with large transient influencing factor is selected according to the sensitivity S, and selected as the coordinate axis of the reduced PDSR.

For the basic operation and a given fault, when the fault duration t, the algorithm flow is as follows in

In this paper, the method proposed in the paper is tested on the New England 10-generator 39-bus system. The wiring diagram is shown in

This system takes the bus 31 as the balancing machine. The fault state is set as three-phase short circuit fault of 15 - 16 line in the system, cleared 0.12 seconds later.

1) The phase trajectory of the generator is obtained by time-domain simulation of the initial operating point;

2) The power angle curve of instability is obtained by increasing the failure time to t = 0.22 s, as shown in

3) In the critical set A, the generator’ active power output is increased to the upper limit. If the system is unstable, the controlling unstable generator G32 is identified;

4) The controlling unstable critical point of generator is searched by Dichotomy;

5) The sensitivity α is obtained based on the critical point as shown in

Generator number | Sensitivityα |
---|---|

G30 G32 G33 G34 G35 G36 G37 G38 G39 | −0.2801 0.3338 −0.1282 −0.1439 −0.2073 −0.1019 −0.2675 −0.1910 −0.3085 |

The PDSR with G32 as the controlling unstable mode is obtained. For the visual representation, the generator G30is selected as the auxiliary axis. The PDSR (shaded area) is shown in the

6) In the two-dimensional security region composed by G30 and G32, we consider the upper limit of active power output of balancing machine G31 and the upper limit of active power output of G30 and G32, and draw the two-di- mensional PDSR.

In order to verify the correctness of the DSR calculation results, several operating points are selected in the injection space of the system. The direction to choose operating points is from the basic operating point toward the G32 controlling boundary. Time-domain simulation is used to prove the accuracy of PDSR.

The corresponding unstable power angle curve as shown in

Operating point | G30/MW | G32/MW | Criterion of PDSR | Time-domain simulation |
---|---|---|---|---|

1 2 3 4 | 240 235 230 220 | 670 680 690 710 | Yes Yes No No | Stable Stable Unstable Unstable |

In order to verify the correctness of the DSR calculation results, in the injection space of the power system, we select a number of operating pointsfrom the basic operating pointto the G32 dominant direction.Time-domain simulation is showed in

In this paper, a method to solve practical dynamic security region is proposed based on the phase trajectory analysis. It studies the variation of the phase angle and angular velocity when the generators’ active power changes. Based on the analysis of phase trajectory stability function f and phase trajectory sensitivity matrix S, an effective relation between the criterion of phase trajectory and dynamic security region is established. The PDSR is obtained by the analysis of the phase trajectory. The result has shown that this method does not need a lot of critical points and the calculation speed is improved greatly.

This work is supported by the National Science Foundation of China (No.51377118) and the project of State Grid Tianjin Electric Power Company (KJ15-1-08).

Gao, Y., Chang, J.T., Qin, C., Zeng, Y., Liu, Y.Y. and Li, S.W. (2017) Practical Dynamic Security Region Based on Phase Trajectory. Energy and Power Engineering, 9, 503-514. https://doi.org/10.4236/epe.2017.94B056