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Abstract

Stochastic disturbances play a profound problem in the power system, which have an important impact on the stability of the power system. The paper proposes the stability analysis of stochastic disturbance bounded value of linear power system, and presents that the stability of power system has bounded value under stochastic disturbance and additional disturbance, and gives the analysis process in combination with stochastic differentiation. The equation theory proposes a numerical solution based on mean stability to calculate the boundedness of infinite systems under the influence of stochastic disturbance and additional disturbance. The results show that the system has bounded value stability under the disturbance.

Keywords

Stochastic disturbance stochastic linear power systems bounded value stability analysis

Introduction

The power system is a multi-dimensional complex system with stochastic disturbances. The quality of power is directly related to the normal development of society. Disturbance is one of the important factors affecting power quality and normal operation of power systems. By studying the impact of noise on grid stability, people can better improve the system and reduce the problems caused by grid instability in production and life. There have been many stochastic stability analysis methods.

For the study of the stability of stochastic power systems, stochastic modeling and stability analysis in wind power systems are proposed, which are based on Markov theory, the response and stability of the power system under small Gaussian stochastic excitation are used.¹ For uncertainty and stochastic ness discussion, probabilistic transients stability assessment tools is proposed in power systems.² There has been considerable recent interest in the analysis of small signal stability problem in power system operation,³ The literature studies the control network system based on Markov delay characteristics.⁴ Stochastic differential equations (SDEs) are employed.⁵ An evaluation of parameter uncertainty for power system dynamics calculated for the feasible method. Some literatures focus on the problem of Gauss stochastic small excitation, and have achieved good results.^6,7

The scale of the interconnected power grid is constantly expanding, and stochastic disturbances caused by loads, faults, etc. are becoming more common, which makes the power system suffers. The stochastic disturbance is more and more obvious, the power system stability is affected to some extent.^8–11 Some literatures use the improved probability method to analyze the dynamic stability under different operating modes,¹² some propose an evaluation.¹³ A feasible method for calculating the influence of parameter uncertainty on the dynamic behavior of power systems are developed.^14,15 The analysis of the stochastic disturbance stability of different power systems are proposed.^4,16–18

This paper firstly describes the current research status of stochastic power system stability, and analyzes the stability of stochastic linear power system. Based on this, the mean stable method is analyzed. Then simulate the system stability and analyze the simulation results. Finally, and the significance of studying the stability of stochastic power systems is illustrated.

Stable interval stability of power system stochastic disturbance

In many practical projects, the disturbance experienced by the system can be approximated as a zero-mean Gaussian process with smooth independent increments. A stochastic linear model and stochastic nonlinear model can be expressed as in general¹⁹:

dx (t) = Ax (t) dt + Bu (t) dt + \sum_{i = 1}^{r} σ_{i} F_{i} (x (t)) d ω_{i} (t)

(1)

dx (t) = f (x (t), t) dt + g (x (t)) u (t) dt + σ (x (t), t) d ω (t)

(2)

Based on this basis, considering the additional disturbance of the stochastic system, a mathematical model of the following Stochastic state is formed,

y = f (x (t)) dt + g (x (t)) d ω (t) + h (t) u (t) + l (v (t))

(3)

Where f(x(t))dt denotes an n-dimensional vector in equation (3). Where $g (x (t)) d ω (t)$ is an n*m matrix $x (t) = {[x_{1} (t), x_{2} (t), \dots, x_{n} (t)]}^{T}$ is N-dimensional vector stochastic state variables. h(t)u(t) represents the stochastic disturbance in the system, affecting the equilibrium point and accuracy of the system. $f (x (t)) dt + g (x (t)) d ω (t)$ represents the stochastic state of the grid when it is running. $ω (t)$ indicates n m-dimensional vector wiener process. l(v(t)) represents the secondary additional disturbance in the system and superimposes with the signal.

Model analysis

The state space equation in a continuous linear system is defined generally¹⁹:

\frac{dx (t)}{dt} = f [x (t)] + g [x (t)] d ω

(4)

As such, linearize the above equation is chosen as follows,

\frac{dx (t)}{dt} = Ax (t) + Bg (t) d ω

(5)

Equation (5) is the object model. Where: x(t) is an n×1 state vector, A is an n × n coefficient matrix. g(t) is the m×1 control vector. t is continuous time. In the absence of disturbance, the power system is usually in stable operation. This paper will study the stability of stochastic power systems with stochastic and additional disturbance.

Stochastic disturbance is a type of disturbance that is unpredictable at a given instant due to the accumulation of large amounts of undulating disturbances that are randomly generated in time. Additional disturbance is a type of disturbance superimposed on a signal, which is often referred to as a secondary additional disturbance, whether there is a signal or not. The amplitude distribution follows the Gaussian distribution, and the spectral power density is uniform, so it can be considered Gaussian noise, and considered as Gaussian noise. Generalized exponential model of additional disturbance⁷ is $y_{t} = Π_{j = 1}^{m} x_{jt}^{θ_{j}} + v_{t}$ , $t = 1, 2, \dots n$ . Introducing the corresponding noise item in the object model,

\begin{matrix} \frac{dx (t)}{dt} = Ax (t) + Bg (t) d ω + ξ (t) + l (v (t)) \end{matrix}

(6)

where ξ(t) represents a stochastic disturbance vector, and l(v(t)) is an additional disturbance vector. It is now necessary to study whether the upper form is bounded and whether the solution of the model converges to equilibrium in continuous time.

Stability analysis

Stochastic stability theory is used to solve stochastic systems where the initial state is not in equilibrium. At time [t_0, ∞), it is judged whether the distance of the solution of the stochastic system to the equilibrium state is bounded, and whether the solution of the stochastic system converges to the equilibrium state at t→∞.

Power system stochastic disturbance bounded value stability model

Equation (12) is considered as the definition of a single-machine infinite system. The equation of rotor motion after adding stochastic disturbance and additional disturbance to the generator can be written as follows:

{\begin{matrix} \overset{\cdot}{δ} = ω - ω_{0} \\ \overset{\cdot}{ω} = \frac{1}{M} [P_{m} - P_{e} - D (ω - ω_{0}) + σ ω (t) + v (t)] dt \end{matrix}

(7)

where: M is the inertia time constant; δ denotes the power angle, ω is the rotational speed, t is the time, D is the damping coefficient, σ represents the intensity of the stochastic disturbance ω(t), and ε>0, such that $‖ σ ‖ \leq ε$ , $v (t)$ is additional disturbance, $P_{e} = \frac{E^{'} U}{X} \sin δ$ denotes electromagnetic power. $P_{m} = \frac{E^{'} U}{X} \sin δ_{0}$ is mechanical power, the value is equal to $P_{e}$ steady state value. Equation (7) can be expressed as following differential equation,

{\begin{matrix} d δ = ω d t \\ d ω = \frac{1}{M} [- \frac{E^{'} U}{X} \cos δ_{0} Δ δ - D ω + σ ω (t) + v (t)] d t \end{matrix}

(8)

Model mean stability

Regarding the stability study of power systems, the numerical simulation of transient differential algebraic equations for power systems under stochastic loads is proposed.²⁰ This paper presents a numerical method of mean stability to solve the stability of the system.

The mean stability is studied¹²: In the literature, $X (t)$ can satisfy $lim_{t \to \infty} E | | X (t) | | 〈 C, C 〉 0$ . The system is in a steady state. Regardless of additional disturbance, Equation (8) is converted into a vector form. The influence of stochastic factors here is the same stochastic item $ω_{i} (t)$ , there is a secondary additional disturbance $Qv (t)$ during the operation of the system. We consider switching modes such as stochastic vibration of prime mover torque, stochastic fluctuation of load, and stochastic fluctuation of renewable energy power generation. When each generator switches mode, the quantity and type are different. $σ$ , $τ$ represents the intensity of stochastic excitation $ω_{i} (t)$ under different switching.

\begin{matrix} (\begin{matrix} d δ \\ d ω \end{matrix}) = (\begin{matrix} 0 & 1 \\ - \frac{E^{'} U}{X} \cos δ_{0} & - \frac{D}{M} \end{matrix}) (\begin{matrix} d δ \\ d ω \end{matrix}) \\ + (\begin{matrix} 0 \\ \frac{σ}{M} \end{matrix}) ω (t) + (\begin{matrix} 0 \\ \frac{τ}{M} \end{matrix}) v (t) \end{matrix}

(9)

The above equations are given by:

\overset{\cdot}{X} = AX + B ω (t) + Qv (t)

(10)

Where, $X = (\begin{matrix} d δ \\ d ω \end{matrix})$ ; $A = (\begin{matrix} 0 & 1 \\ - \frac{E^{'} U}{X} \cos δ_{0} & - \frac{D}{M} \end{matrix})$ ; $B = (\begin{matrix} 0 \\ \frac{σ}{M} \end{matrix})$ ; $Q = (\begin{matrix} 0 \\ \frac{τ}{M} \end{matrix})$ .

According to the stochastic differential equation, the secondary additional stochastic disturbance has the same stochastic attribute:

\begin{matrix} X = e^{At} X_{0} + {\int_{0}^{t} e}^{A (t - s)} (B + Q) dH (s) \end{matrix}

(11)

The relationship between mathematical expectation and variance in probability theory: ${(E | | X | |)}^{2} \leq E [{| | X | |}^{2}] = E [X^{T} (t) X (t)]$ . Properties of stochastic differential equations by Ito is shown in the following, as obtained from equation (11):

\begin{matrix} E [X^{T} (t) X (t)] = E [{(e^{At} X_{0})}^{T} e^{At} X_{0}] \\ + E [{({\int_{0}^{t} e}^{A (t - s)} (B + Q) dH (s))}^{T} {\int_{0}^{t} e}^{A (t - s)} (B + Q) dH (s)] \\ = {| | e^{At} X_{0} | |}^{2} + \int_{0}^{t} {| | e^{A (t - s)} (B + Q) | |}^{2} ds \end{matrix}

(12)

and

\begin{matrix} E [X^{T} (t) X (t)] = ‖ e^{At} X_{0} ‖^{2} + \int_{0}^{t} {‖ e^{A (t - s)} (B + Q) ‖}^{2} ds \\ \leq C^{2} e^{2 λ t} N_{1} + C^{2} \frac{1}{2 λ} (e^{2 λ t} - 1) N_{2} \end{matrix}

(13)

1) $Q > B$

\begin{matrix} ‖ B + Q ‖^{2} = ‖ B ‖^{2} + 2 ‖ B ‖ ‖ Q ‖ + ‖ Q ‖^{2} \leq 4 ‖ B ‖^{2} \\ lim_{t \to \infty} (E ‖ X ‖)^{2} \leq lim_{t \to \infty} E [X^{T} (t) X (t)] \leq \end{matrix}

(14)

\begin{matrix} lim_{t \to \infty} C^{2} e^{2 λ t} N_{1} + C^{2} \frac{1}{2 λ} (e^{2 λ t} - 1) N_{2} = - C^{2} \frac{2 N_{2}}{λ} \\ C_{0} > \sqrt{- C^{2} \frac{2 N_{2}}{λ}} \end{matrix}

(15)

Since the real part of the eigen values of the state matrix of the small disturbance stabilizing system is less than zero, so there is λ < 0. We obtain the differential according to the nature of matrix as follows:

C^{2} > 0, N_{1} = ‖ X_{0} ‖^{2}, N_{2} = ‖ B ‖^{2}, λ < 0,

then,

lim_{t \to \infty} E ‖ X ‖ \leq C_{0}

2) $Q = - B$

\begin{matrix} E [X^{T} (t) X (t)] = ‖ e^{At} X_{0} ‖^{2} + \int_{0}^{t} {‖ e^{A (t - s)} Q ‖}^{2} ds \\ \leq C^{2} e^{2 λ t} N_{1} \end{matrix}

(16)

\begin{matrix} lim_{t \to \infty} (E ‖ X ‖)^{2} \leq lim_{t \to \infty} E [X^{T} (t) X (t)] \leq \\ lim_{t \to \infty} C^{2} e^{2 λ t} N_{1} + C^{2} \frac{1}{2 λ} (e^{2 λ t} - 1) N_{2} = lim_{t \to \infty} C^{2} e^{2 λ t} N_{1} = 0 \end{matrix}

(17)

$C^{2} > 0$ , $N_{1} = ‖ X_{0} ‖^{2}$ ; $λ < 0$ , then,

lim_{t \to \infty} E ‖ X ‖ \leq 0

3) $Q = 0$

\begin{matrix} lim_{t \to \infty} (E ‖ X ‖)^{2} \leq lim_{t \to \infty} E [X^{T} (t) X (t)] \leq \\ lim_{t \to \infty} C^{2} e^{2 λ t} N_{1} + C^{2} \frac{1}{2 λ} (e^{2 λ t} - 1) N_{2} = - C^{2} \frac{N_{2}}{2 λ} \\ C_{0} > \sqrt{- C^{2} \frac{N_{2}}{2 λ}} \end{matrix}

(18)

$C^{2} > 0$ , $N_{1} = ‖ X_{0} ‖^{2}$ , $λ < 0$ , where,

lim_{t \to \infty} E ‖ X ‖ \leq C_{0}

Then, $B = 0$ , the result is the same.

4) $Q = B$

\begin{matrix} lim_{t \to \infty} (E ‖ X ‖)^{2} \leq lim_{t \to \infty} E [X^{T} (t) X (t)] \leq \\ lim_{t \to \infty} C^{2} e^{2 λ t} N_{1} + C^{2} \frac{1}{λ} (e^{2 λ t} - 1) N_{2} = - C^{2} \frac{N_{2}}{λ} \end{matrix}

(19)

C_{0} > \sqrt{- C^{2} \frac{N_{2}}{λ}}

we can obtain C in a series as follows,

K_{0} = C_{0} > \sqrt{- C^{2} \frac{N_{2}}{λ}} K_{0}^{'} = C_{0} > \sqrt{- C^{2} \frac{N_{2}}{2 λ}}

Where,

K_{0} - K_{0}^{'} = \sqrt{- C^{2} \frac{N_{2}}{λ}} - \sqrt{- C^{2} \frac{N_{2}}{2 λ}} = \frac{2 - \sqrt{2}}{\sqrt{2}} \sqrt{- C^{2} \frac{N_{2}}{λ}}

From the above equations, the interval of the boundary value can be deduced from the following:

[- \frac{2 - \sqrt{2}}{\sqrt{2}} \sqrt{- C^{2} \frac{N_{2}}{λ}}, \frac{2 - \sqrt{2}}{\sqrt{2}} \sqrt{- C^{2} \frac{N_{2}}{λ}}]

5) $Q < B$

‖ B + Q ‖^{2} = ‖ B ‖^{2} + 2 ‖ B ‖ ‖ Q ‖ + ‖ Q ‖^{2} \leq 4 ‖ Q ‖^{2}

(20)

\begin{matrix} lim_{t \to \infty} (E ‖ X ‖)^{2} \leq lim_{t \to \infty} E [X^{T} (t) X (t)] \leq \\ lim_{t \to \infty} C^{2} e^{2 λ t} N_{1} + C^{2} \frac{1}{2 λ} (e^{2 λ t} - 1) N_{2} = - C^{2} \frac{2 N_{2}}{λ} \end{matrix}

(21)

C_{0} > \sqrt{- C^{2} \frac{2 N_{2}}{λ}}

$C^{2} > 0$ ; $N_{1} = ‖ X_{0} ‖^{2}$ ; $λ < 0$ , then,

lim_{t \to \infty} E ‖ X ‖ \leq C_{0}

The above equations are used to analyze the stability of the stochastic disturbance bounded value of power system. By comparing the system with stochastic disturbance and additional disturbance, it is concluded that the system with additional disturbance increases than the boundary value of only stochastic disturbance, and the boundary value is quantified. Further analysis shows that stochastic disturbance in power systems that only contain stochastic disturbance is part of a system with both multi-dimensional stochastic and additional disturbance. For a multi-dimensional complex system power system with stochastic disturbances, the stochastic disturbance and additional disturbance of the stochastic linear power system are stable, with stochastic small disturbances, the stochastic power system is mean stable and free from Stochastic and additional disturbance.

Therefore, the small disturbance stochastic stability system is bounded, which proves that the system mean is stable under the disturbance.

Results and discussion

The stability of the synchronous operation of the power system is judged according to the variation law of the system voltage after the disturbance, and this property is also called voltage stability. The simulation model will study the effects of disturbance of different intensities on voltage stability.

A set of data is generated by using a function to simulate a power system site. Setting the sampling frequency to 1000 Hz, the number of sampling points to 600, and the grid frequency to 50 Hz to model the power system signal. Set the Fourier transform data point to 512, and perform Fourier transform on the sampled signal to obtain the spectrum signal, as shown in Figure 1.

Figure 1.

Power system spectrum signal.

During normal and stable operation, the voltage of the system is constant and the frequency domain signal is also in a convergent state.

When disturbance is introduced, the sampled signal receives a change in the disturbance waveform. Similarly, the spectrum signal is obtained after performing Fourier transform on the set data points is as shown in Figure 2.

Figure 2.

Spectral signal under stochastic disturbance.

When the disturbance intensity is 20%, the voltage fluctuation and the Fourier transformed spectrum signal are shown in Figure 3.

Figure 3.

20% stochastic disturbance intensity spectrum signal.

It can be seen that the introduction of 20% disturbance intensity, the voltage is disturbed, the signal fluctuates around the equilibrium point in a small range, and the amplitude of the fluctuation does not exceed ±2 V, which does not have much influence on the equilibrium point of the system. After transforming the time domain signal into a frequency domain signal by Fourier transform, it can be found that the spectrum of the periodic signal is a non-periodic line, called a spectral line. The spectral line amplitude has a decreasing trend as a whole, and the system converges.

It is reflected from Figure 4 that when 50% disturbance intensity is introduced, the voltage fluctuation amplitude gradually increases, but the fluctuation amplitude still does not exceed ±2 V. The fluctuation of the frequency domain signal increases slightly, and the system is still in a stable state.

Figure 4.

50% stochastic disturbance intensity spectrum signal.

When the disturbance intensity is 80%, the voltage fluctuation and the Fourier transformed spectrum signal are shown in Figure 5.

Figure 5.

80% stochastic disturbance intensity spectrum.

When 80% a secondary additional disturbance intensity is introduced, the voltage fluctuation amplitude is large. In the frequency domain signal, the fluctuation of the spectral line also increases, and the system gradually loses stability.

The simulation results show that the power system has average stability when it is interfered by low-intensity a secondary additional disturbance. The signal fluctuates around the equilibrium point and does not affect the equilibrium stability. This will have a great impact on the normal operation of the power system, so a secondary additional disturbance in the production of electric energy should be minimized.

As shown in Figure 6, the difference between stochastic disturbance and additional disturbance is the boundary value of the system, parameter values are shown in Table 1. As shown in the results in Figure 7, the sum between the two is the boundary value of the stochastic linear power system, parameter values are shown in Table 2. Sampling point (SP) means that each group has 10 points. Here, stochastic disturbance difference is used, and there is a bounded value interval for the stochastic disturbance of the linear power system. The system output with stochastic disturbance, and secondary additional disturbance has presence boundary value.

Figure 6.

Boundary difference between stochastic disturbance and additional disturbance.

Table 1.

Difference value table.

Group no	SP1	SP2	SP3	SP4	SP5	SP6	SP7	SP8	SP9	SP10
1	−1.2131	−2.8211	0.4675	−1.4641	−0.2669	0.9576	−1.8596	−1.335	−1.5888	−1.3117
2	−0.0336	−2.903	0.1082	−0.2845	−0.8135	−0.3733	−1.3547	0.3471	−0.8817	−1.5826
3	−1.369	0.4378	−0.9326	−1.228	−0.5135	−1.0856	−0.6275	−1.1538	−0.4847	0.3576
4	−1.4127	−0.3363	−1.4153	−1.3463	−1.0612	−1.2709	−1.583	−2.4115	−2.0664	−0.4446
5	0.4825	−2.2399	−1.4842	−0.2146	−1.095	0.1957	−1.7544	−1.4097	−2.771	−2.0074
6	0.1885	−0.6985	−1.4751	−0.694	−1.6297	−0.4662	−2.8103	0.096	−0.8638	−0.4156
7	−1.6514	−2.2457	−2.281	−2.9353	−0.8639	−2.5033	0.1836	−1.5794	0.2577	−1.0546
8	−2.0264	−0.8912	0.5826	−1.3961	−1.312	−0.6031	−0.5067	−1.4304	−0.3967	−1.7159
9	−1.1489	0.1436	−0.0589	0.4885	−0.3805	−0.2193	−1.6607	−0.5824	−0.4179	−1.1564
10	0.5597	0.4019	−1.3629	−2.1242	−2.5689	−0.8595	−1.8678	−1.6551	0.1947	−1.8963
11	−0.6742	−1.618	−2.2324	0.1454	−0.1326	−0.2909	−0.8294	−1.0575	−0.9893	−0.2098
12	−1.002	−1.7489	−1.7986	0.4648	−0.8846	−2.6234	−1.512	−1.3458	−0.4683	0.4959
13	0.3588	−1.0638	−1.2471	−0.5147	−0.3145	−0.5942	−0.937	−1.7112	−1.7951	−1.058
14	−1.6854	−2.1083	−0.8045	−2.6017	−0.9516	−0.0614	−1.0336	−1.7566	−1.5899	−1.0365
15	−1.3261	−2.0096	−0.187	−0.0049	−1.2012	−0.2322	−1.2058	0.5008	−2.0386	−2.0965
16	−0.0915	−0.2369	−0.9205	−0.1642	0.7294	−1.538	−1.8521	1.3155	−1.5222	−3.3521
17	−0.5311	−2.084	−1.2793	−1.7442	−1.2869	−0.1808	−0.3805	−1.5677	−2.1952	1.109
18	−0.6969	−2.4965	−0.4196	−0.9099	−1.25	−0.4478	−1.5901	−1.1833	−2.6138	−0.8633
19	−1.1484	−3.6203	−3.5522	−0.2672	−1.7689	−1.1682	−2.1607	−1.1196	−2.0009	0.4788
20	−0.2696	−3.1611	0.4586	−1.1584	−0.6902	0.0132	−0.5927	0.1431	−1.8534	−2.1331
21	−0.6586	−0.9502	−1.9514	−1.0673	−2.6851	−0.3232	−0.7439	−0.3356	0.1749	−1.6616
22	−1.4518	−1.0981	−0.3136	0.0646	−0.608	−1.0548	0.1503	−1.4025	0.4285	−2.182
23	−1.2354	−1.4014	−3.3091	−1.7542	−1.1396	−1.4625	−0.2337	−1.5277	−0.36	−1.4172
24	−2.0995	0.122	−0.6042	−2.0141	−0.7414	−1.0347	−1.3992	−1.6092	−0.4906	0.0238
25	−1.3505	0.414	−0.6785	0.0205	−1.5276	−0.4555	−2.8048	1.1951	−1.208	−0.2854
26	0.2535	−0.2265	−2.1193	−1.2718	−1.2889	−0.6156	−0.8448	−0.9196	0.9574	0.9598
27	1.2433	0.1384	−1.8888	−2.7492	−1.7607	−0.2925	−1.0649	−1.0791	−1.164	0.2717
28	0.0258	−1.1176	−0.4779	0.5545	−1.2202	−3.8639	0.0665	−4.0461	−0.8143	−1.0833
29	−1.3118	−1.6737	−0.3967	0.2769	−0.4409	−0.5656	−1.1493	−0.008	−0.4227	−3.1403
30	−0.6479	−0.5142	−1.4877	−1.0609	−1.6656	−1.1422	−1.1512	−0.6813	−2.1188	−0.5167
31	−0.491	−1.8549	−0.9643	−1.3563	−0.151	−0.8646	−0.2401	−2.4784	−0.7802	−2.2054
32	−0.7919	−1.679	−0.3635	−1.8083	−0.9188	−0.8531	−1.229	0.6788	0.2381	−1.6627
33	−2.9932	−3.2385	−1.8608	−3.1272	−1.2457	−0.3492	0.4827	−0.431	−1.6723	−2.3767
34	0.2205	−1.1715	1.8105	−1.0646	−1.5715	−1.5377	0.4388	−1.69	−0.6813	1.336
35	−0.401	−0.7585	0.1982	−0.0891	−1.1637	−0.5546	−1.2257	−1.0788	−1.8187	−1.1427
36	−1.2108	−0.1911	−1.1891	−1.0463	0.3824	−0.0301	−2.0622	1.4016	0.6471	−2.039
37	−0.8017	−0.5518	−0.9124	−0.6014	−1.6758	−0.5074	−1.0398	−0.2786	−0.0511	−0.6229
38	−0.8321	0.5416	−0.7004	−0.849	−1.0845	0.4912	−1.5197	−0.6845	0.2858	−2.4758
39	−1.9972	−0.4806	−0.3644	−0.898	−0.448	−2.5423	−1.424	−1.9962	−1.7126	−2.6749
40	−0.7379	−1.2983	−0.3569	−2.7266	−1.8951	−0.1977	−0.5618	−2.2513	−0.2497	1.6246
41	−0.1658	−0.143	−1.2754	0.982	−1.7954	0.776	−0.6432	0.089	−2.2516	−0.8579
42	−1.758	−0.9478	−1.3355	0.4758	−0.3816	1.6339	0.2481	−0.6695	−0.8997	−3.1005
43	0.1459	−1.1459	−1.1483	−0.7165	−0.9902	−0.3448	−3.7659	−1.919	−1.3989	0.2394
44	−0.5079	1.0563	−2.5298	−0.6035	−1.3391	−3.2864	−0.6934	0.5574	−1.6595	−0.8749
45	−1.4313	−0.2417	1.0325	−2.5039	−0.3218	−1.6528	0.3187	1.026	−1.2819	−3.3599
46	−1.6822	0.8487	−1.3284	−0.8881	−1.2097	−0.7104	−2.0511	−0.4901	−0.2262	−2.0949
47	−0.9982	−1.2396	−1.5917	1.315	−1.2251	−1.7033	−0.2382	−1.7388	−2.0839	−0.0337
48	0.2635	−0.633	−1.2249	−1.2677	−1.6903	−0.5535	−0.2571	0.2645	−3.1291	−3.0452
49	−1.617	−0.8111	−2.1256	−1.4962	−0.1744	−0.6539	−1.6329	−1.4719	−0.6344	1.4809
50	0.0832	−0.1208	−0.678	−1.9326	−2.0902	−2.5519	−2.2675	−0.0334	−1.0414	−0.7009
51	−1.4729	−1.8793	−1.1666	−1.2847	−0.9329	−1.1349	−1.2205	−0.028	−0.9246	−1.5306
52	−2.1289	−0.1136	−0.392	−3.5354	−0.9403	−1.3945	−0.7296	−0.4558	−1.7771	−1.3726
53	−2.8302	−0.8311	−1.636	0.179	−0.4992	−2.1736	−2.2459	−0.3677	−2.7518	−0.9555
54	−1.7708	1.2329	−1.0955	0.1385	−1.6478	−1.6545	−1.5816	−0.7204	−2.9454	−2.3304
55	−0.2356	−2.0927	−1.2109	−1.2701	−0.4868	−2.0419	−0.3089	−2.1226	−2.3888	−1.0844
56	−1.5329	−2.0175	−0.7499	−1.5334	0.0159	−1.2305	−0.4012	0.8641	−2.9176	−2.2155
57	0.292	−2.1788	−1.0686	−0.7639	−0.1572	−2.6118	−1.2717	−1.7958	−0.6913	−1.1561
58	−0.5764	−0.232	−0.9249	−1.7888	0.1572	−2.6682	−0.5735	−0.554	−0.6022	−0.9955
59	−1.3508	−1.0499	−1.8099	−0.9246	−0.1618	−1.7484	0.1823	−1.3542	−2.7466	−1.2513
60	0.4729	1.2374	−1.2594	0.1915	−1.8783	−0.9414	−1.9055	−0.1183	−0.4809	1.5499

Figure 7.

Boundary sum between stochastic disturbance and additional disturbance.

Table 2.

Sum value table.

Group no	SP1	SP2	SP3	SP4	SP5	SP6	SP7	SP8	SP9	SP10
1	0.2691	1.2644	0.8699	2.108	2.3849	1.408	4.3446	0.3456	0.7242	2.1312
2	−0.2323	1.0324	−0.7055	−2.1653	−2.7756	−0.1981	−0.5723	−0.4583	−0.288	−1.2639
3	1.9045	1.9567	0.7565	1.6737	2.2508	2.1403	0.9611	−0.0591	1.4988	−0.4471
4	0.4364	2.1961	−0.9915	0.1598	−0.6556	−1.8701	−0.2438	−0.5479	−0.4724	1.3685
5	1.3009	4.3154	1.5264	2.0803	3.4012	1.7015	3.151	3.2398	1.4998	2.5326
6	2.6837	−1.2304	0.0338	−0.3583	−1.0989	−2.2613	0.5091	0.2376	0.3974	1.3537
7	2.3415	−0.1684	1.2231	1.6485	1.8435	1.6463	2.267	1.6767	3.2007	1.4409
8	1.4116	2.2264	−1.5024	−0.4338	−1.7423	−0.4264	−2.7428	−0.3608	−1.3512	−0.1771
9	1.1583	2.9155	1.7102	2.77	3.2587	1.0755	0.1161	1.4535	2.5553	1.1228
10	2.3515	−0.6869	−0.809	0.3835	−0.3503	−1.8166	−1.7023	−0.5744	−0.4889	−0.5689
11	−0.4545	0.4362	0.4897	2.736	1.2456	1.5295	4.1735	4.4856	4.1823	0.686
12	−0.6947	1.4089	−1.9065	−0.7076	−0.0623	−0.8305	−2.5056	−1.1777	0.6346	0.4223
13	0.8448	0.9566	2.0885	3.7676	2.6543	4.255	3.2798	2.8525	1.379	0.4877
14	0.4689	0.0874	0.4867	0.5222	−1.4749	−2.7347	−2.2349	1.0702	−1.1613	−1.4183
15	0.5092	1.3811	1.8027	−0.0136	1.5559	2.5376	0.5392	3.8812	0.5027	0.6911
16	−0.0008	1.075	0.0939	−1.013	−1.6625	−1.2227	0.1351	−0.5501	−1.0033	1.2241
17	0.2837	0.8358	1.8873	4.9587	5.2442	2.8338	3.5518	2.3566	1.9048	1.033
18	−0.6823	−1.6488	−1.5809	−0.3091	0.4051	−0.7195	−2.9371	−1.2073	−2.0565	1.5086
19	0.4142	0.1671	2.626	2.4327	3.7232	3.3473	1.8813	1.7799	0.7812	2.2824
20	2.1084	−0.9333	−0.5311	−1.7846	−2.4924	−1.8027	0.0527	1.2781	0.5915	−0.2985
21	0.162	1.3723	1.4425	3.5037	1.7357	3.3438	0.6177	2.0254	3.5032	0.8619
22	0.5025	−0.3089	−1.852	−0.188	−1.3465	−1.7273	0.5876	−2.6705	0.543	1.7689
23	0.5516	0.4529	3.5877	1.375	2.8606	1.7344	4.4466	2.5152	2.4745	1.0336
24	2.4181	0.5344	−0.4202	−0.7862	−1.7177	−1.0209	−1.5183	−0.759	2.8699	−0.4318
25	−0.0329	2.237	1.3804	1.9695	0.0983	3.3544	1.2934	2.7629	2.4825	2.0835
26	0.475	−0.8428	0.0071	0.2293	−2.4193	−3.0864	−2.2981	−1.6476	1.1609	−2.7716
27	0.4833	1.0071	1.3115	1.5777	1.6089	3.9921	3.0571	1.5108	1.2022	0.6413
28	1.435	0.0217	−0.2284	−1.5443	−2.1441	−0.1319	−1.4061	0.1157	−0.0213	−0.6955
29	2.021	1.4965	2.6331	1.4508	2.473	3.0365	2.8013	2.2415	0.9501	2.1358
30	0.0476	−0.5624	−0.8313	−0.0253	−0.4519	−1.8291	−1.0371	−1.3937	−1.5359	−0.2215
31	1.5957	1.547	1.9179	4.1333	1.2791	2.1383	1.7747	0.9343	3.1458	2.7597
32	0.2833	−1.7177	0.981	−1.6917	−0.1023	−0.1606	−1.1968	−2.7155	−0.3924	0.6149
33	2.5495	1.0534	1.83	3.1815	2.6229	3.3944	4.1425	2.4031	3.033	1.6684
34	0.5434	0.6416	−0.9539	0.029	−1.3395	−0.7844	−3.2858	−0.0634	−1.448	−0.8285
35	0.9956	2.5813	3.052	2.2775	3.8207	4.9481	1.9211	2.1336	0.7358	0.8358
36	−0.1435	−0.9137	1.133	0.1484	−2.1902	−1.3691	0.2352	−0.5719	0.8924	−1.5248
37	1.1044	1.9123	1.1201	3.91	3.1069	0.5096	1.2026	3.5655	1.9186	2.5413
38	0.6097	0.542	−1.8599	−1.4532	−1.5233	−0.1552	−1.6555	−0.7901	−0.4506	−0.5562
39	1.8608	2.7364	2.1824	2.9052	3.3354	1.439	3.8642	0.3457	0.8805	0.7453
40	1.1057	0.4233	−1.0501	−2.2326	−0.6279	−1.9311	−0.5828	−0.3903	0.7843	−1.0014
41	−0.0628	1.6937	0.7694	1.6596	2.5282	2.4207	3.3757	1.625	2.3953	0.7318
42	−0.2553	1.7539	−0.5596	−1.0686	−1.8111	−1.6959	−0.139	−1.6278	−1.3623	−0.3137
43	−0.8771	1.6416	1.2203	4.5677	3.4293	1.6816	4.6182	2.35	1.244	0.574
44	0.8411	0.0211	0.4264	−0.8272	−1.0507	−2.5232	−2.0036	−1.3542	1.1731	−1.6803
45	−0.3298	0.9198	1.6307	1.9618	1.2457	5.2078	3.6448	0.3497	1.7719	3.133
46	1.2252	0.4511	0.0751	−1.0271	−2.1562	−0.5207	1.1442	−2.3485	−0.1613	0.6757
47	1.7157	1.4382	2.5811	4.4396	2.756	1.3147	0.8891	1.534	2.5372	1.9772
48	1.4949	−0.2987	−2.0928	0.1316	−0.9742	−1.7747	−0.2755	−0.2678	−1.0323	0.6653
49	0.9801	1.3211	−0.2549	1.3082	2.0133	3.832	2.7117	2.5395	1.1379	0.051
50	1.3044	0.7459	−1.1357	−1.7051	−1.9676	−1.5438	−0.2326	−0.674	−1.8356	0.161
51	1.7898	1.6227	2.4486	3.5088	2.2252	2.5413	2.4332	2.3394	1.7642	1.4125
52	1.8924	−1.1526	−1.2474	−1.481	−1.5725	−2.1239	−2.632	−1.0806	−0.976	0.1198
53	0.2635	1.6257	1.0203	1.1196	2.6023	3.9061	1.6393	2.2535	2.687	1.0098
54	1.5785	−0.0798	0.9386	−3.1149	−2.5323	−1.5798	−0.7032	0.0711	0.5716	1.3613
55	2.0659	2.1521	2.2112	1.5288	3.3346	3.4235	0.9589	2.2741	2.2307	1.7626
56	−0.3657	−0.3685	0.1155	−0.6217	−1.4569	−1.415	−1.1881	−2.7304	−0.1079	1.4192
57	−1.3458	1.4261	1.8787	3.0856	2.043	2.4474	3.0241	2.2722	2.9966	2.6675
58	0.3137	0.5777	−0.9625	0.343	−3.0205	−1.9792	1.8538	−0.3112	0.3505	1.004
59	0.4479	0.9558	2.1399	2.6614	2.4995	0.7817	3.5723	1.754	2.8396	1.2964
60	0.6947	−0.4746	−2.0557	−1.319	−0.2644	−3.1025	−0.3832	−0.742	−0.9723	0.6818

Conclusion

In the process of social and economic development, the power system will be affected by more and more factors. The power system is a high-dimensional, multi-objective, associative and decentralized system. The traditional model can not meet the new factor disturbance situation, which makes it impossible to accurately judge the stability of the system. Therefore, modeling issues influenced by more factors are of great significance to power system stability. We will further study the mean stability and mean square stability of stochastic linear power systems under multi variate stochastic disturbances.

This paper studies the stability of the bounded value of a stochastic linear power system under stochastic disturbance and additional disturbance. The causes of various disturbances are expounded, the research status of such problems is analyzed, the mathematical model under stochastic state is proposed, and the model is analyzed and calculated. The mean stable method is used to calculate the system bounded under stochastic disturbance. The system with additional disturbance is introduced. The results show that the system is in a stable state, which verifies the existence of bounded value stability of stochastic disturbance in linear power systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by Anhui Province Natural Science Foundation (1908085MF215), Pre-research of National Natural Science Fund Project of Anhui Polytechnic University (Xjky02201905).

ORCID iD

Jie Xu

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Stable bounded value analysis of disturbance in stochastic linear power systems

Abstract

Keywords

Introduction

Stable interval stability of power system stochastic disturbance

Model analysis

Stability analysis

Power system stochastic disturbance bounded value stability model

Model mean stability

Results and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References