Electricity Market Stochastic Dynamic Model and Its Mean Stability Analysis

Abstract

Based on the deterministic dynamic model of electricity market proposed by Alvarado, a stochastic electricity market model, considering the random nature of demand sides, is presented in this paper on the assumption that generator cost function and consumer utility function are quadratic functions. The stochastic electricity market model is a generalization of the deterministic dynamic model. Using the theory of stochastic differential equations, stochastic process theory, and eigenvalue techniques, the determining conditions of the mean stability for this electricity market model under small Gauss type random excitation are provided and testified theoretically. That is, if the demand elasticity of suppliers is nonnegative and the demand elasticity of consumers is negative, then the stochastic electricity market model is mean stable. It implies that the stability can be judged directly by initial data without any computation. Taking deterministic electricity market data combined with small Gauss type random excitation as numerical samples to interpret random phenomena from a statistical perspective, the results indicate the conclusions above are correct, valid, and practical.

Stability of the Stochastic Model for Power Markets with Interval Parameters

“…If = 0, then the model gets rid of random factors and becomes a deterministic model, which agrees with the theorem in [24]. So we can see that the mean interval stable Theorem 8 and Corollary 9 are a generalization of [24,25].…”

“…Wheñ 1 =̃2 = 0 in Corollary 9, is a certain matrix and the consumers demand elasticity is certain. Then, the stability determine condition of model (12) can be simplified as max ( T 33 3 33 ) < 0; that is to say, all eigenvalues of 3 are negative, which is consistent with Theorem 2 in [10] and theorem in [25]. If = 0, then the model gets rid of random factors and becomes a deterministic model, which agrees with the theorem in [24].…”

“…Finally, the validity of results is verified by using exact power market examples and stimulation analysis in Matlab. The results generate conclusions about power market certainty models [8], interval models [24], and stochastic models [25].…”

“…The random excitation can be deemed as power demand stochastic fluctuation caused by objective factors, and these fluctuation are presumed to be Gaussian Wiener process generally. (12), it can be simplified to the deterministic dynamic model proposed by Alvarado in [8], the interval dynamic system model in [24] and the stochastic dynamic model for power market in [25], respectively. Therefore, the model for power market (12) considering both the randomness of power demand and interval features of demand elasticity for suppliers and consumers, is a generalization of deterministic model, interval model and stochastic model proposed by Alvarado, instead of only considering one of the interval or stochastic features.…”

“…In the power market, considering that demand elasticity is uncertain and ranging within interval, a power market interval model is described, and the conditions to determine the interval stability are provided in [24]. In [25], a class of power market stochastic dynamic model is presented, and the conditions to determine the mean stability are provided considering the randomness of the electricity demanding. Factors including economy increasing, industrial structure, and energy consumption structure, such as the uncertainty of wind power fluctuation and interval, jointly become a factor that makes demand elasticity of both sides and the power demand uncertain and influence the stability of power market.…”