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Probabilistic load flow (PLF) calculation, as a fundamental tool to analyze transmission system behavior, has been studied for decades. Despite a variety of available methods, existing PLF approaches rarely take system control into account. However, system control, as an automatic buffer between the fluctuations in random variables and the variations in system states, has a significant impact on the final PLF result. To consider control actions' influence, this paper proposes the first analytical PLF method for the transmission grid that takes into account primary and secondary frequency controls. This method is based on a high-precision linear power flow model, whose precision is even further improved in this paper by an original correction approach. This paper also proves that if the joint probability distribution (JPD) of random variables is expressed by a Gaussian mixture model (GMM), then the JPD of system states (e.g., nodal voltages) is an infinite GMM. By leveraging this proposition, the proposed method can generate the joint PLF of the whole system, is applicable to random variables obeying any distributions, and is capable of capturing their correlation. The high accuracy and satisfactory efficiency of this method are verified on test cases scaling from 14 to 1354 buses.

Faults in photovoltaic (PV) systems can seriously affect the efficiency, energy yield as well as the security of the entire PV plant, if not detected and corrected quickly. Therefore, fault diagnosis of PV arrays is indispensable for improving the reliability, efficiency, productivity and safety of PV power stations. Instead of conventional thresholding methods and artificial intelligent (AI) machine learning approaches, an innovative Gaussian Mixture Model (GMM) based fault detection approach is proposed in this work. This approach combines the superiority of GMM in modeling stochastic power outputs of PV modules and the flexibility and simplicity of Sandia PV Array Performance Model (SPAM) to accurately detect under-performing modules. Firstly, GMM is proposed to represent the probabilistic distribution functions (PDF) of different PV modules' power outputs, and the parameter sets of which are obtained by Expectation Maximization algorithm. Secondly, a simplified explicit expression for output power of PV modules, which highlights the influences of tilt and azimuth angles, is deduced based on the SPAM. Then, an orientation independent vector C is developed to eliminate the probability distribution differences of power outputs caused by varying azimuth angles and tilt angles. Jensen-Shannon (JS) divergence, which captures the differences between probability density of C of each PV module, are generated and used as fault indicators. Simulation data acquired from the original SPAM are used to assess the performance of the proposed approach. Results show that the proposed approach successfully detects faults in PV systems. This work is especially suitable for the PV modules that have different installation parameters such as azimuth angles and tilt angles, and it does not require the use of irradiance or temperature sensors.

With the proliferation of distributed generators and energy storage systems, traditional passive consumers in power systems have been gradually evolving into the so-called "prosumers", i.e., proactive consumers, which can both produce and consume power. To encourage energy exchange among prosumers, energy sharing is increasingly adopted, which is usually formulated as a generalized Nash game (GNG). In this paper, a distributed approach is proposed to seek the Generalized Nash equilibrium (GNE) of the energy sharing game. To this end, we convert the GNG into an equivalent optimization problem. A Krasnosel'ski{\v{i}}-Mann iteration type algorithm is thereby devised to solve the problem and consequently find the GNE in a distributed manner. The convergence of the proposed algorithm is proved rigorously based on the nonexpansive operator theory. The performance of the algorithm is validated by experiments with three prosumers, and the scalability is tested by simulations using 123 prosumers.