Quadratic probabilistic load flow with linearly modelled dispatch

The multiple solution analysis of probabilistic load flow

Normal-transformation-based probabilistic load flow with correlated wind and load forecast errors

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 variables (e.g., nodal voltages), 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 variables is an infinite GMM. By leveraging this proposition, the proposed method generates the joint PLF of the whole system at the quasi-steady and steady states. The high accuracy and satisfactory efficiency of this method are verified on multiple test systems with different percentages of renewables.

Because of uncertainties of renewable generation resources and load, efficient tools are required for load flow analysis in power systems. Many of the existing papers assume a set of given probability density functions (PDFs) to model uncertainties and develop parametric probabilistic load flow tools. However, the uncertainties might not fall in any standard class of PDFs. Thus, nonparametric tools are needed. This paper presents a method for nonparametric probabilistic load flow analysis to determine PDFs of the load flow outputs. The proposed method is based on the mean value first order saddle point approximation. For a system with $\boldsymbol {n}$ random variables, $\boldsymbol {(n+1)}$ load flow calculations are utilized to establish first order Taylor series expansion and then saddle point approximation is adopted to determine the probabilistic characteristic of desired output variables. The proposed nonparametric estimator provides accurate results while needing a reasonable computational effort. The estimator needs only a limited available data set of random variables without requiring information whether the data set belongs to a certain class of parametric distribution functions. Furthermore, both probability and cumulative distribution functions of load flow outputs are directly established without using integral or differential operators. The proposed method is tested on the IEEE 14-bus and the IEEE 118-bus test systems and promising results are obtained.

Advanced Probabilistic Load Flow Technique in Consideration of Non-Gaussianity and Nodal Correlation of Input Variables

It is an important part for the power system operation risk control to identify the critical lines. Fluctuation of wind power and fluctuation of load will have a great influence on power transmission and distribution, which will lead to the change of critical lines. Based on probabilistic load flow (PLF) analysis, the risk theory is introduced to put forward, and probabilistic flow betweenness considering uncertainty of wind power and load is presented for critical line identification, which reflects the influence of power grid structure characteristics, change of system running state and uncertain factors on power transmission. It can identify critical lines more accurately. Simulation calculation on CEPRI-36 bus system integrated with wind farms and IEEE-39 bus system with wind farms verify effectiveness of this method. It is able to search for the critical lines according to system running state in actual application, so as to adopt security control measures for the system.

The paper extends the methods of evaluating the continuity in supplying electric power systems using comparative methods of the probabilistic load flow and the stochastic load flow. In the first case a binomial distribution for the units generators and a normal one for the loads are accepted. Branch outages are simulated by fictitious power injections at the corresponding nodes. The loss of load probability and expected energy not supplied are estimated with Gauss Laplace integral functions. The results are compared with the ones obtained through a reliability simulation with DigSILENT Power Factory application.

A non-iterative and exact linearization load flow technique for circuit contingency effects in power systems

This chapter deals with probabilistic approaches for the steady-state analysis (probabilistic load flow) of distribution systems with wind farms. The probabilistic analysis is performed taking into account the randomness of both the distribution system loads and the wind energy production. Several approaches are presented to obtain the probability functions of state and dependent variables (e.g., voltage amplitudes and line flows). These approaches are mainly concentrated on wind farm probabilistic models, using one of the classical probabilistic techniques (e.g., Monte Carlo simulation, convolution process, and special distribution functions) to perform the probabilistic load flow. Numerical applications on a 17-bus balanced test distribution system and on an IEEE 34-bus unbalanced test distribution system are presented and discussed, considering the various wind farm models.

In allusion to the uncertainty and correlation of distributed generation (DG) output in the active distribution network, a new method of probabilistic load flow (PLF) analysis was proposed based on improved sequence operation method. This method utilised the non‐parametric kernel density estimation method and established a probabilistic model of DG daily output considering the timing characteristics. Then this method proposed the multivariate joint distribution probabilistic sequence based on the present sequence operation theory to describe the sequence transformation of the probabilistic model. Finally, an improved dependent probabilistic sequence method was proposed and combined with the traditional linear power flow calculation to achieve the probability distribution information of system power flow. Simulation tests were carried out on the modified 33‐bus distribution system. Test results indicated that the proposed method can achieve more accurate results of PLF calculation and has higher computational efficiency.

This paper shows a practice to raise the reliability of an electric power system by the installation of distributed generation, taking gasified biomass as fuel. To calculate the reliability index, a probabilistic load flow was used. This index is determined as the fault probability of the system. The resolution of this probabilistic load flow combines the method of cumulants and Gram–Charlier expansion. To achieve the reliability index, simulating a number of contingencies is required; the greater the number of simulated contingencies, the higher the accuracy of the index obtained. This probabilistic technique uses the random variables as starting information, so the two generators and loads are simulated as random variables. The generators of this distributed generation are biomass-fueled gas engines, commonly found in Spain. The simulations carried out on the IEEE 14-bus Test System, including three biomass generators, show that the inclusion of this type of generation improves the overall reliability indices of the electrical system.

This paper presents a Probabilistic Load Flow (PLF) procedure that allows the impact of distributed generation (DG) on the transmission system to be assessed. By means of the Monte Carlo technique, several system states are simulated, varying the power injection of DG. The load flow solution is performed by the Newton-Raphson method and a distributed slack bus formulation is adopted, in order to distribute the active power mismatch (created by the random generation) on all the thermal units in the system. Tests are carried out both on a small system and on the Italian network. Results lead to conclude that the probabilistic approach to the power system analysis provides more indications than the traditional deterministic techniques, especially considering a high penetration of DG.

In this paper a probabilistic model for the active power produced and the reactive power absorbed by Wind Turbines (WTs) equipped with induction generators is developed which takes into account the probabilistic nature of short-term wind velocity forecasts. The model is incorporated in a radial distribution load flow program which allows probabilistic modeling of loads at the MV/LV substation level and of voltage regulator effects at the beginning of the MV distribution feeder. Using this program probabilistic short- and medium-term predictions of the power flows at the various sections of the feeder and of voltage profiles at all nodes of the network are obtained.

An intrusive spectral method of probabilistic load flow (PLF) is proposed in the paper, which can handle the uncertainties arising from renewable energy integration. Generalized polynomial chaos (gPC) expansions of dependent random variables are utilized to build a spectral stochastic representation of PLF model. Instead of solving the coupled PLF model with a traditional, cumbersome method, a modified stochastic Galerkin (SG) method is proposed based on the P-Q decoupling properties of load flow in power system. By introducing two pre-calculated constant sparse Jacobian matrices, the computational burden of the SG method is significantly reduced. Two cases, IEEE 14-bus and IEEE 118-bus systems, are used to verify the computation speed and efficiency of the proposed method.

Power flow problems can be solved in a variety of ways by using the Newton–Raphson approach. The nonlinear power flow equations depend upon voltages Vi and phase angle δ. An electrical power system is obtained by taking the partial derivatives of load flow equations which contain active and reactive powers. In this paper, we present an efficient seventh-order iterative scheme to obtain the solutions of nonlinear system of equations, with only three steps in its formulation. Then, we illustrate the computational cost for different operations such as matrix–matrix multiplication, matrix–vector multiplication, and LU-decomposition, which is then used to calculate the cost of our proposed method and is compared with the cost of already seventh-order methods. Furthermore, we elucidate the applicability of our newly developed scheme in an electrical power system. The two-bus, three-bus, and four-bus power flow problems are then solved by using load flow equations that describe the applicability of the new schemes.