Statistics of multimoding in optimum compensation of reactive loads of electrical systems

. A stochastic approach has been implemented to account for multimodality to optimize the operating conditions of electrical systems. On the basis of the algorithm for optimization of the instantaneous mode and the stochastic model of the load graphs, a mathematical model of the generalized reduced gradient was obtained. The practical implementation of the algorithm was carried out due to its low labor intensity.


Introduction
Optimal reactive load compensation (RLC) in the grids of electrical systems (ES) can significantly increase the efficiency of their functioning. The problem solving here requires taking into account the aggregate of steady-state modes (multimoding), obtaining and analyzing the integral parameters of this aggregate, such as energy loss in the system, ranges and diagrams of voltages and reactive power changes in its nodes. Direct optimization of each mode separately, generalization and analysis of economic and mode characteristics of the aggregate of modes dramatically complicates the problem at hand, making it excessively cumbersome and time-consuming. Against this background, the most productive is the gradient methods, proved themselves in operational problems, using the probabilistic-statistical (stochastic) approach, which allows solving problems more strictly than deterministic approaches [1]. A statistical model for accounting the set of steady-state modes [2,3] combined with a generalized reduced gradient (GRG) method [4,5] are the basis of the developed optimization method. The basic principles for solving the problem of optimal RLC in terms of the minimum loss of electric energy (EE) is given below.

Statistical Model of Electrical Loads
The optimal RLC is formulated as a nonlinear mathematical programming problem [4]; it belongs to the stochastic optimization since loads of the ES nodes are random variables. The stochastic approach of taking into account multimoding [2,3], caused by the change in electric loads, is implemented based on an analytical model of load changes which is obtained through factor (component) analysis [6,7]. Electrical loads modelling on the basis of factor analysis allows identifying common and most stable patterns of changing the configuration of load curves, compressing the information regarding the multimoding through a small number of generalizing factors with their subsequent application of them when calculating the EE losses and other integral parameters.
The load diagrams modelling using the principal component analysis is as follows. Based on a representative sample N of initial load diagrams, we determine a moment correlation matrix (MCM) and select М of maximum eigenvalues i  and their corresponding eigenvectors i  . These values determine the main factors -generalized (orthogonal) load diagrams (GLD): The modelling GLDs are a set of statistically independent basis vectors oriented so that each of them reflects most of the connection of the initial aggregate of load diagrams and contributes most to the variance of the initial variables. Like eigenvectors, GLDs are orthogonal (statistically independent), uncorrelated (unrelated) quantities. They attribute the properties of linearity and additivity to the models, the statistical method, and the whole process of multimoding modelling. Analysis of the GLD configurations for various realizations of the random process of changing the loads showed the presence of statistical stability of these factor models, i.e. proximity of the numerical characteristics of the corresponding GLDs of various samples of the original diagrams.
This statistical transformation of MCM helps simulate fairly accurately the original curves of electrical loads ij P and ij Q using known mathematical expectations i MP and i MQ and simulated deviations of loads from mathematical expectations in the form of M-linear combinations of statistically stable GLD: (2) Such a representation of the loads turned out to be effective since obtaining model (2) of acceptable accuracy needs only to take into account up to the first three or four GLDs (М << N), which reflect up to 85-95% of the total variance of the initial loads. The modelling error for the interval values of unknown load curves by models (2), being in the range of ± (2-15) %, is not a crucial factor for determining the integral characteristics since its influence decreases as a result of stepwise (iterative) refinement of pseudo-average loads and, accordingly, load diagrams models in the combined algorithm for determining these parameters.
The solution for optimal RLC is based on taking into account the entire set of modes in the form of their integral characteristics, primarily EE losses, which are a target criterion for solving the operational problem with determining the optimal load of existing reactive power sources.

Statistical modelling of the set of steady-state modes [2, 3]
The general expression of the EE load losses in the ES with m-branches is basically determined by accurate summing (integrating) of power losses Р  at all time intervals t  (in all modes) of calculated period Т according to the classical expressions: The EE load losses are the sum of the main component М determined for the average loads mode, and the variance component which takes into account the deviation of the loads from the average values: -second derivatives of the expression of power losses with respect to the corresponding variables, they have been calculated relevantly to the same point; N -the number of nods in the circuit without a slack bus.
The main component of EE losses is determined by calculating the steady-state mode (SSM) for average loads with high reliability. The greatest difficulty is a complete and simple calculation of the multimoding when calculating the variance component, which is a critical factor in the EE loss analysis in general.
The EE loss expression (4) is featured by the correlation moments of moduli V and phases δ of the voltages, which form the MCM of the voltages, which are obtained on the basis of a system of equations written similarly to the linearized equations of nodal voltages (NVEs): where ΔРi, ΔQi, ΔVi, Δδi -deviations of active, reactive powers, moduli, phases of nodal voltages from their mathematical expectations.
Since the deviations of the voltages and wattages from their mathematical expectations are approximately related by the NVE system (5), the centered random parameters (variations of the voltage phases and moduli) formulated by linear combinations of GLD: After substituting deviation of the mode parameters from expressions (2) and (6) into the system (5), where ki ki   , , ki ki    are determined with initial (reconstructed) load diagrams and GLD according to (2) as: Similar to (9) for the elements, MCMs of wattages are: Considering the correlation moments for moduli and phases of voltages (9), the EE load losses (4) are: Analysis of the accuracy of calculating the EE losses by the statistical testing with 2-4 GLD made for samples of test circuits of power grids with 35, 110, 220 kV, a small number of nods (up to 10), and real-world 6-220 kV circuits of power grids within the Krasnoyarsk energy system with up to 25 nods, resulted in finding that these expressions allow to compute load losses with acceptable accuracy: considering significance level of 0.95 for the samples, the average error in calculating EE losses for test circuits was = -(1,2÷1,7) % with σ2 scattering up to 0.70, and ±(0,25÷0,45) % with σ 2 scattering up to 0.64 for real-world circuits.
EE losses of idle running T N of transformers with T i g conduction are specified in the initial and optimal modes in accordance with the obtained voltage diagrams (6): Linear system of equations in SSM (5), factor transforming (1), load model (2) and relations (6) -(12), expressing deviations of dependent variables (δ, V) and their second moments through the corresponding characteristics of independent variables (P, Q), collectively form a statistical model for the SSM analysis and ES multimolding analysis.
Finding the integral characteristics through the stochastic method (7) -(12) does not require interval calculations of the modes; it comes down to one calculation of the SSM of the electrical system for average loads and an additional solution of three to four systems of linear equations (7) with the implacable Jacobian matrix, which drastically simplifies calculations of multimoding and EE losses in general, in comparison with direct dcalculations of the SSM (3) over the intervals for averaging the electrical load diagrams. The method allows to calculate electric power losses and other integral characteristics with accuracy and reliability sufficient for practice. Moreover, in comparison with the deterministic methods, the tolerance to random errors increases. EE load losses can be calculated via any algorithm computing the SSM which can be supplemented by blocks for determining (7)

Mathematical model for stochastic optimization of modes
The basis of the model is the multimode-based GRG constructing apparatus with the statistical accounting of multimoding [8,9]. While solving the operational problem, we define the minimum of the objective function of the total EE losses (11), (12) under the balance nonlinear equality constraints (NVEs) for the mathematical expectation of mode parameters, and simple inequality constraints max min where G is a number of nods with sources of reactive power (RP).
Еру constraints (13) all be applied to the entire time interval, i.e. for each mode, which should be controlled in two ways: 1) modelling with GLD for RP (2) and voltages (6) at each optimization step and checking compliance with the constraints (13); 2) calculating of design ranges for variation of the considered parameters, which, considering variances (9), (10), are determined by Chebyshev's inequalities: Considering antisymmetric (biased) function of distribution density of Vi, Qi , the values of kβ, ensuring the minimum error of the interval analysis on average up to 5- If simple constraints (13) are violated, the basic set changes. This means exchanging corresponding components between vectors and .
The following separation of variables is proposed for the components of eigenvectors and modeling coefficients: In this case, a basic set change is provided only for the variables reflecting multimoding with the first GLD (М = 1).
Using the matrixed linearized system (5), provided the mode is balanced for the active power (ΔРi = 0), following the illustrated separation of variables (16), with the subsequent grouping of the vectors of the dependent and independent variables, we obtain the system of equations reflecting the parameters of SSM corresponding to the loads expectations: In systems (17), (18), the following matrices are used to relate the dependent and independent mode parameters (20) When inserting dependent variables (Δδ, ΔQ1, ΔV2 and ′ , ″ , ″ ) into (17), (20) through independent variables (ΔV1, ΔQ2 and ″ , ″ ) considering objective function F (11) and component (12) upon transition to infinitesimal increments of the variables, the expression of reduced gradient simulating loads only with the first actively constrained GLD (13), will be:

Multimode-based optimal choice of loading RP sources
An objective function of the total EE losses (11), (12) is determined by the mathematical expectations of the mode parameters, the components of the eigenvectors and The variables found during the optimization allow obtaining: 1. A criterion (objective) function of total EE losses and its components (the value of EE load losses (11) and idle running (12) in the initial and optimal states).
2. Ranges of alteration of the optimized mode parameters (14) taking into account expressions (9) and (10) for variances.
3. Diagrams (curves) of loading of the RP sources (2) and voltages (6) in the system nodes in the given time interval.   (17) for fixed values of the RP sources and the further analysis of the parameters of the basic steadystate mode with subsequent verification of interval constraints (13). If the resulting voltages do not satisfy the controlled constraints (13), it is necessary to obtain new values of the controlled variables (RP sources) by decreasing  step or by means of fixing the violated limits on the limit values (change of basis), and then redetermine the dependent variables.
6. Controls of the decline of the objective function at (k + 1) iterative step and the fulfillment of the criteria for the termination of the optimization search.
The calculation cycles for 2-6 are iterated until the optimum condition is satisfied i.e. the minimum of the objective function of the total EE losses.
Software for optimization algorithms. These algorithms form the basis of ORESА stochastic optimization software [10] based on algorithms and OPRES instant mode optimization software [11]. ORESА is intended for optimal distributing of reactive loads of existing RP sources over a time interval according to the criterion of minimum EE losses and aims to solve the problems of ES optimal functioning in various mode planning cycles.

Conclusions
We implemented the proposed modification of the generalized reduced gradient method for stochastic modeling of multimode-based electrical systems. The program provides ranges and diagrams of loadings for RP and voltages changes of RP sources and other ES nodes, EE losses in the initial and optimal states avoiding analyzing and optimizing of the modes at each load stationarity interval. The accuracy of ORESA was assessed via statistical tests as a result of direct reproduction of the totality of typical optimal modes on a variety of circuits of 35-220 kV electrical grids and systems: the accuracy achieved for solving this particular operational problem was sufficient for actual practices.