Prompt determination of the static stability margins in electrical energy systems equipped with distributed generation plants

. The wide use of distributed generation (DG) technologies in electrical energy systems (EES) requires development of new control algorithms in normal, emergency and postemergency modes. The issues of determining the static aperiodic stability (SAS) margin in EES equipped with DG plants are of particular relevance. These plants can be removed from the consumption centers, which can lead to SAS reserves depletion. The article presents the results of studies aimed at the developing SAS reserves express calculation methods in EES equipped with DG plants. An effective technique to determine stability margins was proposed based on one of the modifications of limiting modes equations. The results of determining SAS for an electrical network with distributed generation plants are presented. Additionally, the simulation of transient processes in the studied EES for various points in the space of the mode's controlled parameters has been carried out in the Matlab system.


Introduction
The issues of determining limit load modes and static aperiodic stability (SAS) margins [1][2][3] are relevant when designing and operating electrical energy systems (EES) and have both independent significance and are an integral part of other electrical energy issues related to providing a required level of EES reliability [4].
The issues of determining SAS reserves are of particular relevance in power supply systems equipped with distributed generation (DG) plants, in particular, implemented on the basis of non-traditional renewable energy sources [5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Such plants, as mini hydropower plants and offshore windmill farms can be in a remote location from consumption centers, which leads to SAS reserves depletion [19,20]. This work considers the technique for situational determining static stability margin in electrical networks with DG plants, in which case limit loads equations are used in the above technique. Situational calculations of stability margins for three-node equivalent circuit with two DG plants and EES connection are provided. Additionally, transient processes simulation is performed.

Problem formulation
Static stability margin can be determined as Euclidian norm of K vector [3]: In such problem formulation the static stability margin is a distance (in the metric set by the coefficients i μ ) from the point 0 Y to hypersurface F L ( Figure 1). Each loading direction i Y Δ will correspond to its value i  and the search of loading critical direction is required for reliable assessment of stability margin Fig. 1. With respect to stability margin determining.

Technique for operational determining SAS reserves based on limit loads equations
The problem of assessing the stability margin in the loading critical direction can be formulated as follows: determine under constraints where Formally, in addition to condition (2), it would be necessary to introduce the constraint 0 det    X F . However, as it will be shown below, this constraint has already been set in equations (1) and (2).
To solve the formulated problem (assuming that the limits of stability and the transmitted power coincide), the Lagrange function is written This system has two solutions: 1. A trivial one, when it satisfies the initial mode with parameters 0 X , 0 Y , when 2. Required one, when at least one of the vectors components  and Y D is not equal to 0. In this case, the equation Hence, such a solution satisfies the limiting loads hypersurface Geometrical interpretation of weighting critical direction search.

Equations (3) can be represented as
Since vector  is determined the accuracy of the multiplier, variables replacement can be made 4 E3S Web of Conferences 224, 02004 (2020) TPACEE-2020 Having determined from the first equation and, by substituting into the third equation of system (6), one can obtain a system that is LLE modification designed to search for the limiting mode in the critical direction of loading: If vector components Y D are introduced to the first group of equations (6) linearly, then This takes place when SSE recorded in Cartesian coordinates system can be represented as: where Pi0, Qi0power injections in the initial mode; To solve equations (8), one can use Newton's method; in this case, at each iteration, the following system of linear equations (SLE) is solved: System (8) can be written as a single vector equation This equation will satisfy two solutions: it is possible to reach the points of local extreme of this function. An effective way to overcome the difficulties associated with the presence of a trivial solution to equations (8) can be implemented by including in these equations a condition requiring that R vector at the solution locus should not be not zero [4]. This can be done by introducing an indicator variable α , by representing system (8) as: where In an abridged form, the last system can be written as follows The value of the stability margin  is determined after the end of the iteration process using the formula Determination of the SAS reserves based on the above described approach was carried out in relation to the network [19,20] diagram, the general layout and diagram of which are shown in Figure 3. The DG plants rated powers are assumed equal to 24 MW. Simulation results are provided in Figure 4 and in Table 1.   The results obtained allow us to conclude that an efficient technique for analyzing static aperiodic stability in an EES equipped with DG plants can be implemented on the LLE basis, which is applicable in design problems, as well as operational and emergency control (OC and EC). The nondegeneracy of the LLE Jacobi matrix at the solution locus [4] ensures reliability of results obtaining, which is very important for OC and EC problems solution.

Determining stability margins based on limiting modes equations with increased non-linearity
Parameters of the limiting mode in critical (the most dangerous) loading direction cr Y Δ can be found from solution of the following equations system [4]: However, when using these equations, significant computational difficulties arise due to the presence of a trivial solution 0 X X  , corresponding to the initial mode. To overcome this difficulty, an additional variable and the equation can be introduced into system (15) where c is a variable that determines the stability margin value.
Simulation results are provided in Figure 5 and in Table 2.   Thus, based on equations (16), an alternative approach to solving the problem of SAS reserves operational determining can be implemented.

Transient processes simulation results
Two mini-hydroelectric power plants with a rated power of 24 MW each, operating from the synchronous generators, are used as DG plants in the system under study. The operating modes of the network under study when an additional load is connected at the EES substation, as well as in emergency mode, for which lines 1-3 are exemplified, imply the implementation of power delivery from each generator to the receiving system (node 3 in Figure 3). The diagram of the network model under study, developed using the Simulink and SimPowerSystems simulation packages, is provided in Figure 6. The model used the standard units of the specified Matlab system modeling packages. In this case, thyristor excitation systems of synchronous generators (Excitation System1 and Excitation System2 units) were simulated by the first-order aperiodic link with transfer function

. The
Hydraulic Turbine unit in Figure 6 consists of the main servomotor with the proportionalplus-integral controller, speed limiter and a hydraulic turbine itself. The mathematical model of the main servomotor with the controller is represented by the following transfer function: Power lines (Line units in Figure 6) are simulated by a series-connected RL circuit, the resistances of which are shown in Figure 3. Thus, the dynamic modeling of the electrical network under study with DG plants confirms the stability regions which are calculated using the limit modes equations.

Conclusions
The following conclusions can be formulated based on the performed theoretical analysis and computer-aided modeling: 1. An effective technique is proposed for the operational determining of stability margins regions in an EES equipped with DG plants. The nondegeneracy of the LLE Jacobi matrix of limit loads equations at the solution loci ensures the reliability of results obtaining, which is very important in solving the issues of operational and, especially, emergency control. 2. The results of determining SAS reserves for an electrical network with distributed generation plants are presented. The computer modeling results confirm the correctness of EES stability margins calculation using the proposed technique.
3. An alternative approach to solving the issue of SAS reserves determining can be implemented based on the limiting modes equations of with increased non-linearity.