Хмельник Соломон Ицкович: другие произведения.

Functional for Power Systems

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  • Аннотация:
    The variational optimum principle for electric power systems, which are in essence non-linear electric circuits, is formulated and proven. It is shown that the resulting functional is optimized when the stationary value of the integrand is the equation of power system regime. Based on this principle, a universal method of solution for various problems of power systems analysis and control is presented. There are numerous examples, and some program in MATLAB.

  
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  Contents
  
  Preface \ 5
  1. About Convex Functionals \ 8
  1.1. Gradient Descent Along a Convex Functional \ 8
  1.2. Sufficient Conditions of Convex Functional"s Optimum \ 10
  2. Power in Sinusoidal Current Electric Circuits \ 12
  2.1. Introduction \ 12
  2.2. The Function Lnw \ 12
  2.3. The Function Lnwi \ 14
  2.4. The Functions Weg, Wegi \ 15
  2.5. The Functions Msg \ 16
  3. Nonlinear Alternate Current Electric Circuit \ 17
  4. Unconditional Nonlinear Electric Circuit \ 18
  4.1. The First Functional \ 18
  4.2. The Second Functional \ 18
  5. Computing Algorithm for Unconditional Nonlinear Electric Circuit \ 20
  5.1. The First Functional \ 20
  5.2. The Second Functional\ 21
  6. Generalized Functional for Unconditional Nonlinear Electric Circuit \ 24
  7. Generalized Functional for Unconditional Nonlinear Electric Circuit with sinusoidal currents \ 27
  7.1. The Algorithm \ 27
  7.2. Nonlinear equations \ 30
  7.3. Flux-distribution \ 34
  8. Sufficient Optimum Conditions for Generalized Functional of Unconditional Nonlinear Electric Circuit \ 45
  9. Nonlinear Underdetermined Unconditional Electric Circuit \ 48
  9.1. The First Functional \ 48
  9.2. The second Functional \ 49
  9.3. Generalized Functional \ 50
  9.4. Generalized Functional for Unconditional Nonlinear Underdetermined Electric Circuit with Sinusoidal Currents \ 54
  9.5. Sufficient Optimum Conditions for Generalized Functional of Unconditional Nonlinear Underdetermined Electric Circuit \ 56
  10. Analysis and Control Problems in Power Systems \ 58
  1. Flux-distribution Calculation in a Power System \ 58
  2. The Minimization of Nodal Generated Powers Deviation from their Plan Targets. The First Method \ 59
  3. Minimization of Nodal Generated Powers Deviation from their Plan Targets. The Second Method \ 44
  4. Cost-effective Flux-distribution - Minimization of Full Generation Cost with a Given Total Generation. The First Method \ 59
  5. Cost-effective Flux-distribution - Minimization of Full Generation Cost with a Given Total Generation. The Second Method \ 61
  6. The State Value \ 61
  7. Reactive Power Minimization \ 61
  8. About Total Cost Minimization \ 62
  9. A search for an admissible regime of a power system \ 63
  11. Nonlinear Unconditional Electric Circuit with Fixed Voltage Modules \ 65
  11.1. The First Functional \ 65
  11.2. The Second Functional \ 67
  11.3. The generalized functional \ 68
  11.4. The generalized functional for sinusoidal currents electric circuits \ 70
  12. Nonlinear unconditional electric circuit with fixed current modules \ 72
  12.1. The First Functional \ 72
  12.2. The Second Functional \ 74
  12.3. The generalized functional \ 75
  12.4. The generalized functional for sinusoidal currents electric circuits \ 77
  13. Nonlinear unconditional electric circuit with powers, voltages and currents sources \ 79
  13.1. Generalizations \ 79
  13.2. An algorithm of sinusoidal currents electric circuit design \ 80
  13.3. Control function \ 82
  13.4. A Search for an Admissible Regime of the Power System \ 82
  Reference \ 84
  
  Preface
  
  In [1] it is shown that in direct current electric circuit containing direct power sources, a certain quadratic function of currents is being minimized. In [2] it is shown that in alternate current electric circuit a certain quadratic functional of a function of electric charges is being optimized. Moreover, such a functional exists for every linear electromechanical system. In the present book we proceed to generalize these results, and show that in a power system with power sources (which in essence is a nonlinear electric circuit) also a certain quadratic functional of a function of charges is being optimized. There are no constraints, because they are also included in the quadratic functional. The stationary value of the charges function is the equation of the power system regarded as a nonlinear electric circuit. Thus, the calculation (design) of a given power system may be formulated mathematically as a variational problem of seeking an unconditional optimum of a quadratic functional. This problem always has a solution. This result may be used for the development of universal software system for high-speed power systems design. The universality is due to the fact that the calculations methods are unified and do not depend on * configuration and composition of the active and passive elements, * the types of functions of the power sources and customers.
  
  In this way we achieve
  1. high speed of operation provided by the fact that:
  * the functional is quadratic and has only one global optimum,
  * the second Kirchhoff's law equations are excluded from the problem's constraints, thus reducing its size,
  * the functional has an unconditional optimum;
  2. the existence of a comparatively simple algorithm even for a complex mathematical statement, for example, for design of a power system with complex configuration and aperiodical discontinuous perturbation actions;
  3. reliable search procedure (the iteration process always converges);
  4. iteration process convergence even with incompatible input data (a regime is found, which in a certain sense is closest to the input data by its parameters);
  5. possibility of general compound electromechanical systems design, for instance, of electric circuits with motors, generators, hydraulic transducers, etc.).
   The obtained results are used to develop algorithms for various analysis and control problems in the power systems. In particular, these are such problems as:
  1. Flux-distribution calculation in the power system in traditional formulation.
  2. The minimization of nodal generated capacities deviation from their plan targets. This problem is equivalent to the flux-distribution calculation with known nodal capacities, in the case when the nodal capacities are incompatible for some reasons.
  3. Cost-effective flux-distribution. This problem is equivalent to flux-distribution calculation with unknown nodal capacities and a given total generation with simultaneous minimization of the total generation cost.
  4. The state value. This problem is equivalent to calculating a regime according to the results of imperfect measurements.
  5. Reactive power minimization. This problem is equivalent to the flux-distribution calculation with indeterminate reactive powers.
  6. A search for an admissible regime of a power system.
   In the presented problems a nonlinear electric circuit of a power system is a model for a convex programming problem. The existence of a global optimum permits to use the gradient descent method for this circuit design - namely, the calculation of the currents and potentials. Besides, this circuit may be modified to an unconditional circuit, thus constituting a model of a convex programming problem without constraints - i. e. an unconditional convex programming problem. By choosing an appropriate value of a certain parameter of the unconditional electric circuit we may get the calculation parameters of the basic circuit to be arbitrary close to those of the unconditional circuit. There is an inverse negative relationship between the accuracy and the problem solution time. In practice it means that the dispatcher may quickly look through the approximate optimization variants (varying the set points), and then make a more accurate calculation of the chosen variant.
  
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