Method of symmetrical co-ordinates applied to the solution of polyphase networks

Abstract

OF PAPER In the introduction a general discussion of unsymmetrical systems of co-planar vectors leads to the conclusion that they may be represented by symmetrical systems of the same number of vectors, the number of symmetrical systems required to define the given system being equal to its degrees of freedom. A few trigonometrical theorems which are to be used in the paper are called to mind. The paper is subdivided into three parts, an abstract of which follows. It is recommended that only that part of Part I up to formula (33) and the portion dealing with star-delta transformations be read before proceeding with Part II. Part I deals with the resolution of unsymmetrical groups of numbers into symmetrical groups. These numbers may represent rotating vectors of systems of operators. A new operator termed the sequence operator is introduced which simplifies the manipulation. Formulas are derived for three-phase circuits. Star-delta transformations for symmetrical coordinates are given and expressions for power deduced. A short discussion of harmonics in three-phase systems is given. Part II deals with the practical application of this method to symmetrical rotating machines operating on unsymmetrical circuits. General formulas are derived and such special cases, as the single-phase induction motor, synchronous motor-generator, phase converters of various types, are discussed.

Operation of a hybrid modular multilevel converter during grid voltage unbalance

“…We consider a system where the grid voltage is composed of positive and negative sequence voltages during unbalance. The unbalance factor (β) represents the ratio of the amplitude of the negative to the positive sequence voltages in the circuit [32] [33]. It is considered that the converter is controlled to inject negative sequence voltage proportionate to that in the grid to avoid the circulation of negative sequence current.…”

A Comparative Study of Ground Fault Analysis for a Practical Case of a Transmission Line Equipped with Different Series FACTS Devices

“…The method was originally presented by C.L. Fortescue in 1918 and has been popular ever since [41][42]. Figure 5 shows the equivalent sequence circuits of a transmission line AB whose impedance is Z AB .…”

“…In order to obtain the phase voltages at the fault point in the presence of FACTS device and fault resistance, the following equation is used, [41][42]:…”

Four-Wire Delta Service Sinusoidal Operation and Compensation Simulator

“…The positive, negative and zero-sequence components of the primary winding currents ( ̅ , ̅ , ̅ ) are obtained from Fortescue's theorem [14]. There are no zero-sequence components in the primary winding currents ( ̅ 0), and the same occurs with the set of secondary currents ( ̅ ̅ , ̅ , ̅ ), whose symmetrical components ( ̅ , ̅ , ̅ ) are the primary symmetrical components multiplied by the transformation ratio ( ), according to Equation (8); thus, ̅ 0.…”