Impedance in load flow studies?
- Using admittance matrix over impedance matrix has many advantages. But if I have to summarize in one phrase it would be "Reduced Memory Requirement"
- The thing you need to understand is the load flow studies are done for a large and widely interconnected power systems. It may contains hundreds, thousands and may be millions of power system components (e.g. International Grid) . It makes sense to use admittance matrix for such humongous power systems.
- Now, How does it saves memory ?
- Consider a 5 bus admittance matrix like this:
- Image Source: Electrical Power Systems, CL Wadhwa
- Did you notice many elements are zero in the above matrix ?
- Now imagine a power system with 1000s of buses. How many elements will be zero there ? Well that depends upon the interconnection of buses.
- For example if bus no 1 is not connected to bus 100 , the element a[1][100] will be zero and so will be the element a[100][1].
- You see the beauty of admittance matrix, we get another zero with every zero. So we have twice as number of zeros as non-interconnected buses.
- And the thing is you needn't store zeros in the matrix thus saving a lot of memory.
- Another thing is that Admittance matrix is symmetrical about leading diagonal as mentioned above a[1][100]=a[100][1]. So all we need to do is just store the elements of upper diagonal and the elements below the diagonal can be copied from there.
- So since the memory requirement is reduced. The time complexity of the program doing the load analysis will reduce , for a hypothetical example let's say from O(n^2) to O(n*logn). Program will be work better and faster giving the results in real time. These results will be further used in deciding the state of system (which demands fast and accurate results, that's why PMUs cable of DFT/FFTs are used), calculating instantaneous power,phase angles and voltage magnitudes.
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| impedance representation |
Advantages
- Apart from above major advantage, I found following advantages which makes Load flow studies a cakewalk (at-least in theory ):
- Matrix is really easy to form once you get hold of the algorithm.
- With any type of changes in network, the elements can be changed without actually changing the whole matrix.
- For Medium/Long lines , the shunt elements can be easily added.
- Zbus can simply be obtained once you have Y-bus matrix.
Electrical impedance
Scientists
Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied.
In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current (AC) circuit. Impedance extends the concept of resistance to AC circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude. When a circuit is driven with direct current (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.
It is necessary to introduce the concept of impedance in AC circuits because there are two additional impeding mechanisms to be taken into account besides the normal resistance of DC circuits: the induction of voltages in conductors self-induced by the magnetic fields of currents (inductance), and the electrostatic storage of charge induced by voltages between conductors (capacitance). The impedance caused by these two effects is collectively referred to as reactance and forms the imaginary part of complex impedance whereas resistance forms the real part.
The symbol for impedance is usually Z and it may be represented by writing its magnitude and phase in the form |Z|∠θ. However, cartesian complex number representation is often more powerful for circuit analysis purposes.
The term impedance was coined by Oliver Heaviside in July 1886. Arthur Kennelly was the first to represent impedance with complex numbers in 1893.
Impedance is defined as the frequency domain ratio of the voltage to the current. In other words, it is the voltage–current ratio for a single complex exponential at a particular frequency ω. In general, impedance will be a complex number, with the same units as resistance, for which the SI unit is the ohm (Ω). For a sinusoidal current or voltage input, the polar form of the complex impedance relates the amplitude and phase of the voltage and current. In particular:
The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude.
The phase of the complex impedance is the phase shift by which the current lags the voltage.
The reciprocal of impedance is admittance (i.e., admittance is the current-to-voltage ratio, and it conventionally carries units of siemens, formerly called mhos).
Measurement
The measurement of the impedance of devices and transmission lines is a practical problem in radio technology and other fields. Measurements of impedance may be carried out at one frequency, or the variation of device impedance over a range of frequencies may be of interest. The impedance may be measured or displayed directly in ohms, or other values related to impedance may be displayed; for example, in a radio antenna, the standing wave ratio or reflection coefficient may be more useful than the impedance alone. The measurement of impedance requires the measurement of the magnitude of voltage and current, and the phase difference between them. Impedance is often measured by "bridge" methods, similar to the direct-current Wheatstone bridge; a calibrated reference impedance is adjusted to balance off the effect of the impedance of the device under test. Impedance measurement in power electronic devices may require simultaneous measurement and provision of power to the operating device.
The impedance of a device can be calculated by complex division of the voltage and current. The impedance of the device can be calculated by applying a sinusoidal voltage to the device in series with a resistor, and measuring the voltage across the resistor and across the device. Performing this measurement by sweeping the frequencies of the applied signal provides the impedance phase and magnitude.
The use of an impulse response may be used in combination with the fast Fourier transform (FFT) to rapidly measure the electrical impedance of various electrical devices.
The LCR meter (Inductance (L), Capacitance (C), and Resistance (R)) is a device commonly used to measure the inductance, resistance and capacitance of a component; from these values, the impedance at any frequency can be calculated.
Variable impedance
In general, neither impedance nor admittance can be time varying as they are defined for complex exponentials for –∞ < t < +∞. If the complex exponential voltage–current ratio changes over time or amplitude, the circuit element cannot be described using the frequency domain. However, many systems (e.g., varicaps that are used in radio tuners) may exhibit non-linear or time-varying voltage–current ratios that appear to be linear time-invariant (LTI) for small signals over small observation windows; hence, they can be roughly described as having a time-varying impedance. That is, this description is an approximation; over large signal swings or observation windows, the voltage–current relationship is non-LTI and cannot be described by impedance.
See also
Bioelectrical impedance analysis
Characteristic impedance
Electrical characteristics of dynamic loudspeakers
High impedance
Immittance
Impedance bridging
Impedance cardiography
Impedance matching
Impedance microbiology
Negative impedance converter
Resistance distance
Impedance control
Courtesy of wikipedia....
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