Chapter 8
Models for Probabilistic Network Analysis
P. Caramia, P. Verde, P. Varilone and G. Carpinelli
In this chapter, some models for the probabilistic network analysis of transmission and distribution power systems in presence of harmonics are dealt with. The models considered derive from the classical models for deterministic analysis: direct method, Harmonic Power Flow and Iterative Harmonic Analysis. The chapter is organized so that for each method the theoretical aspects are presented first, then numerical examples on transmission and distribution test power systems follow to illustrate the methods. All models described assume deterministic values for network impedances.
8.1 Introduction
The probabilistic harmonic modeling presented in the previous Sections are concerned with the characterization of injected current harmonics at a given busbar by one or several non-linear loads.
In recent International Standards, however, the interest is devoted not only to current but also to voltage harmonic probability density functions, and in particular to their percentiles and maximum value evaluation [1],[2]: the IEC 1000-3-6 in the assessment procedure refers to 95% probability daily value and to maximum weekly value of voltage and current harmonics while in the European Standard EN 50160 the 95% probability weekly value should not exceed the specified limit. Moreover, probabilistic aspects are also included in the recent draft revision of the IEEE 519 [3],[4]. On the other hand, knowledge of the voltage probability density functions is mandatory when the problem of the effects of distortions on electrical components is dealt with [5], [6].
The probabilistic network analysis of transmission and distribution power systems in presence of harmonics can be effected:
by direct methods, in which the probabilistic evaluation of the voltage harmonics at system busbars is effected neglecting the interactions between the supply voltage distortion and the non linear load current harmonics;
by integrated methods, in which the probabilistic evaluation of the voltage and current harmonics are calculated together, properly taking into account the above mentioned interactions. Only two deterministic integrated methods have been extended in the probabilistic field: the harmonic power flow and the iterative harmonic analysis.
8.2 Probabilistic direct method for network analysis
In the deterministic field, the Direct Method for network analysis consists in two steps [7]. The method starts with the evaluation of the converter current waveform assuming that the voltage at converter terminal is an ideal pure sinusoid; the waveforms are obtained by time domain simulation or with closed form relations. Then, the Fourier analysis is effected and the current harmonics are obtained. This process is repeated for all converters present in the system.
The second step consists in the evaluation of the voltage harmonics at system busbars; the following network harmonic relations are applied:
,
Equation 8.1
where is the (i,j)-term of the harmonic impedance matrix and are the current harmonic of order h injected into the network at bus j and the voltage harmonic of the same order at the same bus, respectively. The current harmonic can be due to the presence of either one or more converters at the same bus j.
The Equation 8.1 are N sum of N phasors and they correspond to the following 2N sum of 2N scalar quantities:
Equation 8.2
being , and .
In the probabilistic field, the input data and output variables of Direct Methods are random variables and therefore they must be characterized by probability functions, in particular probability density functions (pdf’s), either joint or marginal.
The Probabilistic Direct Methods require two steps to be performed.
The pdf’s of the current harmonics injected into the network busbars are evaluated first, for fixed pdf’s of the input data and employing proper harmonic current models (Figure 8.1). The voltage harmonic pdf’s at system busses are then evaluated by assuming the harmonic current pdf’s of step 1 as input data and using the relations (1) as harmonic voltage model.
Figure 8.1. Probabilistic Direct Method
8.2.1 Theoretical Aspects
The harmonic current models and the probabilistic techniques to perform the evaluation of the current harmonics pdf’s (step 1, Figure 8.1) are shown in details in the previous Sections and in [8], which deal with the representation of a random harmonic phasor (only one converter is present at the same bus) or of a sum of random harmonic phasors (more converters are present at the same bus). Generally, the rectangular components or x-y proiections are chosen for phasor representation, because of the convenience they offer when adding phasors; in such cases, once the pdf’s of sum of x-y proiections are known, the pdf of magnitude of sum is obtained. To obtain pdf’s, analytical expressions, approximate solutions, the convolution approach, the Monte Carlo simulation or the Central Limit Theorem are applied.
With reference to the evaluation of the voltage harmonic pdf’s (step 2, Figure 8.1), the analysis of Equation 8.1 clearly shows that each voltage harmonic phasor of order h at system busbars can be obtained as a sum of random phasors, each one being the product of a harmonic current phasor times a harmonic impedance matrix term. In practice, to obtain the voltage harmonic pdf’s, the evaluation of the pdf’s of a sum of phasors must be effected once again, as in the case of current harmonic evaluation. Hence, the probabilistic techniques of the first step (analytical expressions, approximate solutions, the Monte Carlo simulation, the Central Limit Theorem and so on) can be utilized to derive the voltage harmonic pdf’s too.
8.2.2 Numerical Applications
For illustration purposes, follows the probabilistic Direct Method applied to the 18-bus distribution test system of Figure 8.2 [9]. The values of electrical parameters of system components are reported in [10].
The aim is the evaluation of the 5th harmonic voltage probability density functions caused by the presence of one or more static converters at different busbars.
Figure 8.2. 18-bus distribution test system [9]
The following cases will be presented:
1 static converter at bus 14;
8 static converters of the same power at different busbars;
8 static converters at different busbars with one of them of dominant power.
In all cases, for the evaluation of the pdf’s of the fifth harmonic current amplitude and argument we used the following very simple ac/dc six pulse static converter model [11]:
Equation 8.3
being V the RMS of the converter voltage, a the converter firing angle, R the dc converter resistance.
Assuming the firing angle as the only input random variable [11] with an uniform pdf, the fifth harmonic amplitude pdf is shown in Figure 8.3 a)[1]; the pdf of the harmonic current argument is a uniform pdf, as clearly indicated by Equation 3.
(a) (b)
Figure 8.3. 5th harmonic current (a) and voltage (b)amplitude probability density functions
(a) (b) (c)
Figure 8.4. 5th harmonic voltage probability density functions without dominant converters: real component (a), imaginary component (b), amplitude (c)
(a) (b) (c)
Figure 8.5. 5th harmonic voltage probability density functions with a dominant converter: real component (a), imaginary component (b), amplitude (c)
With reference to Case (i), in Figure 8.3b) the pdf’s of the fifth harmonic voltage amplitude at busses j = 1, j = 11 and j = 14 are shown.
From the analysis of Figure 8.3 b) it can be deduced that the mean values and variances are increasing when the converter bus 14 is approached. This result could be forecasted analytically by recalling that the harmonic impedance matrix term modulus increases as the converter bus is approached with obvious consequences on mean values and variances.
With reference to Case (ii), in Figure 8.4 the pdf’s of the real component, imaginary component and amplitude of the 5th harmonic voltage at busbars j = 1 and j = 14 are shown, assuming that eight converters are placed at busbars 3, 4, 7, 10, 13, 14, 15 and 16.
From the analysis of Figure 8.4 it clearly appears that no contribution to real or imaginary components of the harmonic voltage is dominant so that these components tend to approach a normal distribution, as suggested by the Central Limit Theorem application. This is confirmed by the inspection of all the pdf’s (not shown in the figures) of terms of the sum on the right hand of Equation 8.2; for example, the pdf’s of the eight terms on the right hand of the real or imaginary component of the voltage phasor at busbar 14 have similar minimum and maximum values.
With reference to Case (iii), in Figure 8.5 the pdf’s of the real component, imaginary component and amplitude of the 5th harmonic voltage at busbars j = 1 and j = 14 are shown, assuming that one power prevailing MW converter is placed at bus 14 and seven lower power converters are placed at busbars 3, 4, 7, 10, 13, 15 and 16.
From the analysis of Figure 8.5 it clearly appears that one contribution to real or imaginary components of the harmonic voltage is dominant and, then, the real and imaginary components of the voltage harmonic are far from the normal distribution. This is confirmed by the inspection of all the pdf’s of terms of the sum on the right hand of Equation 8.2: for example, the pdf’s of 7 terms on the right hand of the real or imaginary component of the voltage amplitude at busbar 14 have similar minimum and maximum values, all included in the interval [-0.15, 0.1], while the eighth term has minimum and maximum values included in the interval [-1.5, 1].
8.3 Probabilistic Harmonic Power Flow
The Probabilistic Harmonic Power Flow (PHPF) represents the natural extension to the probabilistic field of the Harmonic Power Flow firstly proposed by Heydth et alii [12].
8.3.1 Theoretical Aspects
The Harmonic Power Flow model here considered is expressed by the following equations:
Equation 8.4
Equation 8.5
Equation 8.6
Equation 8.7
Equation 8.8
,
Equation 8.9
where in the case of probabilistic harmonic power flow:
input random vectors of active and reactive powers specified at fundamental for each linear load busbar,
input random vectors of total active and apparent powers specified for each non linear load busbar,
input random vector of active power specified at fundamental for each generator busbar without the slack,
input random vector of voltage magnitude at fundamental specified for each generator busbar,
U, F, X input random vectors of voltage magnitude and argument at all harmonics and at fundamental, of angles of commutation and of the auxiliary parameters ai and bi,
U1, F1 input random vectors of voltage magnitude and argument at the fundamental.
The Equation 8.4 and Equation 8.5 represent the power balance equations (active, reactive and apparent) at linear and non linear load busbars, the Equation 8.6 represent the active power and voltage regulation balance equations at generator busbars, the Equation 8.7 represent the harmonic current balance equations at linear load busbars, the Equation 8.8 represent the harmonic and fundamental current balance equations at non linear load busbars and, finally, the Equation 8.9 represent the commutation angle equations at non linear load busbars.
From Equation 8.4 to Equation 8.9 it was assumed that in the converter busbars the firing angle ai and the dc load auxiliary parameter bi (i.e. the resistance R in case of dc passive load and the voltage E in case of active load) are unknown and that total active and apparent powers are specified[2].
From Equation 8.4 to Equation 8.9 can be expressed in a compact form as:
Equation 8.10
where Tb is the input random vector including active, reactive and apparent powers, total or at fundamental, and N is the state random vector including amplitude and arguments of voltages, at fundamental and at harmonic orders.
Several probabilistic techniques have been applied to the non linear system of Equation 8.10 in order to obtain the pdf’s of the state random vector starting from the knowledge of the pdf’s of the input random vector [13]-[16]. They are:
· Non Linear Monte Carlo simulation;
· Linear Monte Carlo simulation;
· Convolution Process approach;
· Approximate Distribution approach.
8.3.1.1 Non linear Monte Carlo Simulation
The Monte Carlo (MC) procedure requires the knowledge of probability density functions (pdf’s) of the input variables; for each random input datum, a value is generated according to its proper probability density function.
According to the generated input values, the operating steady-state conditions of the harmonic power flow system are evaluated solving the non-linear system of Equation 8.10 by means of an iterative numerical method. Once the convergence is achieved, the state of the multiconvertor power system is completely known and the values of all the variables of interest are stored.
The preceding procedure is repeated a sufficient number L of times to obtain a good estimate of the probability density functions of the output variables according to stated accuracy [17]-[20]. A useful stopping criterion to be adopted can be based on the use of a coefficient of variation tolerance, as proposed in [21].
8.3.1.2 Linear Monte Carlo Simulation
If the vector m(Tb) is the expected values of Tb and a deterministic harmonic power flow is calculated using m (Tb) as input data, the solution of Equation 8.10 will give the state vector N0 such that:
gS(N0) = m( Tb).
Equation 8.11
Linearizing Equation 8.10 around the point N0 and recalling Equation 8.11, it results:
N @ N0 + A D Tb = N’0 + A Tb
Equation 8.12
where:
D Tb = Tb- m ( Tb)
N’0 = N0 - A m ( Tb),
being the matrix A the inverse of the Jacobian matrix evaluated in N0.
The linear Equation 8.12 can be included in a Monte Carlo simulation to obtain approximate probability density functions of the output N vector components starting from the pdf’s of the input Tb vector components.
It should be noted that the use of the linear MC simulation allows a drastic reduction of computational efforts in comparison to non linear MC.
However, it should also be noted that since the harmonic power flow equations are linearized around an expected value region, any movement away from this region produces an error. The errors can grow with the variance of the input random vector components, with entity linked to the non linear behavior of the equation system.