Probabilistic Power Flow

Prof. Alberto Berizzi and Duong LeDeparment of EnergyPolitecnico di Milano

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1. WHY PROBABILISTIC POWER FLOW?

• The conventional deterministic power flow (PF) is the most widely used tool in power systemanalysis, operations, planning, and control.• Deterministic PF uses the specified values of power generation and load, and the parameters of

the network topology to compute system steady-state operating conditions without taking intoaccount any sources of uncertainty affecting the power system.• Today, the deepening penetration of renewable energy sources, such as wind and photovoltaic

solar, has introduced additional uncertainty into various areas of power system. This addeduncertainty, together with the conventional sources of uncertainty such as the loads and theavailability of resources and transmission assets, makes clear the limitations of the conventionaldeterministic PF in analysis and computation of power system. Consequently, the deterministicPF is no more sufficient for power system analysis and computation.• A probabilistic approach needs to be used, where sources of uncertainty are explicitly repre-

sented by random variables. This approach is referred to as Probabilistic Power Flow (PPF).

2. MATHEMATICAL BACKGROUND

2.1. Probability of stochastic events

In probability theory and statistics, a stochastic event is an event that may or may not occur. Prob-ability is a measure for the possibility of occurrence of stochastic events. The probability of event Ais usually denoted as PA:

0 ≤ PA ≤ 1 (1)

When PA = p, the stochastic event A is likely to occur with the probability p.

In particular, if:

• PA = 1: occurrence of event A is certain;• PA = 0: non-occurrence of event A is certain.

The set of all possible outcomes of a random phenomenon is called sample space, denoted as Ω.

The sum of probabilities of all possible outcomes of an event is equal to 1: PΩ = 1.

Example 1: Tossing a coin, there are two possible outcomes, i.e., head (H) and tail (T ), in thesample space: Ω = H,T. If the coin is balanced, the probabilities for getting either head or tailwill be: PH = PT = 0.5.

2.2. Random variable and its distribution

A random variable (r.v.) is a function that assigns a number to each point in its sample space. Thereare basically two types of r.v.s: continuous and discrete r.v.s.

Generally, if a r.v. can take on any value in the sample space, it is called a continuous r.v.; other-wise, called a discrete r.v.

For every real number x, the cumulative distribution function (c.d.f.) FX(x) of a r.v. X is givenby:

FX(x) = PX ≤ x (2)

and if X is continuous, FX(x) can be defined in terms of its probability density function (p.d.f.)

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fX(x) as follows:

FX(x) =

x∫−∞

fX(x)dx (3)

If X is discrete with possible values xi, its c.d.f. is defined as:

FX(x) =∑xi≤x

pi (4)

where pi is the probability corresponding to the possible value xi: pi = PX = xi.

It is noted that, in probability theory and statistics, probability mass function (p.m.f.) is usuallyused for discrete r.v.s instead of p.d.f .

The p.m.f. is:

fX(x) =

PX = xi if x = xi0 if x 6= xi

(5)

Example 2: In power systems, load can vary and get any value between its minimum and maximumvalues so that it can be modeled as a continuous r.v. Similarly, nodal voltages and power flowsare also continuous r.v.s. On the contrary, components in power system such as transmission lines,transformers, and generators have random phenomena with two, i.e., running and outage, or morestates, so that their operating states can be modeled as discrete r.v.s.

2.3. Characteristic function

For a r.v. X with the c.d.f. FX(·), its characteristic function ψX(t) is given as follows [1]:

ψX(t) = E(ejtX) =

+∞∫−∞

ejtxdFX(x) (6)

where t is a real variable, j is the imaginary unit, and E(·) is the mathematical expectation operator.

2.4. Moments

Given a continuous r.v. X , the rth (r ∈ 1, 2, 3, . . . ) order moment is defined as [1]:

mXr = E(Xr) =

+∞∫−∞

xrfX(x)dx (7)

The quantity mX = mX1 is called the expectation (mean) of X:

mX = mX1 = E(X) =

+∞∫−∞

xfX(x)dx (8)

From the mean mX , the rth order central moment is defined as:

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µXr = E[(X −mX)r] =

+∞∫−∞

(x−mX)rfX(x)dx (9)

The second central moment µX2 is called the variance and it is usually denoted by σ2X

, where σX isthe standard deviation of X .

If X is discrete with ν possible values, the rth order moment is:

mXr = E(Xr) =ν∑i=1

pixri (10)

where pi is the probability corresponding to value xi of X . The mean is consequently defined as in(11):

mX = mX1 = E(X) =ν∑i=1

pixi (11)

Analogously, the rth order central moment is:

µXr =ν∑i=1

pi(xi −mX)r (12)

Central moments are used more often than moments, because they relate only to the spread andshape of the distribution, rather than to its location.

2.5. Cumulants

The cumulants of a probability distribution are an alternative to the moments in representing r.v.s.Their algebraic properties are very interesting and allow easier calculations.

Using McLaurin’s series to expand lnψX(t) of X , the result is:

lnψX(t) =n∑r=1

kXr

r!(jt)r + en (13)

where kXr is defined as the rth (r ∈ 1, 2, 3, . . . ) order cumulant of X , and en is the error of theorder expansion.

Cumulants kXr can be obtained from moments mXr and vice versa [2]: kX1 = mX1

kXr+1 = mXr+1 −r∑i=1

CirmXikXr−i+1

(14)

and mX1 = kX1

mXr+1 = kXr+1 +r∑i=1

CirmXikXr−i+1

(15)

where Cir =

r!

i!(r − i)!.

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2.6. Examples of distributions

2.6.1. Uniform distribution

A uniform distribution is a distribution that has constant probability in a certain interval, e.g., [a, b].The general formula for the p.d.f. of a uniform r.v. X is:

fX(x) =

1

b− aa ≤ x ≤ b

0 otherwise(16)

where a is the location parameter and (b− a) is the scale parameter.

A uniform distribution with interval [a, b] is usually denoted as U(a, b). In particular, when a = 0and b = 1, it is called standard uniform distribution.

The c.d.f. is:

FX(x) =

0 x < ax− ab− a

a ≤ x ≤ b

1 x > b

(17)

Example 3: The p.d.f. and c.d.f. of uniform distribution U(a, b) are shown in figures 1 and 2,respectively.

Figure 1: p.d.f. of uniform distribution U(a, b)

Figure 2: c.d.f. of uniform distribution U(a, b)

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2.6.2. Normal distribution

The normal distribution (or Gaussian distribution) is a continuous probability distribution. It is oneof the most widely known and used distributions.

The p.d.f. of a normal r.v. X is:

fX(x) =1√

2πσXexp(−

(x−mX)2

2σ2X

) (18)

where mX is the mean, σ2X

is the variance, and σX is the standard deviation.

In probability theory and statistics, a normal distribution is usually denoted by N(m,σ2) for short.

The c.d.f. of a normal distribution is:

FX(x) =1√

2πσX

x∫−∞

exp(−(t−mX)2

2σ2X

)dt (19)

In particular, when mX = 0 and σ2X

= 1, the normal distribution is called standard normal distri-bution with its p.d.f. and c.d.f. as follows:

φX(x) =1√2π

exp(−1

2x2) (20)

ΦX(x) =1√2π

x∫−∞

exp(−1

2t2)dt (21)

Example 4: The p.d.f.s and c.d.f.s of normal distributions, for example, with different values of mand σ are shown in figures 3 and 4, respectively.

Figure 3: p.d.f.s of normal distributions

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Figure 4: c.d.f.s of normal distributions

2.6.3. Binomial distribution

The binomial distribution is a discrete probability distribution. A binomial r.v. X is the number ofsuccesses x in n repeated trials of the so-called Bernoulli experiment.

Let perform a Bernoulli experiment. Assume that there are only two possible outcomes for a trial:success with probability p (the same on every trial) and failure with probability (1 − p). The trial isrepeated n times. Also assume that all trials are independent, i.e., the outcome of any trial has noeffect on others. If the number of occurrences of event success is x, the corresponding probabilitieswill be px for event success and (1− p)n−x for event failure.

The p.m.f. of a binomial distribution is:

fX(x;n, p) =

(n

x

)px(1− p)n−x (22)

where(n

x

)=

n!

x!(n− x)!, x = 0, 1, 2, ..., n, and the c.d.f. is:

FX(x;n, p) =x∑i=0

(n

i

)pi(1− p)n−i (23)

The mean and variance of a binomial distribution are as follows:

mX = np (24)

σ2X

= np(1− p) (25)

Example 5: In power systems, power plants are usually included several identical units. Each unithas two states: running (denoted as 1) and outage (denoted as 0). The power output can be modeledby a binomial distribution with the occurrences of failure event (outage state): q = FOR (ForcedOutage Rate).

For example, a power plant consists of two identical units, each rated at 10 MW with FOR = 0.08(Fig. 5).

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~ ~0 10 20

0

0.2

0.4

0.6

0.8

1

Pg (MW)

Prob

abili

ty

Figure 5: Example of a power plant

+ The probability of outage state: q = FOR = 0.08

+ The probability of running state: p = 1− q = 0.92

+ Let x be the number of units with running state; using (22) with n = 2 and p = 0.92, we havecorresponding probabilities:

f(x = 0) =

(2

0

)0.920(1− 0.92)2−0 = 0.0064

f(x = 1) =

(2

1

)0.921(1− 0.92)2−1 = 0.1472

f(x = 2) =

(2

2

)0.922(1− 0.92)2−2 = 0.8464

Assume all units operating at the rated power, the power outputs of the power plant are:

Pg(x = 0) = 0MW

Pg(x = 1) = 10MW

Pg(x = 2) = 20MW

Finally, the power output of the power plant can be modeled by a binomial r.v. Pg with the possiblevalues and corresponding probabilities as in Table 1 and the p.m.f. plotting in Fig. 5.

Table 1: Possible values and corresponding probabilities of power output

state value (MW) probability

x=0 0 0.0064

x=1 10 0.1472

x=2 20 0.8464

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2.7. Applying properties of cumulants to a linear combination of random variables

If Y is a r.v., a linear combination of N independent r.v.s Xi (i = 1, 2, ...,N ):

Y = a1X1 + a2X2 + · · ·+ aNXN (26)

then the rth order cumulant of Y is calculated as [3]:

kY r = ar1kXr1

+ ar2kXr2

+ · · ·+ arNkXrN

(27)

2.8. Series expansions

The Gram-Charlier series expansion is a popular method to obtain the p.d.f. and c.d.f. of desiredr.v.s from their moments or cumulants [3]. In particular, this expansion gives an approximation of ap.d.f. and c.d.f. of a continuous r.v. around the normal distribution.

Considering an arbitrary continuous r.v. Y with mean value mY and standard deviation σY , ex-pressed in standard normalized form as X = (Y −mY )/σY , the c.d.f. of the standardized variableX can be written as:

FX(x) = ΦX(x) +C1

1!

dΦX(x)

dx+C2

2!

d2ΦX(x)

dx2+ · · · (28)

and its p.d.f. can be written as:

fX(x) = φX(x) +C1

1!

dφX(x)

dx+C2

2!

d2φX(x)

dx2+ · · · (29)

where φX(·) and ΦX(·) represent the p.d.f. and c.d.f. of standard normal distribution (see (20) and(21)), respectively.

The coefficients Ci can be calculated from its central moments [3, 4], for example, the first eightcoefficients are calculated as follows:

C1 = 0 C5 = −µX5

σ5X

+ 10µX3

σ3X

C2 = 0 C6 =µX6

σ6X

− 15µX4

σ4X

+ 30

C3 = −µX3

σ3X

C7 = −µX7

σ7X

+ 21µX5

σ5X

− 105µX3

σ3X

C4 =µX4

σ4X

− 3 C8 =µX8

σ8X

− 28µX6

σ6X

+ 210µX4

σ4X

− 315

(30)

Central moments can be calculated as functions of cumulants [5]:

µX2 = kX2

µX3 = kX3

µX4 = kX4 + 3k2X2

µX5 = kX5 + 10kX3kX2

µX6 = kX6 + 15kX4kX2 + 10k2X3 + 15k3

X2

µX7 = kX7 + 21kX5kX2 + 25kX4kX3 + 105kX3k2X2

µX8 = kX8 + 28kX6kX2 + 56kX5kX3 + 35k2X4

+ 210kX4k2X2 + 280kX2k

2X3 + 105k4

X2

(31)

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Therefore, if cumulants of a r.v. are known, the Gram-Charlier expansion makes it possible toapproximate its c.d.f. and p.d.f. using (28) and (29).

Beside Gram-Charlier expansion, there are some other expansions, e.g., Edgeworth and Cornish-Fisher, which are also popularly used to obtain c.d.f. and p.d.f. from moments or cumulants of ar.v.

3. PROBABILISTIC POWER FLOW METHODS

A general classification of PPF calculation is as follows: numerical methods and analytical meth-ods.

3.1. Numerical methods

- Monte Carlo Simulation (MCS) is a typical numerical method.

- MCS has now spread to a large number of scientific fields such as physics, mathematics, andengineering, etc.

- MCS is a method of computation that relies on repeated random sampling to obtain results. Thismethod is most suited to calculation using computer and applies for complex and non-linear modelswhich are difficult and even infeasible to obtain an exact result with a deterministic algorithm.

- In MCS method, the state of each component, e.g., generator, transmission line, transformer,etc., in power system are obtained by sampling. Power injections at each bus in electric networks aresampled according to their distributions. Finally, random numbers and probabilities are used as inputsof MCS to solve a probabilistic problem.

- Pros and cons: Although the MCS method is able to provide accurate results, it is really time-consuming. This is the main drawback of MCS method so that several less burdensome PPF methodshave been developed to obtain computational efficiency.

- Figure 6 shows the diagram of a simple MCS.

Input data(Predefined the number of samples: NS)

Set initial sample times: N = 0

Deterministic PF

Store results: state variables & power flows for each sampling

N < NS

Sampling power injections at each bus, state of each component

N = N +1

Yes

No

Output(state variables & power flows in

terms of p.d.f. and/or c.d.f.)

Figure 6: MCS diagram

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3.2. Analytical methods

- Adopting convolution techniques [7,8], point estimate method [9], and cumulant method [4], etc.

- In this section, a cumulant-based PPF method is presented as a representative for analytical ap-proaches: the method is based on the series expansions and sensitivity matrices. Continuous anddiscrete distributions are dealt with together during the computations.

- The form of the PF equations generally is:

Real and reactive injected powers:

Pi = Vi

n−1∑k=0

Vk(Gik cos θik +Bik sin θik) (32)

Qi = Vi

n−1∑k=0

Vk(Gik sin θik −Bik cos θik) (33)

Real and reactive line power flows:

Pik = ViVk(Gik cos θik +Bik sin θik)− tikGikV2i (34)

Qik = ViVk(Gik sin θik −Bik cos θik) + tikBikV2i −B′ikV 2

i (35)

where:

• Pi and Qi are real and reactive power injections at bus i,• Pik and Qik are real and reactive power power flows on the branch ik,• Vi and Vk are voltage magnitudes at buses i and k,• θik is voltage angle difference between buses i and k,• Gik and Bik are the real and imaginary parts of the admittance matrix of branch ik,• B′ik is the haft of susceptance of branch ik,• tik is the transformation ratio of branch ik,• n is the total number of buses in the system.

- The above equations can be expressed by matrix form as

w = g(x) (36)

z = h(x) (37)

where:

• w is the vector of nodal power injections,• x is the vector of state variables,• z is the vector of line power flows,

and

• g(x) are the power flow equations,• h(x) are the functions to compute line power flows.

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- The solution x and z of the above equations is obtained by solving a deterministic PF.

- Using Taylor series expansion to linearize the above equations around the solution point gives:

∆x = [S]|x∆w (38)

∆z = [T ]|x∆w (39)

where [S]|x is the inverse of the Jacobian matrix and [T ]|x is the sensitivity matrix of power flowswith respect to nodal power injections, both computed at the solution point x:

[S]

=[J]−1

=

∂P

∂θ

∂P

∂V∂Q

∂θ

∂Q

∂V

−1

(40)

[T]

=[G] [

S]

=

∂Pik∂θ

∂Pik∂V

∂Qik

∂θ

∂Qik

∂V

∂P

∂θ

∂P

∂V∂Q

∂θ

∂Q

∂V

−1

(41)

Size of matrices:[J ] and [S]: (n− 1 + `)× (n− 1 + `)[G] and [T ]: (2n)× (n− 1 + `)(` is the total number of load buses)

- To solve the PPF, each element of w is considered as the realization (observed value; the valueassigned to the r.v.) of the r.v. associated with each nodal power injection W, ∈ 1, ..., n− 1 + `(assume bus 0 is the slack bus in single slack bus model; elements from 1 to (n−1) are relevant toreal power equations; the others to reactive power equations). From relationships in (38) and (39),elements of x and z are, therefore, the realizations of corresponding r.v.s Xχ, χ ∈ 1, 2, ..., n−1+`(similarly, elements from 1 to (n−1) are related to angle terms, while the rest are voltage magnitudeterms) and Zζ , ζ ∈ 1, 2, ..., 2b (b is the total number of branches), respectively. Consequently, wehave the relationships of r.v.s as follows:

X1

X2...

Xn−1+`

= [S]|x

W1

W2...

Wn−1+`

(42)

Z1

Z2...Z2b

= [T ]|x

W1

W2...

Wn−1+`

(43)

- Based on (42), (43), and (27), the rth order cumulants of state variables and line power flows canbe obtained from the rth order cumulants of nodal power injections as

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kXr

1

kXr2...

krXn−1+`

= [S(r)]|x

kW r

1

kW r2...

kW rn−1+`

(44)

kZr

1

kZr2...

kZr2b

= [T (r)]|x

kW r

1

kW r2...

kW rn−1+`

(45)

where [S(r)] and [T (r)] are the matrices with elements S(r)ij = srij and T

(r)ij = trij , respectively.

- Assume that all input r.v.s are independent, the calculation procedure for this method can bedescribed as follows:

(1) Run the deterministic PF calculation to obtain the expected values x and z and the sensitivitymatrices [S]|x and [T ]|x;

(2) Calculate the cumulants of nodal power injections by summing the cumulants of the distribu-tions associated with loads, generators, and other injected powers at the same bus;

(3) Compute cumulants of state variables and line power flows using (44) and (45);

(4) Obtain the c.d.f.s and/or p.d.f.s of the outputs of interest using approximation methods basedon series expansions (e.g., (28) and/or (29)).

4. EXAMPLE

PPF calculation for 4-bus test system

4.1. Data

The single line diagram for 4-bus test system is shown in Fig. 7, while network data are providedin Table 2 and 3 (all information taken from [10]).

Figure 7: Single line diagram of 4-bus test system

The system has:

- two generators are connected to buses 1 and 4;- the system has four lines and four buses: bus 1 is the slack bus (reference bus or swing bus),bus 4 is PV bus (generator bus), buses 2 and 3 are load buses (PQ buses);

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- the base MVA: 100 MVA;- the base voltage: 230 kV.

Table 2: Line data for 4-bus test system

from-bus to-bus R X G B total charging Y/2

(p.u.) (p.u.) (p.u.) (p.u.) (MVAr) (p.u.)

1 2 0.01008 0.05040 3.815629 -19.078144 10.25 0.05125

1 3 0.00744 0.03720 5.169561 -25.847809 7.75 0.03875

2 4 0.00744 0.03720 5.169561 -25.847809 7.75 0.03875

3 4 0.12720 0.06360 3.023705 -15.118528 12.75 0.06375

Table 3: Generation and load data for 4-bus test system

generation load

bus P (MW) Q (MVAr) P (MW) Q (MVAr) V (p.u.) remark

1 - - 50 30.99 1.00, 00 Slack

2 0 0 170 105.35 1.00, 00 PQ

3 0 0 200 123.94 1.00, 00 PQ

4 318 - 80 49.58 1.02, 00 PV

Assumption for probabilistic data: loads have normal distributions characterized by mean (expectedvalue) and standard deviation, expected value = nominal power, standard deviation = 8% nominalpower, as presented in Table 4.

Table 4: Probabilistic data for loads

bus mean standard deviation variance

P (MW)

1 50 4 16

2 170 13.6 184.96

3 200 16 256

4 80 6.4 40.96

Q (MVAr)

1 30.99 2.4792 6.1464

2 105.35 8.4280 71.0312

3 123.94 9.9152 98.3112

4 49.58 3.9664 15.7323

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4.2. MCS

Example for sampling power injections (Matlab code):

MU = [50 170 200 80 30.99 105.35 123.94 49.58];SIGMA = [16 184.96 256 40.96 6.1416 71.0312 98.3112 15.7323];Nsample = 10000; %Number of samplesW normal = mvnrnd(MU,SIGMA,Nsample);% Multivariate normal distribution with mean MU and covariance SIGMA, size: 10000× 8

Examples for plotting (Matlab code):

figure(1); %Figure 8hist(W normal(:,1),20); %Histogram plot for real power injection of load at bus 1xlabel(’PL1 (MW)’);ylabel(’Frequency’);

figure(2); %Figure 9cdfplot(W normal(:,1)); %Empirical CDF plot for load (real power) at bus 1xlabel(’PL1 (MW)’);ylabel(’Cumulative probability’);

35 40 45 50 55 60 650

500

1000

1500

PL1 (MW)

Freq

uenc

y

Figure 8: Histogram of real power injection of load at bus 1

15

PL1 (MW)

35 40 45 50 55 60 650

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P (MW)

Cum

ulat

ive

prob

abili

ty

Figure 9: c.d.f. of real power injection of load at bus 1

Deterministic PF

Pik Qik θi Vi

4.3. Cumulant method

- Run deterministic PF; obtain x, z, [S]|x, and [T ]|x as follows

x: state variables (Table 5)

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Table 5: State variables

bus voltage angle

(p.u.) (rad)

1 1.0000 0.0000

2 0.9814 -0.0169

3 0.9678 -0.0325

4 1.0200 0.0266

z: real and reactive power flows (Table 6)

Table 6: Real and reactive power flows

line Pik Qik

(p.u.) (p.u.)

1-2 0.3873 0.2678

1-3 0.9818 0.6637

2-4 -1.3151 -0.7473

3-4 -1.0288 -0.5914

sensitivity matrices:

[S]|x =

0.03843 0.01079 0.02835 −0.00362 −4.68956× 10−5

0.01074 0.03131 0.01824 −2.06× 10−5 −0.0037583260.02878 0.01861 0.04890 −5.51× 10−5 −8.09× 10−5

0.00504 −0.00013 −0.00033 0.02241 5.48× 10−7

−0.00019 0.00603 −0.00032 3.62× 10−7 0.02516

[T]|x =

−0.73948 −0.20226 −0.53137 −0.01032 0.00088−0.26940 −0.81399 −0.45772 0.00052 −0.014280.26608 −0.20091 −0.52784 −0.00763 0.00087−0.26597 0.19816 −0.45189 0.00051 −0.007260.03520 0.03941 0.10353 −0.44125 −0.000170.04988 −0.02566 0.08475 −9.55× 10−5 −0.669890.06329 0.04613 0.12120 0.57333 −0.000200.06702 0.03533 0.11386 −0.00013 0.36614

- Vector of nodal power injections (p.u.):

P2

P3

P4

Q2

Q3

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- The second order cumulants of nodal power injections (p.u.):kP 2

2

kP 23

kP 24

kQ22

kQ23

=

0.0184960.0256

0.0040960.007103120.00983112

- Calculate the cumulants of state variables and line power flows (p.u.):

+ The first order cumulants (p.u.): (equal to x and z)kΘ1

2

kΘ13

kΘ14

kV 12

kV 13

=

−0.0169−0.03250.02660.98140.9678

kP 11−2

kP 11−3

kP 12−4

kP 13−4

kQ11−2

kQ11−3

kQ12−4

kQ13−4

=

0.38730.9818−1.3151−1.02880.26780.6637−0.7473−0.5914

+ The second order cumulants (p.u.): (using (44) and (45))

kΘ22

kΘ23

kΘ24

kV 22

kV 23

= [S(2)]|x ×

kP 2

2

kP 23

kP 24

kQ22

kQ23

=

0.3368× 10−4

0.2873× 10−4

0.3398× 10−4

0.4037× 10−5

0.7154× 10−5

kP 21−2

kP 21−3

kP 22−4

kP 23−4

kQ21−2

kQ21−3

kQ22−4

kQ23−4

= [T (2)]|x ×

kP 2

2

kP 23

kP 24

kQ22

kQ23

=

0.01230.01920.00350.00320.00150.00450.00250.0015

Note: In this example, we consider only the normal distribution of loads. For normal distribution,

the first and second cumulants can characterize totally the distribution (the higher orders of cumulants

18

are equal to zero).

- Adopting (28) and (29), we can obtain c.d.f.s and/or p.d.f.s of state variables and line powerflows. Figures 10 and 11 show, for example, the p.d.f. and c.d.f. of r.v. V2 (voltage magnitude at bus2).

0.974 0.976 0.978 0.98 0.982 0.984 0.986 0.988 0.99 0.9920

0.01

0.02

0.03

0.04

0.05

0.06

0.07

V2 (p.u.)

MCSCumulant

V2 (p.u.)

Prob

abili

ty

Figure 10: p.d.f. of voltage magnitude at bus 2

V2 (p.u.)

Cum

ulat

ive

prob

abili

ty

0.974 0.976 0.978 0.98 0.982 0.984 0.986 0.988 0.99

0

0.2

0.4

0.6

0.8

1

CumulantMCS

Figure 11: c.d.f. of voltage magnitude at bus 2

Note: In this case, all distributions obtained are normal with the mean values (the first order cumu-lants) and the variances (the second order cumulants) calculated above.

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Summary results:

+ MCS method:

mean variance

bus voltage angle voltage angle

(p.u.) (rad) (p.u.) (×10−5) (rad) (×10−4)

1 1.0000 0.0000 0.0000 0.0000

2 0.9814 -0.0170 0.4080 0.3388

3 0.9678 -0.0325 0.7157 0.2907

4 1.0200 0.0265 0.0000 0.3414

mean variance

line Pik Qik Pik Qik

(p.u.) (p.u.) (p.u.) (p.u.)

1-2 0.3895 0.2683 0.0124 0.0015

1-3 0.9823 0.6638 0.0194 0.0045

2-4 -1.3156 -0.7478 0.0036 0.0025

3-4 -1.0277 -0.5914 0.0032 0.0015

Time consuming: 39.12 seconds

+ Cumulant method:

mean variance

bus voltage angle voltage angle

(p.u.) (rad) (p.u.) (×10−5) (rad) (×10−4)

1 1.0000 0.0000 0.0000 0.0000

2 0.9814 -0.0169 0.4037 0.3368

3 0.9678 -0.0325 0.7154 0.2873

4 1.0200 0.0266 0.0000 0.3398

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mean variance

line Pik Qik Pik Qik

(p.u.) (p.u.) (p.u.) (p.u.)

1-2 0.3873 0.2678 0.0123 0.0015

1-3 0.9818 0.6637 0.0192 0.0045

2-4 -1.3151 -0.7473 0.0035 0.0025

3-4 -1.0288 -0.5914 0.0032 0.0015

Time consuming: 0.16 seconds

Further consideration:

Let assume that the power output of generator at bus 4 is binomial distribution.Data for generator at bus 4:+ Number of units: 6+ Rated power of each unit: 53 MW+ q = FOR = 0.09How can we model? And take into account in the PPF calculation?

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Cited references

[1] M. Kendall and A. Stuart, The Advanced Theory of Statistics. London, U.K.: C. Griffin, 4thedition, vol. 1, 1977.

[2] W. D. Tian, D. Sutanto, Y. B. Lee, and H. R. Outhred, ”Cumulant based probabilistic powersystem simulation using Laguerre polynomials,” IEEE Trans. Energy Convers., vol. 4, no. 4,pp. 567-574, Dec. 1989.

[3] H. Cramer, Mathematical Methods of Statistics. Princeton University Press, 1945.

[4] P. Zhang and S. T. Lee, ”Probabilistic load flow computation using the method of combinedcumulants and Gram-Charlier expansion,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 676-682,Feb. 2004.

[5] A. Stuart and J. K. Ord, Kendall’s Advanced Theory of Statistics. London: Charles Griffin andCompany Limited, 5th edition, vol. 1, 1987.

[6] J. E. Kolassa, Series Approximation Methods in Statistics. New York: Springer, 3rd edition, 2006.

[7] R. N. Allan and M. R. G. Al-Shakarchi, ”Probabilistic a.c. load flow”, Proceedings of the Institu-tion of Electrical Engineers, vol. 123, no. 6, pp. 531-536, Jun. 1976.

[8] R. N. Allan and M. R. G. Al-Shakarchi, ”Probabilistic techniques in a.c. load-flow analysis”,Proceedings of the Institution of Electrical Engineers, vol. 124, no. 2, pp. 154-160, Feb. 1977.

[9] C. L. Su, ”Probabilistic load-flow computation using point estimate method,” IEEE Trans. PowerSyst., vol. 20, no. 4, pp. 1843-1851, Nov. 2005.

[10] J. J. Grainger and JR. W. D. Stevenson, Power System Analysis. McGraw-Hill, Inc. P329-375,1994.

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