5

Computational Methods for Electric Power Systems

Mariesa L. Crow

Missouri University of Science and Technology

5.1    Power Flow

Admittance Matrix • Newton–Raphson Method

5.2    Optimal Power Flow

Steepest Descent Algorithm • Limitations on Independent Variables • Limitations on Dependent Variables

5.3    State Estimation

Electric power systems are some of the largest human-made systems ever built. As with many physical systems, engineers, scientists, economists, and many others strive to understand and predict their complex behavior through mathematical models. The sheer size of the bulk power transmission system forced early power engineers to be among the first to develop computational approaches to solving the equations that describe them. Today’s power system planners and operators rely very heavily on the computational tools to assist them in maintaining a reliable and secure operating environment.

The various computational algorithms were developed around the requirements of power system operation. The primary algorithms in use today are the power flow (also known as load flow), optimal power flow (OPF), and state estimation. These are all “steady-state” algorithms and are built up from the same basic approach to solving the nonlinear power balance equations. These particular algorithms do not explicitly consider the effects of time-varying dynamics on the system. The fields of transient and dynamic stability require a large number of algorithms that are significantly different from the powerflow-based algorithms and will therefore not be covered in this chapter.

5.1    Power Flow

The underlying principle of a power flow problem is that given the system loads, generation, and network configuration, the system bus voltages and line flows can be found by solving the nonlinear power flow equations. This is typically accomplished by applying Kirchoff’s law at each power system bus throughout the system. In this context, Kirchoff’s law can be interpreted as the sum of the powers entering a bus must be zero, or that the power at each bus must be conserved. Since power is comprised of two components, active power and reactive power, each bus gives rise to two equations—one for active power and one for reactive power. These equations are known as the power flow equations:

$0=\Delta P_i=P_i^{inj}-V_i{\displaystyle \sum _{j=1}{V}_j{Y}_{ij}\cos \left({\text{θ}}_i-{\text{θ}}_j-{\varphi }_{ij}\right)}$

(5.1)

$0=\Delta Q_i=Q_i^{inj}-V_i{\displaystyle \sum _{j=1}^N{V}_j{Y}_{ij}\sin \left({\text{θ}}_i-{\text{θ}}_j-{\varphi }_{ij}\right)i=1,\dots ,N}$

(5.2)

where $P_i^{inj}$ and $Q_i^{inj}$ are the active power and reactive power injected at the bus i, respectively. Loads are modeled by negative power injection. The values V_i and V_j are the voltage magnitudes at bus i and bus j, respectively. The values θ_i and θ_j are the corresponding phase angles. The value Y_ij∠φ_ij is the (ij)th element of the network admittance matrix Y_bus. The constant N is the number of buses in the system. The updates ΔP_i and ΔQ_i of Equations 5.1 and 5.2 are called the mismatch equations because they give a measure of the power difference, or mismatch, between the calculated power values, as functions of voltage and phase angle, and the actual injected powers.

The formulation in Equations 5.1 and 5.2 is called the polar formulation of the power flow equations. If Y_ij∠ϕ_ij is instead given by the complex sum g_ij + jb_ij, then the power flow equations may be written in rectangular form as

$0=\Delta P_i=P_i^{inj}-V_i{\displaystyle \sum _{j=1}^N{V}_j\left(g_{ij}\cos \left({\text{θ}}_i-{\text{θ}}_j\right)+b_{ij}\sin \left({\text{θ}}_i-{\text{θ}}_j\right)\right)}$

(5.3)

$\begin{array}{cc}0=\Delta Q_i=Q_i^{inj}-V_i{\displaystyle \sum _{j=1}^N{V}_j\left(g_{ij}\sin \left({\text{θ}}_i-{\text{θ}}_j\right)+b_{ij}\cos \left({\text{θ}}_i-{\text{θ}}_j\right)\right)}& i=1,\dots ,N\end{array}$

(5.4)

There are, at most, 2N equations to solve. This number is then further reduced by removing one power flow equation for each known voltage (at voltage controlled buses) and the swing bus angle. This reduction is necessary since the number of equations must equal the number of unknowns in a fully determined system. In either case, the power flow equations are a system of nonlinear equations. They are nonlinear in both the voltage and phase angle. Once the number and form of the nonlinear power flow equations have been determined, the Newton–Raphson method may be applied to numerically solve the power flow equations.

5.1.1    Admittance Matrix

The first step in solving the power flow equations is to obtain the admittance matrix Y for the system. The admittance matrix of a passive network (a network containing only resistors, capacitors, and inductors) may be found by summing the currents at every node in the system. An arbitrary bus i in the transmission network is shown in Figure 5.1. The bus i can be connected to any number of other buses in the system through transmission lines. Each transmission line between buses i and j is represented by a π-circuit with series impedance R_ij + jX_ij where R_ij is the per unit resistance of the transmission line. The per unit line reactance is X_ij = ω_s L_ij where ω_s is the system base frequency and L_ij is the inductance of the line. In the π-circuit, the line-charging capacitance is represented by two lumped admittances jB_ij/2 placed at each end of the transmission line. Note that although each transmission-line parameter is in per unit and therefore a unitless quantity, the impedance is given in Ω/Ω whereas the admittance is given as Ω/Ω. The series impedances can also be represented by their equivalent admittance value where

$y_{ij}\angle {\varphi }_{ij}=\frac{1}{R_{ij}+j{X}_{ij}}$

Summing the currents at node i yields

$I_{i,inj}=I_{i1}+I_{i2}+\cdots +I_{iN}+I_{i10}+I_{i20}+\cdots +I_{iN0}$

(5.5)

FIGURE 5.1  Bus i in a power transmission network.

$\begin{array}{l}=\frac{\left({\widehat{V}}_i-{\widehat{V}}_1\right)}{R_{i1}+j{X}_{i1}}+\frac{\left({\widehat{V}}_i-{\widehat{V}}_2\right)}{R_{i2}+j{X}_{i2}}+\cdots +\frac{\left({\widehat{V}}_i-{\widehat{V}}_{iN}\right)}{R_{iN}+j{X}_{iN}}\hfill \\ +\widehat{V}\left(j\frac{B_{i1}}{2}+j\frac{B_{i2}}{2}+\cdots +j\frac{B_{iN}}{2}\right)\hfill \end{array}$

(5.6)

$\begin{array}{l}=\left({\widehat{V}}_i-{\widehat{V}}_1\right)y_{i1}\angle {\varphi }_{i1}+\left({\widehat{V}}_i-{\widehat{V}}_2\right)y_{i2}\angle {\varphi }_{i2}+\cdots +\left({\widehat{V}}_i-{\widehat{V}}_N\right)y_{iN}\angle {\varphi }_{iN}\hfill \\ +{\widehat{V}}_i\left(j\frac{B_{i1}}{2}+j\frac{B_{i2}}{2}+\cdots +j\frac{B_{iN}}{2}\right)\hfill \end{array}$

(5.7)

where ${\widehat{V}}_i=V_i\angle {\text{θ}}_i$.

Gathering the voltages yields

$\begin{array}{ll}{I}_{i,inj}\hfill & =-{\widehat{V}}_1{y}_{i1}\angle {\varphi }_{i1}-{\widehat{V}}_2{y}_{i2}\angle {\varphi }_{i2}+\cdots +{\widehat{V}}_i\left(y_{i1}\angle {\varphi }_{i1}+y_{i2}\angle {\varphi }_{i2}Z+\cdots +y_{iN}\angle {\varphi }_{iN}r+j\frac{B_{i1}}{2}+j\frac{B_{i2}}{2}+\cdots +j\frac{B_{iN}}{2}\right)\hfill \\ \hfill & +\cdots -{\widehat{V}}_N{y}_{iN}\angle {\varphi }_{iN}\hfill \end{array}$

(5.8)

By noting that there are N buses in the system, there are N equations similar to Equation 5.8. These can be represented in matrix form

$\left[\begin{array}{c}{I}_{1,inj}\\ I_{2,inj}\\ ⋮\\ I_{i,inj}\\ ⋮\\ I_{N,inj}\end{array}\right]=\left[\begin{array}{cccccc}{Y}_{11}& -y_{12}\angle {\varphi }_{12}& \dots & -y_{1i}\angle {\varphi }_{1i}& \dots & -y_{1N}\angle {\varphi }_{1N}\\ -y_{21}\angle {\varphi }_{21}& Y_{22}& \dots & -y_{2i}\angle {\varphi }_{2i}& \dots & -y_{2N}\angle {\varphi }_{2N}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ -y_{1i}\angle {\varphi }_{i1}& -y_{i2}\angle {\varphi }_{i2}& \dots & Y_{ii}& \dots & -y_{iN}\angle {\varphi }_{iN}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ -y_{N1}\angle {\varphi }_{N1}& -y_{N2}\angle {\varphi }_{N2}& \dots & -y_{Ni}\angle {\varphi }_{Ni}& \dots & Y_{NN}\end{array}\right]\left[\begin{array}{c}{\widehat{V}}_1\\ {\widehat{V}}_2\\ ⋮\\ {\widehat{V}}_1\\ ⋮\\ {\widehat{V}}_N\end{array}\right]$

(5.9)

where

$Y_{ii}=\left(y_{i1}\angle {\varphi }_{i1}+y_{i2}\angle {\varphi }_{i2}+\cdots +y_{iN}\angle {\varphi }_{iN}+j\frac{B_{i1}}{2}+j\frac{B_{i2}}{2}+\cdots +j\frac{B_{iN}}{2}\right)$

(5.10)

The matrix relating the injected currents vector to the bus voltage vector is known as the system admittance matrix and is commonly represented as Y. A simple procedure for calculating the elements of the admittance matrix is

 

Y(i,j) negative of the admittance between buses i and j

Y(i,i) sum of all admittances connected to bus i

noting that the line-charging values are shunt admittances and are included in the diagonal elements.

5.1.2    Newton–Raphson Method

The most common approach to solving the power flow equations is to use the iterative Newton–Raphson method. The Newton–Raphson method is an iterative approach to solving continuous nonlinear equations in the form

$F\left(x\right)=\left[\begin{array}{c}{f}_1\left(x_1,x_2,\dots ,x_n\right)\\ f_2\left(x_1,x_2,\dots ,x_n\right)\\ ⋮\\ f_n\left(x_1,x_2,\dots ,x_n\right)\end{array}\right]=0$

(5.11)

An iterative approach is one in which an initial guess (x⁰) to the solution is used to create a sequence x⁰, x¹, x²,… that (hopefully) converges arbitrarily close to the desired solution vector x^⋆ where F(x^⋆) = 0.

The Newton–Raphson method for n-dimensional systems is given as

$x^{k+1}=x^k-{\left[J\left(x^k\right)\right]}^{-1}F\left(x^k\right)$

(5.12)

where

$\begin{array}{l}x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ ⋮\\ x_n\end{array}\right]\hfill \\ F\left(x^k\right)=\left[\begin{array}{c}{f}_1\left(x^k\right)\\ f_2\left(x^k\right)\\ f_3\left(x^k\right)\\ ⋮\\ f_n\left(x^k\right)\end{array}\right]\hfill \end{array}$

and the Jacobian matrix [J(x^k)] is given by

$\left[J\left(x^k\right)\right]=\left[\begin{array}{ccccc}\frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2}& \frac{\partial f_1}{\partial x_3}& \dots & \frac{\partial f_2}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1}& \frac{\partial f_2}{\partial x_2}& \frac{\partial f_2}{\partial x_3}& \dots & \frac{\partial f_2}{\partial x_n}\\ \frac{\partial f_3}{\partial x_1}& \frac{\partial f_3}{\partial x_2}& \frac{\partial f_3}{\partial x_3}& \dots & \frac{\partial f_3}{\partial x_n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ \frac{\partial f_n}{\partial x_1}& \frac{\partial f_n}{\partial x_2}& \frac{\partial f_n}{\partial x_3}& \dots & \frac{\partial f_n}{\partial x_n}\end{array}\right]$

Typically, the inverse of the Jacobian matrix [J(x^k)] is not found directly, but rather through LU factorization. Convergence is typically evaluated by considering the norm of the function

$\Vert F\left(x^k\right)\Vert <\text{ε}$

(5.13)

Note that the Jacobian is a function of x^k and is therefore updated every iteration along with F(x^k).

In this formulation, the vector F(x) is the set of power flow equations and the unknown x is the vector of voltage magnitudes and angles. It is common to arrange the Newton–Raphson equations by phase angle followed by the voltage magnitudes as

$\left[\begin{array}{cc}{J}_1& J_2\\ J_3& J_4\end{array}\right]\left[\begin{array}{c}\Delta {\text{δ}}_1\\ \Delta {\text{δ}}_2\\ \Delta {\text{δ}}_3\\ ⋮\\ \Delta {\text{δ}}_N\\ \Delta V_1\\ \Delta V_2\\ \Delta V_3\\ ⋮\\ \Delta Q_N\end{array}\right]=-\left[\begin{array}{c}\Delta P_1\\ \Delta P_2\\ \Delta P_3\\ ⋮\\ \Delta P_N\\ \Delta Q_1\\ \Delta Q_2\\ \Delta Q_3\\ ⋮\\ \Delta Q_N\end{array}\right]$

(5.14)

where

$\begin{array}{l}\Delta {\text{δ}}_i={\text{δ}}_i^{k+1}-{\text{δ}}_i^k\hfill \\ \Delta V_i=V_i^{k+1}-V_i^k\hfill \end{array}$

and ΔP_i and ΔQ_i are as given in Equations 5.1 and 5.2 and are evaluated at δ^k and V^k. The Jacobian is typically divided into four submatrices, where

$\left[\begin{array}{cc}{J}_1& J_2\\ J_3& J_4\end{array}\right]=\left[\begin{array}{cc}\frac{\partial \Delta P}{\partial \text{δ}}& \frac{\partial \Delta P}{\partial V}\\ \frac{\partial \Delta Q}{\partial \text{δ}}& \frac{\partial \Delta Q}{\partial V}\end{array}\right]$

(5.15)

Each submatrix represents the partial derivatives of each of the mismatch equations with respect to each of the unknowns. These partial derivatives yield eight types—two for each mismatch equation, where one is for the diagonal element and the other is for off-diagonal elements. The derivatives are summarized as

$\frac{\partial \Delta P_i}{\partial {\text{δ}}_i}=V_i{\displaystyle \sum _{j=1}^N{V}_j{Y}_{ij}\sin \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)+V_i^2{Y}_{ii}\sin {\varphi }_{ii}}$

(5.16)

$\frac{\partial \Delta P_i}{\partial {\text{δ}}_j}=-V_i{V}_j{Y}_{ij}\sin \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)$

(5.17)

$\frac{\partial \Delta P_i}{\partial V_i}=-{\displaystyle \sum _{i=1}^N{V}_j{Y}_{ij}\cos \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)-V_i{V}_{ii}\cos {\varphi }_{ii}}$

(5.18)

$\frac{\partial \Delta P_i}{\partial V_j}=-V_i{V}_{ij}\cos \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)$

(5.19)

$\frac{\partial \Delta Q_i}{\partial {\text{δ}}_i}=-V_i{\displaystyle \sum _{j=1}^N{V}_j{Y}_{ij}\cos \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)+V_i^2{Y}_{ii}\cos {\varphi }_{ii}}$

(5.20)

$\frac{\partial \Delta Q_i}{\partial {\text{δ}}_j}=V_i{V}_j{Y}_{ij}\cos \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)$

(5.21)

$\frac{\partial \Delta Q_i}{\partial V_i}=-{\displaystyle \sum _{j=1}^N{V}_j{Y}_{ij}\sin \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)+V_i{Y}_{ii}\sin {\varphi }_{ii}}$

(5.22)

$\frac{\partial \Delta Q_i}{\partial V_j}=-V_i{Y}_{ij}\sin \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)$

(5.23)

A common modification to the power flow solution is to replace the unknown update ΔV_i by the normalized value ΔV_i/V_i. This formulation yields a more symmetric Jacobian matrix as the Jacobian submatrices J₂ and J₄ are now multiplied by V_i to compensate for the scaling of ΔV_i by V_i. All partial derivatives of each submatrix then become quadratic in voltage magnitude.

The Newton–Raphson method for the solution of the power flow equations is relatively straightforward to program since both the function evaluations and the partial derivatives use the same expressions. Thus it takes little extra computational effort to compute the Jacobian once the mismatch equations have been calculated.

One of the underlying assumptions of the Newton–Raphson iteration is that the higher order terms of the Taylor series expansion upon which the iteration is based are negligible only if the initial guess is sufficiently close to the actual solution to the nonlinear equations. Under most operating conditions, the voltages throughout the power system are within ±10% of the nominal voltage and therefore fall in the range 0.9 ≤ V_i ≤ 1.1 per unit. Similarly, under most operating conditions the phase angle differences between adjacent buses are typically small. Thus if the swing bus angle is taken to be zero, then all phase angles throughout the system will also be close to zero. Therefore in initializing a power flow, it is common to choose a “flat start” initial condition. That is, all voltage magnitudes are set to 1.0 per unit and all angles are set to zero.

Iteration 1

Evaluating the Jacobian and the mismatch equations at the flat start initial conditions yields

$\begin{array}{l}\left[J^0\right]=\left[\begin{array}{ccc}-13.2859& 9.9010& 0.9901\\ 9.9010& -20.000& -1.9604\\ -0.9901& 2.0000& -19.4040\end{array}\right]\hfill \\ \left[\begin{array}{c}\Delta P_2^0\\ \Delta P_3^0\\ \Delta Q_3^0\end{array}\right]=\left[\begin{array}{c}0.5044\\ -1.1802\\ -0.2020\end{array}\right]\hfill \end{array}$

Solving

$\left[J^0\right]\left[\begin{array}{c}\Delta {\text{δ}}_2^1\\ \Delta {\text{δ}}_3^1\\ \Delta V_3^1\end{array}\right]=\left[\begin{array}{c}\Delta P_2^0\\ \Delta P_3^0\\ \Delta Q_3^0\end{array}\right]$

by LU factorization yields

$\left[\begin{array}{c}\Delta {\text{δ}}_2^1\\ \Delta {\text{δ}}_3^1\\ \Delta V_3^1\end{array}\right]=\left[\begin{array}{c}-0.0096\\ -0.0621\\ -0.0163\end{array}\right]$

Therefore

$\begin{array}{l}{\text{δ}}_2^1={\text{δ}}_2^0+\Delta {\text{δ}}_2^1=0-0.0096=-0.0096\hfill \\ {\text{δ}}_3^1={\text{δ}}_3^0+\Delta {\text{δ}}_3^1=0-0.0621=-0.0621\hfill \\ V_3^1=V_3^0+\Delta V_3^1-=1-0.0163=0.9837\hfill \end{array}$

Note that the angles are given in radians and not degrees. The error at the first iteration is the largest absolute value of the mismatch equations, which is

${\text{ε}}^1=1.1802$

One quick check of this process is to note that the voltage update $V_3^1$ is slightly less than 1.0 per unit, which would be expected given the system configuration. Note also that the diagonals of the Jacobian are all equal or greater in magnitude than the off-diagonal elements. This is because the diagonals are summations of terms, whereas the off-diagonal elements are single terms.

Iteration 2

Evaluating the Jacobian and the mismatch equations at the updated values ${\text{δ}}_2^1$, ${\text{δ}}_3^1$, and $V_3^1$ yields

$\begin{array}{l}\left[J^1\right]=\left[\begin{array}{ccc}-13.1597& 9.7771& 0.4684\\ 9.6747& -19.5280& -0.7515\\ -1.4845& 3.0929& -18.9086\end{array}\right]\hfill \\ \left[\begin{array}{c}\Delta P_2^1\\ \Delta P_3^1\\ \Delta Q_3^1\end{array}\right]=\left[\begin{array}{c}0.0074\\ -0.0232\\ -0.0359\end{array}\right]\hfill \end{array}$

Solving for the update yields

$\left[\begin{array}{c}\Delta {\text{δ}}_2^2\\ \Delta {\text{δ}}_3^2\\ \Delta V_3^2\end{array}\right]=\left[\begin{array}{c}0.0005\\ -0.0014\\ -0.0021\end{array}\right]$

and

$\left[\begin{array}{c}{\text{δ}}_2^2\\ {\text{δ}}_3^2\\ V_3^2\end{array}\right]=\left[\begin{array}{c}-0.0101\\ -0.0635\\ 0.9816\end{array}\right]$

where

${\text{ε}}^2=0.0359$

Iteration 3

Evaluating the Jacobian and the mismatch equations at the updated values ${\text{δ}}_2^2$, ${\text{δ}}_3^2$, and $V_3^2$ yields

$\begin{array}{l}\left[J^2\right]=\left[\begin{array}{ccc}-13.1392& 9.7567& 0.4600\\ 9.6530& -19.4831& -0.7213\\ -1.4894& -3.1079& -18.8300\end{array}\right]\hfill \\ \left[\begin{array}{c}\Delta P_2^0\\ \Delta P_3^0\\ \Delta Q_3^0\end{array}\right]=\left[\begin{array}{c}0.1717\\ -0.5639\\ -0.9084\end{array}\right]\times {10}^{-4}\hfill \end{array}$

Solving for the update yields

$\left[\begin{array}{c}\Delta {\text{δ}}_2^2\\ \Delta {\text{δ}}_3^2\\ \Delta V_3^2\end{array}\right]=\left[\begin{array}{c}-0.1396\\ -0.3390\\ -0.5273\end{array}\right]\times {10}^{-5}$

and

$\left[\begin{array}{c}{\text{δ}}_2^3\\ {\text{δ}}_3^2\\ V_3^3\end{array}\right]=\left[\begin{array}{c}-0.0101\\ -0.0635\\ 0.9816\end{array}\right]$

where

${\text{ε}}^3=0.9084\times {10}^{-4}$

At this point, the iterations have converged since the mismatch is sufficiently small and the values are no longer changing significantly.

The last task in power flow is to calculate the generated reactive powers, the swing bus active power output and the line flows. The generated powers can be calculated directly from the power flow equations:

$\begin{array}{ll}{P}_i^{inj}\hfill & =V_i{\displaystyle \sum _{j=1}^N{V}_j{Y}_{ij}\cos \left({\text{θ}}_i-{\text{θ}}_j-{\varphi }_{ij}\right)}\hfill \\ Q_i^{inj}\hfill & =V_i{\displaystyle \sum _{j=1}^N{V}_j{Y}_{ij}\sin \left({\text{θ}}_i-{\text{θ}}_j-{\varphi }_{ij}\right)}\hfill \end{array}$

Therefore

$\begin{array}{l}{P}_{gen,1}=P_1^{inj}=0.7087\hfill \\ Q_{gen,1}=Q_1^{inj}=0.2806\hfill \\ Q_{gen,2}=Q_2^{inj}=-0.0446\hfill \end{array}$

The total active power losses in the system are the difference between the sum of the generation and the sum of the loads, in this case:

$P_{\text{loss}}={\displaystyle \sum P_{\text{gen}}-{\displaystyle \sum P_{\text{load}}=0.7087+0.5-1.2=0.0087\text{pu}}}$

(5.32)

The line losses for line i–j are calculated at both the sending and receiving ends of the line. The apparent power leaving bus i to bus j on line i–j is

$S_{ij}=V_i\angle {\text{δ}}_i{\left(I_{ij}+j\frac{B_{ij}}{2}{V}_i\angle {\text{θ}}_i\right)}^{\star }$

(5.33)

$=V_i\angle {\text{δ}}_i{\left(\left(\frac{V_i\angle {\text{θ}}_i-V_j\angle {\text{θ}}_j}{R_{ij}+j{X}_{ij}}\right)+j\frac{B_{ij}}{2}{V}_i\angle {\text{θ}}_i\right)}^{\star }$

(5.34)

and the power received at bus j from bus i on line i–j is

$S_{ji}=V_j\angle {\text{δ}}_j{\left(I_{ji}-j\frac{B_{ij}}{2}{V}_j\angle {\text{θ}}_j\right)}^{\star }$

(5.35)

Thus

$P_{ij}=\mathrm{Re}\left\{S_{ij}\right\}=V_i{V}_j{Y}_{ij}\cos \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)-V_i^2{Y}_{ij}\cos {\varphi }_{ij}$

(5.36)

$Q_{ij}=\mathrm{Im}\left\{S_{ij}\right\}=V_i{V}_j{Y}_{ij}\sin \left({\text{δ}}_i-{\text{δ}}_j-{\varphi }_{ij}\right)-V_i^2\left(Y_{ij}\sin {\varphi }_{ij}-\frac{B_{ij}}{2}\right)$

(5.37)

Similarly, the powers P_ji and Q_ji can be calculated. The active power loss on any given line is the difference between the active power sent from bus i and the active power received at bus j. Calculating the reactive power losses is more complex since the reactive power generated by the line-charging (shunt capacitances) must also be included.

5.2    Optimal Power Flow

The basic objective of the OPF is to find the values of the system state variables and/or parameters that minimize some cost function of the power system. The types of cost functions are system dependent and can vary widely from application to application and are not necessarily strictly measured in terms of dollars. Examples of engineering optimizations can range from minimizing

•  Active power losses

•  Particulate output (emissions)

•  System energy

•  Fuel costs of generation

to name a few possibilities. The basic formulation of the OPF can be represented as minimizing a defined cost function subject to any physical or operational constraints of the system:

minimize

$\begin{array}{cc}f\left(x,u\right)& x\in R^n\\ & u\in R^n\end{array}$

(5.38)

subject to

$\begin{array}{cc}g\left(x,u\right)=0& \text{equality constraints}\end{array}$

(5.39)

$\begin{array}{cc}h\left(x,u\right)=0& \text{inequality constraints}\end{array}$

(5.40)

where

x is the vector of system states

u is the vector of system parameters

The basic approach is to find the vector of system parameters that when substituted into the system model will result in the state vector x that minimizes the cost function f (x,u).

In an unconstrained system, the usual approach to minimizing the cost function is to set the function derivatives to zero and then solve for the system states from the set of resulting equations. In the majority of applications, however, the system states at the unconstrained minimum will not satisfy the constraint equations. Thus, an alternate approach is required to find the constrained minimum. One approach is to introduce an additional set of parameters λ, frequently known as Lagrange multipliers, to impose the constraints on the cost function. The augmented cost function then becomes

$\text{minimize }f\left(x,u\right)-\text{λ}g\left(x,u\right)$

(5.41)

The augmented function in Equation 5.41 can then be minimized by solving for the set of states that result from setting the derivatives of the augmented function to zero. Note that the derivative of Equation 5.41 with respect to λ effectively enforces the equality constraint of Equation 5.39.

5.2.1    Steepest Descent Algorithm

1.  Let k = 0. Guess an initial vector u^k = u⁰.

2.  Solve the (possibly nonlinear) system of Equation 5.45 for a feasible solution x.

3.  Calculate C^k+1 and ∇C^k+1 from Equation 5.50. If ║C^k+1 – C^k║ is less than some predefined tolerance, stop.

4.  Calculate the new vector u^{k+1 =} u^k – γ∇C, where γ is a positive number, which is the user-defined “stepsize” of the algorithm.

5.  k = k + 1. Go to step 2.

In the steepest descent method, the u vector update direction is determined at each step of the algorithm by choosing the direction of the greatest change of the augmented cost function C^⋆. The direction of steepest descent is perpendicular to the tangent of the curve of constant cost. The distance between adjustments is analogous to the stepsize γ of the algorithm. Thus the critical part of the steepest descent algorithm is the choice of γ. If γ is chosen small, then convergence to minimum value is more likely, but may require many iterations, whereas a large value of γ may result in oscillations about the minimum.

Many power system applications, such as the power flow, offer only a snapshot of the system operation. Frequently, the system planner or operator is interested in the effect that making adjustments to the system parameters will have on the power flow through lines or system losses. Rather than making the adjustments in a random fashion, the system planner will attempt to optimize the adjustments according to some objective function. This objective function can be chosen to minimize generating costs, reservoir water levels, or system losses, among others. The OPF problem is to formulate the power flow problem to find system voltages and generated powers within the framework of the objective function. In this application, the inputs to the power flow are systematically adjusted to maximize (or minimize) a scalar function of the power flow state variables. The two most common objective functions are minimization of generating costs and minimization of active power losses.

The time frame of OPF is on the order of minutes to 1 h; therefore it is assumed that the optimization occurs using only those units that are currently on-line. The problem of determining whether or not to engage a unit, at what time, and for how long is part of the unit commitment problem and is not covered here. The minimization of active transmission losses saves both generating costs and creates a higher generating reserve margin.

Often the steepest descent method may indicate that either states or inputs lie outside of their physical constraints. For example, the algorithm may result in a power generation value that exceeds the physical maximum output of the generating unit. Similarly, the resulting bus voltages may lie outside of the desired range (usually ±10% of unity). These are violations of the inequality constraints of the problem. In these cases, the steepest descent algorithm must be modified to reflect these physical limitations. There are several approaches to account for limitations and these approaches depend on whether or not the limitation is on the input (independent) or on the state (dependent).

5.2.2    Limitations on Independent Variables

If the application of the steepest descent algorithm results in an updated value of input that exceeds the specified limit, then the most straightforward method of handling this violation is simply to set the input state equal to its limit and continue with the algorithm except with one less degree of freedom.

5.2.3    Limitations on Dependent Variables

In many cases, the physical limitations of the system are imposed upon states that are dependent variables in the system description. In this case, the inequality equations are functions of x and must be added to the cost function. Examples of limitations on dependent variables include maximum line flows or bus voltage levels. In these cases, the value of the states cannot be independently set, but must be enforced indirectly. One method of enforcing an inequality constraint is to introduce a penalty function into the cost function. A penalty function is a function that is small when the state is far away from its limit, but becomes increasingly larger the closer the state is to its limit. Typical penalty functions include

$\begin{array}{cc}p\left(h\right)=e^{kh}& k>0\end{array}$

(5.75)

$\begin{array}{cc}p\left(h\right)=x^{2n}{e}^{kh}& n,k>0\end{array}$

(5.76)

$\begin{array}{cc}p\left(h\right)=a{x}^{2n}{e}^{kh}+b{e}^{kh}& n,k,a,b>0\end{array}$

(5.77)

and the cost function becomes

$C\star :C\left(u,x\right)+{\text{λ}}^{T}g\left(u,x\right)+p\left(h\left(u,x\right)-h^{\max }\right)$

(5.78)

This cost equation is then minimized in the usual fashion by setting the appropriate derivatives to zero. This method not only has the advantage of simplicity of implementation, but also has several disadvantages. The first disadvantage is that the choice of penalty function is often a heuristic choice and can vary by application. A second disadvantage is that this method cannot enforce hard limitations on states, i.e., the cost function becomes large if the maximum is exceeded, but the state is allowed to exceed its maximum. In many applications this is not a serious disadvantage. If the power flow on a transmission line slightly exceeds its maximum, it is reasonable to assume that the power system will continue to operate, at least for a finite length of time. If, however, the physical limit is the height above ground for an airplane, then even a slightly negative altitude will have dire consequences. Thus the use of penalty functions to enforce limits must be used with caution and is not applicable for all systems.

In the case where hard limits must be imposed, an alternate approach to enforcing the inequality constraints must be employed. In this approach, the inequality constraints are added as additional equality constraints with the inequality set equal to the limit (upper or lower) that is violated. This in essence introduces an additional set of Lagrangian multipliers. This is often referred to as the dual-variable approach, because each inequality has the potential of resulting in two equalities: one for the upper limit and one for the lower limit. However, the upper and lower limit cannot be simultaneously violated; thus, out of the possible set of additional Lagrangian multipliers only one of the two will be included at any given operating point and thus the dual limits are mutually exclusive.

5.3    State Estimation

In many physical systems, the system operating condition cannot be determined directly by an analytical solution of known equations using a given set of known, dependable quantities. More frequently, the system operating condition is determined by the measurement of system states at different points throughout the system. In many systems, more measurements are made than are necessary to uniquely determine the operating point. This redundancy is often purposely designed into the system to counteract the effect of inaccurate or missing data due to instrument failure. Conversely, not all of the states may be available for measurement. High temperatures, moving parts, or inhospitable conditions may make it difficult, dangerous, or expensive to measure certain system states. In this case, the missing states must be estimated from the rest of the measured information of the system. This process is often known as state estimation and is the process of estimating unknown states from measured quantities. State estimation gives the “best estimate” of the state of the system in spite of uncertain, redundant, and/or conflicting measurements. A good state estimation will smooth out small random errors in measurements, detect and identify large measurement errors, and compensate for missing data. This process strives to minimize the error between the (unknown) true operating state of the system and the measured states.

The set of measured quantities can be denoted by the vector z, which may include measurements of system states (such as voltage and current) or quantities that are functions of system states (such as power flows). Thus,

$z^{\text{true}}=Ax$

(5.87)

where

x is the set of system states

A is usually not square

The error vector is the difference between the measured quantities z and the true quantities:

$e=z-z^{\text{true}}=z-Ax$

(5.88)

Typically, the minimum of the square of the error is desired to negate any effects of sign differences between the measured and true values. Thus, a state estimator endeavors to find the minimum of the squared error, or a least squares minimization:

$\text{minimize}{\Vert e\Vert }^2=e^T\cdot e={\displaystyle \sum _{i=1}^m{\left[z_i-{\displaystyle \sum _{j=1}^m{a}_{ij}{x}_j}\right]}^2}$

(5.89)

The squared error function can be denoted by U(x) and is given by

$U\left(x\right)=e^T\cdot e={\left(z-Ax\right)}^T\left(z-Ax\right)$

(5.90)

$=\left(z^T-x^T{A}^T\right)\left(z-Ax\right)p$

(5.91)

$=\left(z^{T}z-z^{T}Ax-x^T{A}^{T}z+x^T{A}^{T}Ax\right)$

(5.92)

Note that the product z^T Ax is a scalar and so it can be equivalently written as

$z^{T}Ax={\left(z^{T}Ax\right)}^T=x^T{A}^{T}zp$

Therefore the squared error function is given by

$U\left(x\right)=z^{T}z-2x^T{A}^{T}z+x^T{A}^{T}Ax$

(5.93)

The minimum of the squared error function can be found by an unconstrained optimization where the derivative of the function with respect to the states x is set to zero:

$\frac{\partial U\left(x\right)}{\partial x}=0-2A^{T}z+2A^{T}z+2A^{T}Ax$

(5.94)

Thus,

$A^{T}Ax=A^{T}z$

(5.95)

Thus, if b = A^Tz and  = A^T A, then

$\widehat{A}x=b$

(5.96)

This state vector x is the best estimate (in the squared error) to the system operating condition from which the measurements z were taken. The measurement error is given by

$e=z^{\text{meas}}-Ax$

(5.97)

In power system state estimation, the estimated variables are the voltage magnitudes and the voltage phase angles at the system buses. The input to the state estimator is the active and reactive powers of the system, measured either at the injection sites or on the transmission lines. The state estimator is designed to give the best estimates of the voltages and phase angles minimizing the effects of the measurement errors. All instruments add some degree of error to the measured values, but the problem is how to quantify this error and account for it during the estimation process.

If all measurements are treated equally in the least squares solution, then the less accurate measurements will affect the estimation as significantly as the more accurate measurements. As a result, the final estimation may contain large errors due to the influence of inaccurate measurements. By introducing a weighting matrix to emphasize the more accurate measurements more heavily than the less accurate measurements, the estimation procedure can then force the results to coincide more closely with the measurements of greater accuracy. This leads to the weighted least squares estimation:

$\text{minimize}{\Vert e\Vert }^2=e^T\cdot e={\displaystyle \sum _{i=1}^m{w}_i}{\left[z_i-{\displaystyle \sum _{j=1}^m{a}_{ij}{x}_j}\right]}^2$

(5.98)

where w_i is a weighting factor reflecting the level of confidence in the measurement z_i.

In general, it can be assumed that the introduced errors have normal (Gaussian) distribution with zero mean and that each measurement is independent of all other measurements. This means that each measurement error is as likely to be greater than the true value as it is to be less than the true value. A zero mean Gaussian distribution has several attributes. The standard deviation of a zero mean Gaussian distribution is denoted by σ. This means that 68% of all measurements will fall within ±σ of the expected value, which is zero in a zero mean distribution. Further, 95% of all measurements will fall within ±2σ and 99% of all measurements will fall within ±3σ. The variance of the measurement distribution is given by σ². This implies that if the variance of the measurements is relatively small, then the majority of measurements are close to the mean. One interpretation of this is that accurate measurements lead to small variance in the distribution.

This relationship between accuracy and variance leads to a straightforward approach from which a weighting matrix for the estimation can be developed. With measurements taken from a particular meter, the smaller the variance of the measurements (i.e., the more consistent they are), the greater the level of confidence in that set of measurements. A set of measurements that have a high level of confidence should have a higher weighting than a set of measurements that have a larger variance (and therefore less confidence). Therefore, a plausible weighting matrix that reflects the level of confidence in each measurement set is the inverse of the covariance matrix. Thus, measurements that come from instruments with good consistency (small variance) will carry greater weight than measurements that come from less accurate instruments (high variance). Thus, one possible weighting matrix is given by

$W=R^{-1}=\left[\begin{array}{cccc}\frac{1}{{\text{σ}}_1^2}& 0& \dots & 0\\ 0& \frac{1}{{\text{σ}}_2^2}& \dots & 0\\ ⋮& ⋮& ⋮& ⋮\\ 0& 0& \dots & \frac{1}{{\text{σ}}_m^2}\end{array}\right]$

(5.99)

where R is the covariance matrix for the measurements.

As in the linear least squares estimation, the nonlinear least squares estimation attempts to minimize the square of the errors between a known set of measurements and a set of weighted nonlinear functions:

$\text{minimize}f={\Vert e\Vert }^2=e^T\cdot e={\displaystyle \sum _{i=1}^m\frac{1}{{\text{σ}}^2}{\left[z_i-h_i\left(x\right)\right]}^2}$

(5.100)

where x ∈ Rⁿ is the vector of unknowns to be estimated, z ∈ R^m is the vector of measurements, ${\text{σ}}_i^2$ is the variance of the ith measurement, and h(x) is the nonlinear function vector relating x to z, where the measurement vector z can be a set of geographically distributed measurements, such as voltages and power flows

In state estimation, the unknowns in the nonlinear equations are the state variables of the system. The state values that minimize the error are found by setting the derivatives of the error function to zero:

$F\left(x\right)=H_x^T{R}^{-1}\left[z-h\left(x\right)\right]=0$

(5.101)

where

$H_x=\left[\begin{array}{cccc}\frac{\partial h_1}{\partial x_1}& \frac{\partial h_1}{\partial x_2}& \dots & \frac{\partial h_1}{\partial x_n}\\ \frac{\partial h_2}{\partial x_1}& \frac{\partial h_2}{\partial h_2}& \dots & \frac{\partial h_2}{\partial x_n}\\ ⋮& ⋮& ⋮& ⋮\\ \frac{\partial h_m}{\partial x_1}& \frac{\partial h_m}{\partial x_2}& \dots & \frac{\partial h_m}{\partial x_n}\end{array}\right]$

(5.102)

Note that Equation 5.101 is a set of nonlinear equations that must be solved using Newton–Raphson or another iterative numerical solver. In this case, the Jacobian of F(x) is

$J_F\left(x\right)=H_x^T\left(x\right)R^{-1}\frac{\partial }{\partial x}\left[z-h\left(x\right)\right]$

(5.103)

$=-H_x^T\left(x\right)R^{-1}{H}_x\left(x\right)$

(5.104)

and the Newton–Raphson iteration becomes

$\left[H_x^T\left(x^k\right)R^{-1}{H}_x\left(x^k\right)\right]\left[x^{k-1}-x^k\right]=H_x^T\left(x^k\right)R^{-1}\left[z-h{\left(x\right)}^k\right]$

(5.105)

At convergence, xk + 1 is equal to the set of states that minimize the error function f of Equation 5.100.

This concludes the discussion of the most commonly used computational methods for power system analysis. This chapter describes only the basic approaches to power flow, OPF, and state estimation. These methods have been utilized for several decades, yet improvements in accuracy and speed are constantly being proposed in the technical literature.