A novel approach for harmonic responsibility assessment based on an optimization model

Abstract

This paper presents a novel approach for harmonic responsibility assessment based on an optimization problem solved by safety barrier interior point method. In the proposed methodology, a power quality monitor is installed at the point of common coupling of a distribution system Norton equivalent circuit in order to measure voltage and current phasors continuously obtained from the network in regular time intervals. In order to determine customer and utility contributions, a state variables vector is formed by harmonic currents and impedances from both sides being modeled in rectangular coordinates. The objective function to be minimized is the sum of the quadratic differences between measured current and voltage phasors and their corresponding values calculated as function of the state variables. Uncertainties associated with the network impedances are handled as inequality constraints of the optimization model. Based on superposition principle, indices are proposed for determining harmonic responsibility sharing based on the estimated variables. Results are validated by different case studies considering the effect of measurement accuracy, uncertain parameters and number of samples used for the estimation process.

Appendix A: Safety barrier interior point method

Generally, the optimization problem can be formulated as Eq. (41):

$$\begin{aligned} \begin{array}{llll} \hbox {min} \ J(x^h) &{}\\ \mathbf{subject} to (s.t.): &{}\\ e(x^h)=0&{}\\ l\le x^h \le u&{}\\ \end{array} \end{aligned}$$

(41)

where $x^h$ is the state variables vector for a given harmonic order h, J is the objective function, $e(x^h)$ represents equality constraints and l and u are lower and upper bounds of the inequality constraints, respectively. The inequality constraints are transformed to equality constraints by introducing slack variables $s_{l}$ and $s_{u}$ as described by equations (42):

$$\begin{aligned} \begin{array}{llllll} \hbox {min} \ J(x^h) &{}\\ \mathbf{s}.t.: &{}\\ e(x^h)=0&{}\\ x^h-s_{l}=l&{}\\ x^h+s_{u}=l&{}\\ s_{l} \ge 0, \ s_{u} \ge 0&{}\\ \end{array} \end{aligned}$$

(42)

where $s_{l}$ and $s_{u}$ are slack variables associated with lower and upper limits. The constraints are eliminated by adding a logarithmic barrier function to the objective function as in (43). It means that variables $s_{l}$ and $s_{u}$ must be greater than zero and inequality constraints can never assume values on the border.

$$\begin{aligned} \begin{array}{llllll} \hbox {min} \ J(x^h)-\mu \sum \limits _{j=1}^{n} \hbox {ln}(s_{l,j})- \mu \sum \limits _{j=1}^{n} \hbox {ln}(s_{u,j}) &{}\\ \mathbf{s}.t.: &{}\\ e(x^h)=0&{}\\ x^h-s_{l}=l&{}\\ x^h+s_{u}=l&{}\\ s_{l}> 0, \ s_{u} > 0&{}\\ \end{array} \end{aligned}$$

(43)

where n is the number of inequality constraints of the problem. Initially, the barrier parameter ($\mu$) assumes a value greater than zero, and at the end of the iterative process, it must be near to zero. The Safety barrier Interior Point method introduces the new safety barrier parameter $\rho$ in (44). Its value is always positive and is initially user defined. Then, the new optimization problem can be written as:

$$\begin{aligned} \begin{array}{llllll} \hbox {min} J(x^h)-\mu \sum \limits _{j=1}^{n} \hbox {ln}(s_{l,j}+\rho )- \mu \sum \limits _{j=1}^{n} \hbox {ln}(s_{u,j}+\rho ) &{}\\ \mathbf{s}.t.: &{}\\ e(x^h)=0&{}\\ x^h-s_{l}=l&{}\\ x^h+s_{u}=l&{}\\ s_{l}> 0, \ s_{u}> 0, \rho > 0&{}\\ \end{array} \end{aligned}$$

(44)

In the formulation, $\rho$ ensures the elimination of the barrier proximity problems and the constraints allows the slack variables $s_{l}$ and $s_{u}$ to assume zero values. The state variables can assume the exact limit values (l or u) to find the optimal solution. Then, the Lagrangian function can be defined as the following expression (45):

$$\begin{aligned} \begin{array}{llll} L = J (\mathbf{x })-\sum \limits _{i=1}^{m}\lambda _i e_i(\mathbf{x })-\mu \sum \limits _{j=1}^{n} \hbox {ln}(s_{l,j}+\rho )\\ \qquad -\mu \sum \limits _{j=1}^n \hbox {ln}(s_{u,j}+\rho )\\ \qquad -\sum \limits _{j=1}^n \pi _{l,j}(x_j-s_{l,j}-l_{l})-\sum \limits _{j=1}^n \pi _{u,j}(x_j+s_{u,j}-l_{l}) \end{array} \end{aligned}$$

(45)

where m is the number of equality constraints, Lagrange multipliers vector associated with the equality constraints are $\lambda$, and $\pi _{l}$ and $\pi _{u}$ are Lagrange multipliers vectors associated with the lower and upper bound variables, and can be calculated as (46):

$$\begin{aligned} \begin{array}{ll} \pi _{l,j}=\frac{\mu }{(s_{l,j}+\rho )}&{}\\ \pi _{u,j}=\frac{\mu }{(s_{u,j}+\rho )}&{}\\ \end{array} \end{aligned}$$

(46)

The first-order optimality conditions for the optimization problem associated with the Newton–Raphson method result in (47):

$$\begin{aligned} \left[ \begin{array}{ll} H_{x^h}&{} F^{T}\\ F &{} 0 \\ \end{array}\right] \left[ \begin{array}{c} \Delta x^h\\ \Delta \lambda \\ \end{array}\right] = \left[ \begin{array}{c} G_{x^h}\\ e(x^h) \\ \end{array}\right] \end{aligned}$$

(47)

where

$$\begin{aligned}&H_{x^h}=\omega (x^h,\lambda )+\sum _{j}^{}\frac{\mu }{(s_{l,j}+\rho )^2}+\sum _{j}^{}\frac{\mu }{(s_{u,j}+\rho )^2} \end{aligned}$$

(48)

$$\begin{aligned}&\omega (x^h,\lambda )=\nabla ^2_{x^h} J(x^h)-\sum \limits _{i=1}^{m}\lambda _i \nabla ^2_{x^h} e_i(x^h) \end{aligned}$$

(49)

$$\begin{aligned}&G_{x^h}=r(x^h,\lambda )+\sum _{j}^{}\frac{\mu }{(s_{l,j}+\rho )}-\sum _{j}^{}\frac{\mu }{(s_{u,j}+\rho )} \end{aligned}$$

(50)

$$\begin{aligned}&r(x^h,\lambda )=-\nabla J(x^h)-\nabla e(x^h)^T \lambda \end{aligned}$$

(51)

The Hessian matrix ($H_{x^h}$) as well as to the Gradient vector($H_{x^h}$) are defined from Eqs. (48)–(51) in which $\nabla$ and $\nabla ^2$ are the partial derivative and the gradient of a given function, respectively. From the solution of the system, $\Delta x^h$ and $\Delta \lambda$ are obtained at each iteration of the process, while the search directions $\Delta s_{l}, \Delta s_{u},\Delta \pi _{l,j},\Delta \pi _{u,j}$ are calculated from Eqs. (52–55)

$$\begin{aligned}&\Delta x^h - \Delta s_{l}=0 \end{aligned}$$

(52)

$$\begin{aligned}&\Delta x^h + \Delta s_{u}=0\end{aligned}$$

(53)

$$\begin{aligned}&\Delta \pi _{l,j}=-\frac{\mu \Delta x^h}{(s_{l,j}+\rho )^2}+ \frac{\mu \Delta \pi _{l,j}}{(s_{l,j}+\rho )^2} \end{aligned}$$

(54)

$$\begin{aligned}&\Delta \pi _{u,j}=-\frac{\mu \Delta x^h}{(s_{u,j}+\rho )^2}+ \frac{\mu \Delta \pi _{u,j}}{(s_{u,j}+\rho )^2} \end{aligned}$$

(55)

The step length of the primal-dual variables is given by (56):

$$\begin{aligned} \begin{array}{cc} \alpha _p=\hbox {min}\left\{ \hbox {min}_{\Delta s_{l,j}<0}\frac{s_{l,j}}{|\Delta s_{l,j}|}, \hbox {min}_{\Delta s_{u,j}<0}\frac{s_{u,j}}{|\Delta s_{u,j}|},1\right\} &{}\\ \\ \alpha _d=\hbox {min}\left\{ \hbox {min}_{\Delta \pi _{l,j}<0}\frac{\pi _{l,j}}{|\Delta \pi _{l,j}|}, \hbox {min}_{\Delta \pi _{u,j}<0}\frac{-\pi _{u,j}}{|\Delta \pi _{u,j}|},1\right\} \end{array} \end{aligned}$$

(56)

The variables are then updated for the next iteration (it.) according to (57):

$$\begin{aligned} \begin{array}{llllll} x^h_{it.+1}=x^h_{it.}+\alpha _p \Delta x^h&{}\\ s_{l,(it.+1)}=s_{l,(it.)+\alpha _p \Delta s_{l}}&{}\\ s_{u,(it.+1)}=s_{u,(it.)+\alpha _p \Delta s_{u}}&{}\\ \lambda _{(it.+1)}=\lambda _{(it.)+\alpha _d \Delta \lambda }&{}\\ \pi _{l,(it.+1)}=\pi _{l,(it.)+\alpha _d \Delta \pi _{l}}&{}\\ \pi _{u,(it.+1)}=\pi _{u,(it.)+\alpha _d \Delta \pi _{u}}&{}\\ \end{array} \end{aligned}$$

(57)

The barrier parameter is updated, considering the duality gap (GAP), and the parameter $\beta$ is introduced to control the decay of the barrier parameter to improve the convergence process as in (58).

$$\begin{aligned} \begin{array}{ll} \mu _{it.+1}=\frac{\beta (\hbox {GAP})+\rho \sum \nolimits _{j=1}^{n}(\pi _{l,j}-\pi _{l,j})}{2n} &{} \\ \hbox {GAP}={\sum \nolimits _{j=1}^{n} (s_{l,j}\pi _{l,j}-s_{u,j}\pi _{u,j})} \end{array} \end{aligned}$$

(58)

The safety barrier parameter is also updated by a factor $\tau$ as in Eq. (59), meaning that the value of the parameter is decreased for the next iteration of the algorithm.

$$\begin{aligned} \rho _{it.+1}=\tau \rho _{it.} \end{aligned}$$

(59)

The initial value of the barrier parameter ($\mu =5$) and minimum value ($\mu =10^{-8}$) are adopted. The average of the minimum and maximum values of $x^h$ was set as the initial value of the primal variables, $x^h_o$. The slack variables $s_{l}$ and $s_{u}$ are calculated from $x^h_o-s_{l}=l$ and $x^h_o-s_{u}=u$ for the process initialization. Lagrange multipliers $\lambda$ are initially set to one, and initial values of the dual variables $\pi _{l,j}$ and $\pi _{u,j}$ are calculated from equations.

The safety barrier ($\rho$) parameter has initial value of $10^{-2}$ and minimum value of $10^{-10}$. Values for $\tau$ and $\beta$ are 0.3 and 0.5, respectively. The convergence criteria adopted is $\mu =10^{-5}$, and the optimization algorithm stops when the change in the state variables vector is less than a tolerance of $10^{-6}$.