Interval harmonic power flow for microgrids: an approach using the second-order Taylor series

Abstract

The analysis of harmonic distortions and their propagation in electric power systems is crucial for power quality assessment. Harmonic power flow (HPF) is a widely used technique for this purpose. However, most existing research on HPF focuses on deterministic algorithms, and only a few studies have considered uncertainties associated with harmonic sources connected to microgrids. Due to the lack of recent approaches for evaluating the impact of uncertain data on HPF results, this paper proposes a new methodology for calculating harmonic voltages and distortions in microgrids by assuming uncertain parameters for both loads and harmonic sources. To obtain interval results for harmonic voltages, the load flow equations are expanded up to the second-order terms using Taylor series, considering all partial derivatives of the system variables with respect to a percentage uncertainty value. The proposed method is evaluated using simulations on a 33-bus distribution system, considering the microgrid in both islanded mode and connected to a main power substation. As demonstrated, the interval results are similar to those obtained from Monte Carlo simulations, proving the efficiency of the proposed method with robust solution and reduced computational time.

A First and second derivatives

The first derivatives of the active and reactive powers ($P_k$ and $Q_k$) in relation to the state variables that make up the Jacobian matrix are presented from Eqs. (35) to (46) where $G_{km}$ is the conductance of the line connecting the bus k to m; $B_{km}$ is the susceptibility of the line connecting the bus k to m; $V_k$ is the magnitude of the voltage on bus k and $\theta _k$ are the stress angles on bus k.

$$\begin{aligned}{} & {} \frac{{\partial P}_k}{{\partial \theta }_m}=\ V_k{V_m(G}_{{km}_{\ }}{\textrm{sen} {\theta }_{km}\ }-\ B_{km}{\textrm{cos} {\theta }_{km}\ }) \end{aligned}$$

(35)

$$\begin{aligned}{} & {} \frac{{\partial P}_k}{{\partial \theta }_k}=\ -{V_k}^2B_{kk}-Q_k \end{aligned}$$

(36)

$$\begin{aligned}{} & {} \frac{{\partial P}_k}{{\partial V}_m}=\ V_k{(G}_{{km}_{\ }}{\textrm{cos} {\theta }_{km}\ }+\ B_{km}{\textrm{sen} {\theta }_{km}\ }) \end{aligned}$$

(37)

$$\begin{aligned}{} & {} \frac{{\partial P}_k}{{\partial V}_k}=\ \frac{P_k+{V_k}^2G_{kk}}{V_k} \end{aligned}$$

(38)

$$\begin{aligned}{} & {} \frac{{\partial P}_k}{{\partial P_{g_k}}}=-1 \end{aligned}$$

(39)

$$\begin{aligned}{} & {} \frac{{\partial P}_k}{{\partial Q_{g_k}}}=\frac{{\partial P}_k}{{\partial f}}=0 \end{aligned}$$

(40)

$$\begin{aligned}{} & {} \frac{{\partial Q}_k}{{\partial \theta }_m}=\ {-V}_k{V_m(G}_{{km}_{\ }}{\textrm{cos} {\theta }_{km}\ }+\ B_{km}{\textrm{sen} {\theta }_{km}\ }) \end{aligned}$$

(41)

$$\begin{aligned}{} & {} \frac{{\partial Q}_k}{{\partial \theta }_k}=\ P_k-{V_k}^2G_{kk} \end{aligned}$$

(42)

$$\begin{aligned}{} & {} \frac{{\partial Q}_k}{{\partial V}_m}=\ V_k{(G}_{{km}_{\ }}{\textrm{sen} {\theta }_{km}\ }-\ B_{km}{\textrm{cos} {\theta }_{km}\ }) \end{aligned}$$

(43)

$$\begin{aligned}{} & {} \frac{{\partial Q}_k}{{\partial V}_k}=\ \frac{Q_k-{V_k}^2.B_{kk}}{V_k} \end{aligned}$$

(44)

$$\begin{aligned}{} & {} \frac{{\partial Q}_k}{{\partial Q_{g_k}}}=-1 \end{aligned}$$

(45)

$$\begin{aligned}{} & {} \frac{{\partial Q}_k}{{\partial P_{g_k}}}=\frac{{\partial Q}_k}{{\partial f}}=0 \end{aligned}$$

(46)

The active power generation at PV buses is modelled conform in which Eq. (1) which considers the steady state droop frequency regulation mathematical for all generating units. Thus, the partial derivative of the frequency control equation in relation to the active power generated by the DG and in with respect to the system frequency corresponds to (47) and (48), respectively.

$$\begin{aligned}{} & {} \frac{{\partial y}'_k}{{\partial P_{g_k}}}=1 \end{aligned}$$

(47)

$$\begin{aligned}{} & {} \frac{{\partial y}'_k}{{\partial f}}=K_f \end{aligned}$$

(48)

As $y'_k$, is not a function of $\theta _k$, $V_k$ and $Q_g$ then the corresponding partial derivatives are given by:

$$\begin{aligned} \frac{{\partial y}'_k}{{\partial \theta _k}}=\frac{{\partial y}'_k}{{\partial V_k}}=\frac{{\partial y}'_k}{{\partial Q_{g_k}}}=0 \end{aligned}$$

(49)

The partial voltage derivatives of each DG in the system correspond to:

$$\begin{aligned}{} & {} \frac{{\partial V}'_k}{{\partial V_k}}=1 \end{aligned}$$

(50)

$$\begin{aligned}{} & {} \frac{{\partial V}'_k}{{\partial \theta _k}}=\frac{{\partial V}'_k}{{\partial P_{g_k}}}=\frac{{\partial V}'_k}{{\partial Q_{g_k}}}=\frac{{\partial V}'_k}{{\partial f}}=0 \end{aligned}$$

(51)

The slack bus is a single bus in the network and can be chosen for any bar i, where the partial derivatives with respect to $\theta _i$, $V_k$, $P_{g_k}$, $Q_{g_k}$ and f are given by:

$$\begin{aligned}{} & {} \frac{{\partial \theta }'_i}{{\partial \theta _i}}=1 \end{aligned}$$

(52)

$$\begin{aligned}{} & {} \frac{{\partial \theta }'_i}{{\partial V_k}}=\frac{{\partial \theta }'_i}{{\partial P_{g_k}}}=\frac{{\partial \theta }'_i}{{\partial Q_{g_k}}}=\frac{{\partial \theta }'_i}{{\partial f}}=0 \end{aligned}$$

(53)

The residues are formed according to Eqs. (54) to (58).

$$\begin{aligned}{} & {} \Delta P_k=P_k^{esp}-P_k^{calc} \end{aligned}$$

(54)

$$\begin{aligned}{} & {} \Delta Q_k=Q_k^{esp}-Q_k^{calc} \end{aligned}$$

(55)

$$\begin{aligned}{} & {} \Delta y'_{ng}=P_{g_{ng}}^{esp}-P_{g_{ng}}^{calc}-k_f \cdot (f^{esp}-f) \end{aligned}$$

(56)

$$\begin{aligned}{} & {} \Delta V'_{ng}=V_{ng}^{esp}-V_{ng}^{calc} \end{aligned}$$

(57)

$$\begin{aligned}{} & {} \Delta \theta '_i=\theta _{i}^{esp}-\theta _i^{calc} \end{aligned}$$

(58)

Equation (2) is then solved and the state variables are updated at each iteration it, as seen from Eqs. (59) - (63), as follows:

$$\begin{aligned}{} & {} {\theta _k}^{(it+1)}={\theta _k}^{(it)}+{\Delta \theta _k}^{(it)} \end{aligned}$$

(59)

$$\begin{aligned}{} & {} {V_k}^{((it)+1)}={V_k}^{(it)}+{\Delta V_k}^{(it)} \end{aligned}$$

(60)

$$\begin{aligned}{} & {} {P_{g_k}}^{(it+1)}={P_{g_k}}^{(it)}+{\Delta P_{g_k}}^{(it)} \end{aligned}$$

(61)

$$\begin{aligned}{} & {} {Q_{g_k}}^{(it+1)}={Q_{g_k}}^{(it)}+{\Delta Q_{g_k}}^{(it)} \end{aligned}$$

(62)

$$\begin{aligned}{} & {} f^{(it+1)}=f^{(it)}+ \Delta f^{(it)} \end{aligned}$$

(63)

The second derivatives of the active and reactive powers with respect to the state variables are provided by Equations from (64) to (117):

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _k^2}=-P_k+V_k^2G_{kk} \end{aligned}$$

(64)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _k \partial \theta _m}=\frac{\partial ^2 P_k}{\partial \theta _m \partial \theta _k}=V_kV_m(G_{km}cos\theta _{km}+B_{km}sen\theta _{km})\nonumber \\ \end{aligned}$$

(65)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial V_k \partial \theta _k}=-\frac{Q_k}{V_k}-V_kB_{kk} \end{aligned}$$

(66)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial V_m \partial \theta _k}=\frac{\partial ^2 P_k}{\partial \theta _k \partial V_m}=V_k(-G_{km}sen\theta _{km}+B_{km}cos\theta _{km})\nonumber \\ \end{aligned}$$

(67)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _m^2}=V_kV_m(-G_{km}cos\theta _{km}-B_{km}sen\theta _{km}) \end{aligned}$$

(68)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial V_k \partial \theta _m}=\frac{\partial ^2 P_k}{\partial \theta _m \partial V_k}=V_m(G_{km}sen\theta _{km}-B_{km}cos\theta _{km})\nonumber \\ \end{aligned}$$

(69)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial V_m \partial \theta _m}=\frac{\partial ^2 P_k}{\partial \theta _m \partial V_m}=V_k(G_{km}sen\theta _{km}-B_{km}cos\theta _{km})\nonumber \\ \end{aligned}$$

(70)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial V_k^2 }=2G_{kk} \end{aligned}$$

(71)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial V_m \partial V_k}=\frac{\partial ^2 P_k}{\partial V_k \partial V_m}=(G_{km}cos\theta _{km}+B_{km}sen\theta _{km}) \end{aligned}$$

(72)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial V_m^2 }=0 \end{aligned}$$

(73)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial V_m \partial f}=\frac{\partial ^2 P_k}{\partial V_k \partial f}=\frac{\partial ^2 P_k}{\partial f \partial V_m}=\frac{\partial ^2 P_k}{\partial f \partial V_k}=0\nonumber \\ \end{aligned}$$

(74)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _k \partial Q_{g_k}}=\frac{\partial ^2 P_k}{ \partial Q_{g_k} \partial \theta _k}=-1 \end{aligned}$$

(75)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _k \partial P_{g_k}}=\frac{\partial ^2 P_k}{\partial P_{g_k}\partial \theta _k}=0 \end{aligned}$$

(76)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _k \partial P_{g_m}}=\frac{\partial ^2 P_k}{\partial P_{g_m} \partial \theta _k}=0 \end{aligned}$$

(77)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _k \partial Q_{g_k}}=\frac{\partial ^2 P_k}{ \partial Q_{g_k} \partial \theta _k}=0 \end{aligned}$$

(78)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _m \partial P_{g_k}}=\frac{\partial ^2 P_k}{\partial P_{g_k}\partial \theta _m}=0 \end{aligned}$$

(79)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _m \partial P_{g_m}}=\frac{\partial ^2 P_k}{\partial P_{g_m} \partial \theta _m}=0 \end{aligned}$$

(80)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _m \partial Q_{g_k}}=\frac{\partial ^2 P_k}{ \partial Q_{g_k} \partial \theta _m}=0 \end{aligned}$$

(81)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _m \partial Q_{g_m}}=\frac{\partial ^2 P_k}{ \partial Q_{g_m} \partial \theta _m}=0 \end{aligned}$$

(82)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{ \partial V_k \partial P_{g_k} }=\frac{\partial ^2 P_k}{ \partial P_{g_k} \partial V_k}=\frac{1}{V_k} \end{aligned}$$

(83)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{ \partial V_k \partial P_{g_m} }=\frac{\partial ^2 P_k}{ \partial P_{g_m} \partial V_k}=0 \end{aligned}$$

(84)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{ \partial V_k \partial Q_{g_k} }=\frac{\partial ^2 P_k}{ \partial Q_{g_k} \partial V_k}=0 \end{aligned}$$

(85)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{ \partial V_k \partial Q_{g_m} }=\frac{\partial ^2 P_k}{ \partial Q_{g_m} \partial V_k}=0 \end{aligned}$$

(86)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{ \partial V_m \partial P_{g_k} }=\frac{\partial ^2 P_k}{ \partial P_{g_k} \partial V_m}=0 \end{aligned}$$

(87)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{ \partial V_m \partial P_{g_m} }=\frac{\partial ^2 P_k}{ \partial P_{g_m} \partial V_m}=0 \end{aligned}$$

(88)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{ \partial V_m \partial Q_{g_k} }=\frac{\partial ^2 P_k}{ \partial Q_{g_k} \partial V_m}=0 \end{aligned}$$

(89)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{ \partial V_m \partial Q_{g_m} }=\frac{\partial ^2 P_k}{ \partial Q_{g_m} \partial V_m}=0 \end{aligned}$$

(90)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _k^2}=-Q_k-V_k^2B_{kk} \end{aligned}$$

(91)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _k \partial \theta _m}=\frac{\partial ^2 Q_k}{\partial \theta _m \partial \theta _k}=V_kV_m(G_{km}sen\theta _{km}-B_{km}cos\theta _{km})\nonumber \\ \end{aligned}$$

(92)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial V_k \partial \theta _k}=\frac{P_k}{V_k}-V_kG_{kk} \end{aligned}$$

(93)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial V_m \partial \theta _k}=\frac{\partial ^2 Q_k}{\partial \theta _k \partial V_m}=V_k(G_{km}cos\theta _{km}+B_{km}sen\theta _{km})\nonumber \\ \end{aligned}$$

(94)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _m^2}=V_kV_m(-G_{km}sen\theta _{km}+B_{km}cos\theta _{km}) \end{aligned}$$

(95)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial V_k \partial \theta _m}=\frac{\partial ^2 Q_k}{\partial \theta _m \partial V_k}=V_m(-G_{km}cos\theta _{km}-B_{km}sen\theta _{km})\nonumber \\ \end{aligned}$$

(96)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial V_m \partial \theta _m}=\frac{\partial ^2 Q_k}{\partial \theta _m \partial V_m}=V_k(-G_{km}cos\theta _{km}-B_{km}sen\theta _{km})\nonumber \\ \end{aligned}$$

(97)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial V_k^2 }=2B_{kk} \end{aligned}$$

(98)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial V_m \partial V_k}=\frac{\partial ^2 Q_k}{\partial V_k \partial V_m}=(G_{km}sen\theta _{km}-B_{km}cos\theta _{km}) \end{aligned}$$

(99)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial V_m^2 }=0 \end{aligned}$$

(100)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial V_m \partial f}=\frac{\partial ^2 Q_k}{\partial V_k \partial f}=\frac{\partial ^2 Q_k}{\partial f \partial V_m}=\frac{\partial ^2 Q_k}{\partial f \partial V_k}=0 \end{aligned}$$

(101)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _k \partial P_{g_k}}=\frac{\partial ^2 Q_k}{ \partial P_{g_k} \partial \theta _k}=1 \end{aligned}$$

(102)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _k \partial P_{g_m}}=\frac{\partial ^2 Q_k}{\partial P_{g_m}\partial \theta _k}=0 \end{aligned}$$

(103)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _k \partial Q_{g_m}}=\frac{\partial ^2 Q_k}{\partial Q_{g_m} \partial \theta _k}=\frac{\partial ^2 P_k}{\partial \theta _k \partial Q_{g_k}}=\frac{\partial ^2 P_k}{ \partial Q_{g_k} \partial \theta _k}=0\nonumber \\ \end{aligned}$$

(104)

$$\begin{aligned}{} & {} \frac{\partial ^2 P_k}{\partial \theta _k \partial Q_{g_k}}=\frac{\partial ^2 P_k}{ \partial Q_{g_k} \partial \theta _k}=0 \end{aligned}$$

(105)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _m \partial P_{g_k}}=\frac{\partial ^2 Q_k}{\partial P_{g_k}\partial \theta _m}=0 \end{aligned}$$

(106)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _m \partial P_{g_m}}=\frac{\partial ^2 Q_k}{\partial P_{g_m} \partial \theta _m}=0 \end{aligned}$$

(107)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{\partial \theta _m \partial Q_{g_m}}=\frac{\partial ^2 Q_k}{ \partial Q_{g_k} \partial \theta _m}=0 \end{aligned}$$

(108)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial Q_{g_m} \partial \theta _m}=\frac{\partial ^2 Q_k}{\partial \theta _m \partial Q_{g_m} }=0 \end{aligned}$$

(109)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial V_k \partial Q_{g_k} }=\frac{\partial ^2 Q_k}{ \partial Q_{g_k} \partial V_k}=\frac{1}{V_k} \end{aligned}$$

(110)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial V_k \partial P_{g_m} }=\frac{\partial ^2 Q_k}{ \partial P_{g_m} \partial V_k}=0 \end{aligned}$$

(111)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial V_k \partial P_{g_k} }=\frac{\partial ^2 Q_k}{ \partial P_{g_k} \partial V_k}=0 \end{aligned}$$

(112)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial V_k \partial Q_{g_m} }=\frac{\partial ^2 Q_k}{ \partial Q_{g_m} \partial V_k}=0 \end{aligned}$$

(113)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial V_m \partial P_{g_k} }=\frac{\partial ^2 Q_k}{ \partial P_{g_k} \partial V_m}=0 \end{aligned}$$

(114)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial V_m \partial P_{g_m} }=\frac{\partial ^2 Q_k}{ \partial P_{g_m} \partial V_m}=0 \end{aligned}$$

(115)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial V_m \partial Q_{g_k} }=\frac{\partial ^2 Q_k}{ \partial Q_{g_k} \partial V_m}=0 \end{aligned}$$

(116)

$$\begin{aligned}{} & {} \frac{\partial ^2 Q_k}{ \partial V_m \partial Q_{g_m} }=\frac{\partial ^2 Q_k}{ \partial Q_{g_m} \partial V_m}=0 \end{aligned}$$

(117)