A Sensitivity-Based Three-Phase Weather-Dependent Power Flow Algorithm for Networks with Local Controllers— Supplementary

: This supplementary document is part of the original 2-Part manuscript titled “ A Sensitivity-Based Three-Phase Weather-Dependent Power Flow Algorithm for Networks with Local Controllers .” This research focuses on proposing a novel sensitivity-based three-phase weather-dependent power flow algorithm for distribution networks with local voltage controllers (LVCs). The proposed algorithm has four distinct characteristics: a) it considers the three-phase unbalanced nature of distribution systems, b) the operating state of LVCs is calculated using sensitivity parameters, which accelerates the convergence speed of the algorithm, c) it considers the precise switching sequence of LVCs based on their reaction time delays, and d) the nonlinear influence of weather variations in the power flow is also taken into consideration. In this supplementary document, the relevant derivations of the sensitivity parameters are presented to complement the original 2-Part manuscript. In this supplementary document, we present the derivation of the sensitivity parameters. Let us reconsider the simplified network with LVCs presented in Fig. S1 below. It includes an OLTC connected between the buses h - w , three single-phase capacitors connected to the three phases of bus q , and a DG connected to bus g . Furthermore, an SVR is also connected through the 3-bus equivalent circuit to buses p , m , s .

The vector of LVC control variables ( ) and voltages ( ) are defined as: where •

= [
] includes the tap variables of each phase of the OLTC transformer. It is a 3×1 vector, which directly regulates the vector = �| | � � | |� that includes the magnitude of the three-phase voltages of bus w.
where 1 , 2 , 3 are the series windings of phases a, b, c, respectively, while 1 , 2 , 3 are the shunt windings of phases a, b, c, respectively. (S12) A perturbation ( ) at the tap of phase a ( ) of the SVR will cause a perturbation ( ) of the current according to (S13): In Equation (S13), ( ) is calculated in Equation (S14) below: where and are the A and K matrices with unperturbed taps, as shown in Equations (S15) -(S16).
Moreover, in Equation (S13), ( ) is calculated in Equation (S17) below: where and are the A and K matrices with perturbed tap ratios, as follows: In a similar manner, we calculate the perturbations of all current sources, due to the tap perturbation of each phase e.g , , , , , , , . Each of these are 3×1 in dimension. are derived in a similar manner as in the case of SVR described above, using the current sources of the OLTC model proposed in reference [28] of the manuscript. Therefore, a detailed derivation is not deemed necessary here for the OLTC.

C. Relation between the perturbations of and , , of the capacitor
The current represents the three-phase currents flowing through the capacitors. It is expressed as follows: where is the voltage of phase f = {a, b, c} of bus q.
A small perturbation of capacitance of phase a of bus q will perturb the current , as shown in Equation (S21): The perturbation of due to the perturbation of is equal to Similarly, could be calculated in a similar manner.

D. Relation between the perturbations of and 1
The three-phase positive-sequence apparent power of DG of bus g ( 3 ℎ− ) is calculated, as shown in Equation (S22): where and are the complex positive-sequence voltage and current of DG, respectively.
Based on Equation (S22), a small perturbation at the positive sequence reactive power of DG ( 3 ℎ− ) will cause a perturbation at the positive-sequence current, as follows: The perturbation of Equation (S23) will cause a perturbation at the positive sequence components of three-phase currents ( 1 , 1 , 1 ) of DG, as shown in Equation (S24): where is the phasor rotation operator = 2 3 • .

II. DERIVATION OF THE ELEMENTS OF , MATRIX
The derivation of each element of the , is explained below:

Derivation of
Initially, a small tap perturbation is assumed at each phase of OLTC, which causes a perturbation at the three-phase current sources of OLTC by , as follows: where γ = {h, w} and is the perturbation of current of bus γ due to the perturbation of OLTC tap of phase f = {a, b, c}, calculated according to Section I. B of the supplementary material. In is a 3×3 matrix, because is a 3×1vector.
The perturbation of the three-phase voltages of OLTC bus w due to and is given by Equation (S27).
where − is the perturbation of the voltage (complex value) of phase l = {a, b, c} of bus w, due to the perturbation of OLTC tap of phase r = {a, b, c}. is the 3x3 submatrix of the impedance matrix of the network (see Equation (14) of Part I) that corresponds to the w and h buses.
is the 3x3 submatrix of the impedance matrix of the network that corresponds to the w bus.
The sensitivity of the absolute value of phase voltage of OLTC bus w with respect to the tap of each phase of OLTC transformer is given in Equation (S28).

Derivation of
The perturbation of the three-phase currents of DG, due to the perturbation of positive sequence reactive power ( 1 ) is given by Equation (S30): where 1 , 1 , 1 are derived, as explained in Equations (S22) -(S24). The perturbation of the three-phase voltages of OLTC bus w due to 1 is given by Equation (S31).
where − 1 is the perturbation of the voltage of phase l = {a, b, c} of bus w, due to the perturbation of 1 . The sensitivity of absolute value of phase voltage of OLTC bus w due to the perturbation of 1 is given in Equation (S32).
The sensitivity parameter is given in Equation (S33).

Derivation of
Initially, a small perturbation equal to is assumed at the real and imaginary components of negative-and zero-sequence currents of bus g such that 2 = 2 = 0 = 0 = , where 2 , 2 , 0 , 0 is the real component of negative sequence current, the imaginary component of negative sequence current, the real component of zero-sequence current, the imaginary component of zero-sequence current, respectively.
The perturbation caused at the currents of phases a, b, c ( 2 , 2 , 2 ) due to the perturbation 2 is calculated by Equation (S34).
where for example 2 is the perturbation of current of phase a due to the perturbation of 2 . Similarly, for the other elements.
The perturbation caused at the currents of phases a, b, c ( 2 , 2 , 2 ) due to the perturbation of 2 is calculated by Equation (S35).
The perturbation caused at the currents of phases a, b, c ( 0 , 0 , 0 ) due to the perturbation of 0 is calculated by Equation (S36).
The perturbation caused at the currents of phases a, b, c ( 0 , 0 , 0 ) due to the perturbation of 0 is calculated by Equation (S37).
The perturbation of the three-phase voltages of OLTC bus w due to is given by Equation (S39).
Similarly for the other elements. The sensitivities of absolute value of phase voltages of OLTC bus w due to the perturbation of negative-and zero-sequence current components of DG are given in Equation (S40).
The sensitivity parameter is given in Equation (S41).

Derivation of
Initially, a small perturbation at each tap of SVR equal to dTap is assumed. This perturbation perturbs the SVR current sources of The sensitivity of absolute value of phase voltage of bus w due to the perturbation of SVR taps is given in Equation (S44).
The sensitivity parameter is given in Equation (S45).

Derivation of
Initially, a small perturbation at each phase capacitor equal to is assumed. This perturbation perturbs the current of each phase of bus q by as follows: are calculated as explained in Equation (S21). The perturbation of the three-phase voltages of OLTC bus w due to is given by Equation (S47).
where − is the perturbation of the voltage of phase l = {a, b, c} of bus w, due to the perturbation of capacitor of phase r.
The sensitivity of absolute value of phase voltage of OLTC bus w with respect to each phase capacitor of bus q are given in Equation (S48).
The sensitivity parameter is given in Equation (S49).

Derivation of
Initially, a small tap perturbation is assumed at each phase of OLTC, which causes a perturbation at the current sources of Fig. 6 by as follows: where γ = {h, w} and is calculated according to Equation (S13).
The perturbation of the three-phase voltages of DG bus g due to is given by Equation (S51).
The perturbation of the positive-sequence voltage of DG bus g is given by Equation (S52).
where for example  The perturbation of the three-phase currents of DG, due to the perturbation of positive sequence reactive power ( 1 ) is given by Equation (S55): where 1 , 1 , 1 are derived, as explained in Equations (S22) -(S24). The perturbation of the three-phase voltages of bus g due to is given by Equation (S56).
where is the 3x3 sub-matrix of impedance matrix of Equation (13) of Part I, corresponding to the voltage of g th bus and current of g th bus.
The perturbation of the positive sequence voltage of DG bus g due to the perturbation of The variation of the magnitude of the positive-sequence voltage is calculated by Equation (S58).
Finally, the sensitivity parameter is calculated by Equation (S59).
Initially, a small perturbation equal to is assumed at the real and imaginary components of negative-and zero-sequence currents such that is the real component of negative sequence current, the imaginary component of negative sequence current, the real component of zero-sequence current, the imaginary component of zero-sequence current, respectively.
The perturbation caused at the currents of phases a, b, c ( 2 , 2 , 2 ) due to the perturbation 2 is calculated by Equation (S60).
where for example 2 is the perturbation of current of phase a due to the perturbation of 2 . Similarly for the other elements.
The perturbation caused at the currents of phases a, b, c ( 2 , 2 , 2 ) due to the perturbation of 2 is calculated by Equation (S61).
The perturbation caused at the currents of phases a, b, c ( The perturbation of the three-phase voltages of DG bus g due to is given by Equation (S65): where for example − 2 is the perturbation of the voltage of phase a of bus g, due to the perturbation of 2 . Similarly, for the other elements.
The perturbation of the positive-sequence voltage of DG bus g with respect to the current components is given by Equation (S66).
where for example 1− 2 is the perturbation of the positive-sequence voltage of DG bus g due to the perturbation of 2 .
The sensitivity of absolute value of positive-sequence voltage of DG bus g with respect to each current component is given in Equation (S67).
The perturbation of the positive-sequence voltage of DG bus g due to the tap perturbation is given in Equation (S71).
The sensitivity of absolute value of positive-sequence voltage of bus g with respect to each tap of SVR is given in Equation (S72).
The sensitivity parameter is given in Equation (S73).

Derivation of
Initially, a small perturbation at the capacitor of each phase equal to is assumed. This perturbation perturbs the current of capacitors by as follows: are calculated as explained in Equation (S21). Equation (S75) calculates the perturbation of phase voltages, due to the capacitor perturbation. More specifically, − expresses the perturbation of the voltage of phase ξ = {a, b, c} of bus g, due to the perturbation of capacitor of phase f = {a, b, c}.
The perturbation of the positive-sequence voltage of bus g with respect to capacitor's perturbation is given by Equation (S76).
The sensitivity of absolute value of positive-sequence voltage of DG bus g with respect to each capacitor is given in Equation (S77).
The sensitivity parameter is given in Equation (S78).

Derivation of
Initially, a small perturbation is assumed at each phase of OLTC, which causes a perturbation at the current sources of Fig. S1 by as follows: where γ = {h, w} and is calculated according to Equation (S13). The perturbation of the three-phase voltages of bus g due to is given by Equation (S80).
The perturbation of the negative-and zero sequence-voltage of bus g due to the tap perturbation is given in Equation (S81): where for example 2− is the perturbation of negative-sequence voltage of bus g ( 2 ), due to the tap perturbation of phase a ( ). Similarly for the other elements.
Finally, the sensitivity parameters are calculated by dividing the real and imaginary components of Equation (S81) by the tap variation, as shown in Equation (S82).
where for example 2− is the perturbation caused at the real component of negative-sequence voltage of bus g due to the perturbation of . Similarly, for the other elements.

Derivation of
The perturbation of the three-phase currents of DG, due to the perturbation of positive sequence reactive power ( 1 ) is given by Equation (S83): where 1 , 1 , 1 are derived, as explained in Equations (S22) -(S24). The perturbation of the three-phase voltages of bus g due to is given by Equation (S84).
The perturbation of the negative-and zero sequence-voltage of bus g due to the reactive power perturbation is given in Equation (S85)

Derivation of
Initially, a small perturbation equal to is assumed at the real and imaginary components of negative-and zero-sequence currents such that is the real component of negative sequence current, the imaginary component of negative sequence current, the real component of zero-sequence current, the imaginary component of zero-sequence current, respectively.
The perturbation caused at the currents of phases a, b, c ( 2 , 2 , 2 ) due to the perturbation 2 is calculated by Equation (S87).
where for example 2 is the perturbation of current of phase a due to the perturbation of 2 . Similarly for the other elements.
The perturbation caused at the currents of phases a, b, c ( 2 , 2 , 2 ) due to the perturbation of 2 is calculated by Equation (S88).
The perturbation caused at the currents of phases a, b, c ( The perturbation of the three-phase voltages of DG bus g due to is given by Equation (S92).
The perturbation of the negative-and zero sequence-voltage of bus g due to the perturbation of negative-and zero-sequence current component of DG is given in Equation (S93).

Derivation of
Initially, a small perturbation is assumed at the tap of each phase of SVR equal to dTap. This perturbation perturbs the SVR current sources of Fig. S1 by as follows: where γ = {p, m, s} and is calculated for γ = p and f = a in Equation (S13). Equation (S96) calculates the perturbation of the phase voltages of bus g, due to the tap perturbation of SVR. More specifically, The perturbation of the negative-and zero sequence-voltage of bus g due to the SVR tap perturbation is given in Equation (S97)

Derivation of
Initially, a small perturbation at each capacitor of each phase equal to is assumed. This perturbation perturbs the phase current of capacitors by as follows: where , , are calculated as explained in Equation (S21). Equation (S100) calculates the perturbation of phase voltages of bus g, due to the capacitor perturbations: where − is the perturbation of the voltage of phase l of bus g, due to the perturbation of capacitor of phase r.
The perturbation of the negative-and zero sequence-voltage of bus g due to the capacitor perturbation is given in Equation (S101): where for example 2− is the perturbation of negative-sequence voltage of DG bus g ( 2 ), due to the perturbation of capacitor ( ). Similarly for the other elements.
Finally, the sensitivity parameters are calculated by dividing the real and imaginary components of Equation (S101) by the capacitor's perturbation, as shown in Equation (S102):