UPFC setting to avoid active power flow loop considering wind power uncertainty

Abstract

The active power loop flow (APLF) may be caused by impropriate network configuration, impropriate parameter settings, and/or stochastic bus powers. The power flow controllers, e.g., the unified power flow controller (UPFC), may be the reason and the solution to the loop flows. In this paper, the critical existence condition of the APLF is newly integrated into the simultaneous power flow model for the system and UPFC. Compared with the existing method of alternatively solving the simultaneous power flow and sensitivity-based approaching to the critical existing condition, the integrated power flow needs less iterations and calculation time. Besides, with wind power fluctuation, the interval power flow (IPF) is introduced into the integrated power flow, and solved with the affine Krawcyzk iteration to make sure that the range of active power setting of the UPFC not yielding the APLF. Compared with Monte Carlo simulation, the IPF has the similar accuracy but less time.

Graphical abstract

Keywords

active power loop flow (APLF) / unified power flow controller (UPFC) / wind power uncertainty / interval power flow (IPF)

1 Introduction

The active power loop flow (APLF) is the active power flowing along a closed transmission path. It does not contribute to power supply, but causes the additional power loss. The relay protection along the loop may undesirably trip the lines and the loads. Although the system operator may not realize the loop flow and the unnecessary transmission loss, they are not rare in the meshed grid.

The loop flow is caused by impropriate network configuration or parameters, but may be avoided with the power flow adjustment. The power flow controllers, e.g., the on-load tap changer, the series capacitor, the phase shifting transformer, and the unified power flow controller (UPFC), are capable of controlling the power flow and improving the dynamic performance of the system [1–4], thus may be applied to eliminate the loop flow [5–8].

On the other hand, the power flow controllers with impropriate settings may also lead to the loop flow. A simple case is that a UPFC is installed on one of the parallel lines connecting a wind farm. When the active power setting of the UPFC P_ref is larger than the active power output P_wt of the wind farm, the APLF will be yielded [9,10]. More often, the UPFC setting suitable for one scenario may cause the loop flow in other scenarios.

It is possible that the optimal power flow (OPF) for power flow re-distribution may avoid the loop flow, but to set the UPFC, it is difficult to express the critical existence condition of the APLF and include it in the objective function or constraints of the OPF. Li and Wang proposed a method to quantify the impact of the power setting of the UPFC on the APLF and make sure the adjustment range of the UPFC not yielding the loop flow [11]. At first, simultaneous solution to the power flow of the system with UPFC is derived. Then, the sensitivity of active power flow of the critical line to the active power setting of the UPFC is derived to predict and approach the zero-power-flow of the critical line. Two processes are alternatively performed thus the calculation effort is a little boring, and it is based on the assumption that the relation of UPFC setting to the power flow of the critical line is monotonous.

Both the OPF and the alternative solution are suitable for the given scenario only. They cannot include the uncertainty. The wind power from the wind turbine generators (WTGs), e.g., doubly-fed induction generators (DFIGs) or permanent-magnet synchronous generators (PMSGs), fluctuates more severely and unpredictably than that of the load. The fluctuation may change the path and amount of the loop flow, thus it is more difficult to eliminate it. To describe the uncertainty, the Monte Carlo sampling may be applied [12] with huge calculation effort. The interval power flow (IPF) based on the interval arithmetic (IA) may find the limit of the unknowns following the limit of uncertainty with much less calculation effort [13], but has not yet been applied for loop flow control.

To use the IPF to avoid the loop flow in the wind power system with the UPFC, some problems are to be considered.

(1) The system configuration and the location of UPFC are relatively fixed, hence the adjustment range of power setting of the UPFC is decided by the stochastic bus power. Before solving the IPF, it is necessary to find the range of the wind power affecting the adjustment range of the UPFC.

(2) The inverse matrix of the Jacobian matrix of the IPF is an interval matrix [14]. It is difficult to derive the corresponding power flow sensitivity.

(3) The IPF may be solved by using the optimization or iteration method. The former regards the power flow with the interval variables as inequality constraint, and solves the minimum or maximum problem [15,16]. The latter, based on deterministic power flow, is easy to implement. The most widely used method to solve the nonlinear interval equations is Krawczyk iteration [17].

(4) The interval arithmetic ignores the correlations among the interval variables, therefore the affine arithmetic (AA) is applied to reduce the conservatism [18,19]. Since the initial intervals affect the convergence of the Krawczyk method, the power injection increment method proposed by Pereira et al. may be applied to initialize the iterative process and improve the convergence [20,21].

In this paper, the critical existence condition of the active power loop flow is newly integrated to the simultaneous power flow model of the power system with the UPFC, then extended to the IPF model, and solved by the affine Krawczyk iteration to make sure the adjustment range of the power setting of the UPFC not yielding the loop flow with wind power uncertainty. The numerical results on the New England system are provided and compared with the previous model and the Monte Carlo simulation to validate the accuracy and efficiency of the proposed method.

2 Determination of loop flow

For a system with N_a buses, define A = {a₁, a₂, …, a_Na} and E = {(a_i, a_j)|a_i, a_j

\in

A} as the sets of buses and branches respectively. Using the Floyd-Warshall algorithm, the shortest path between each pair of the buses are defined based on the distance matrix D = [d_ij]_Na×Na, and the path matrix R = [r_ij]_Na×Na. The initial values of D and R, i.e., d_{ij}^{\left(\mathrm{0}\right)} and r_{ij}^{\left(\mathrm{0}\right)}, are defined as

(1)

(2)r_{ij}^{\left(\mathrm{0}\right)}=\{\begin{array}{l}i,d_{ij}^{\left(\mathrm{0}\right)}\ne \infty ,\hfill \\ \infty ,d_{ij}^{\left(\mathrm{0}\right)}=\infty .\hfill \end{array}

At kth iteration (k = 1, …, N_a), D^(k) and R^(k) are updated by

(3)

(4)

After N_a iterations, R^(N_a) and the shortest distance matrix D^(N_a) are derived.

A loop consists of at least 3 buses, thus at least 3 pairs of off-diagonal elements r_{ij}^{\left(N_a\right)} and r_{ji}^{\left(N_a\right)} are not ∞. Decided by R^(N_a), the shortest path set M_ij stores all buses along the shortest path from i to j, and M_ji stores all buses along the shortest path from j to i. Therefore, the set {M_ij, M_ji} is denoted as the loop L.

3 Integrated power flow with the critical existence condition of APLF

As shown in Fig. 1, the UPFC is located at Bus p of line pq, and Bus m is newly introduced for the series converter. The subscripts se and sh denote the series and shunt converters respectively. The active power setting to the UPFC is P_ref.

The controlled line flow of the UPFC is given by Eq. (5). The active power balance between the series and shunt converters is given by Eq. (6). The reactive constraint of p given by Eq. (7) is included since V_sh is an unknown.

(5)\{\begin{array}{l}\mathrm{\Delta }{P}_{\mathrm{ref}}=P_{\mathrm{ref}}-\mathrm{Re}\left[{\dot{V}}_p{\left({\dot{V}}_p-{\dot{V}}_{\mathrm{se}}-{\dot{V}}_m\right)}^{*}{y}_{\mathrm{se}}^{*}\right]=\mathrm{0},\hfill \\ \mathrm{\Delta }{Q}_{\mathrm{ref}}=Q_{\mathrm{ref}}-\mathrm{Im}\left[{\dot{V}}_p{\left({\dot{V}}_p-{\dot{V}}_{\mathrm{se}}-{\dot{V}}_m\right)}^{*}{y}_{\mathrm{se}}^{*}\right]=\mathrm{0},\hfill \end{array}

(6)\mathrm{Re}\left[{\dot{V}}_{\mathrm{se}}{\dot{I}}_{\mathrm{se}}^{*}\right]+\mathrm{Re}\left[{\dot{V}}_{\mathrm{sh}}{I}_{\mathrm{sh}}^{*}\right]=\mathrm{0},

\mathrm{\Delta }{Q}_p=Q_{\mathrm{G}p}-Q_{\mathrm{L}p}-\mathrm{Im}\left({\dot{V}}_p{\displaystyle \sum _{j=\mathrm{1}}^{Na+\mathrm{2}}{Y}_{pj}^{*}{\dot{V}}_j^{*}}\right)

(7)-\mathrm{Im}\left({\dot{V}}_p{\dot{V}}_{\mathrm{se}}^{*}{y}_{\mathrm{se}}^{*}+{\dot{V}}_p{\dot{V}}_{\mathrm{sh}}^{*}{y}_{\mathrm{sh}}^{*}\right)=\mathrm{0},

where {\dot{I}}_{\mathrm{se}} and {\dot{I}}_{\mathrm{sh}} are the currents of the series and shunt branches, respectively; the subscripts G and L denote the generator and the load; Y = [y] is the bus admittance matrix; and the superscript * denotes conjugate. The constraints of the UPFC are solved simultaneously using the power flow equations in Ref. [22].

It is found that the critical existence condition of the APLF is that the active power flow of the critical branch ij in the loop, e.g., P_ij, equals 0, thus the corresponding P_ref is defined as the marginal setting P_crit. Since the power flow of the controlled line has two directions, the adjustment range of the UPFC is decided by its maximum and minimum marginal settings.

Assume that the direction of the controlled power flow is from Bus p to Bus q, i.e., P_ref<0. If there are N_l loop flows (the subscript l denotes the loop), the maximum value of N_l marginal settings may be set as the lower limit P_ref,lower of the UPFC. If the direction of the controlled line flow is opposite, i.e., P_ref>0, the minimum value of N_l marginal settings may be set as the upper limit P_ref,upper.

A method to find the critical existence condition of the loop flows is proposed in Ref. [11], which is a sensitivity-based alternative solution. In this paper, a novel method to decide P_crit is proposed which is more straightforward and easier to implement.

By seeing P_ref as an unknown, the critical existence condition of the loop flow (P_ij = 0) is included in the power flow equations to form the integrated power flow model (8) (Fig. 2), and linearized to Eq. (9).

(8)\{\begin{array}{l}\mathrm{\Delta }\mathit{P}(\mathit{\theta },\mathit{V},\mathit{U},P_{\mathrm{ref}})=\mathrm{0}\hfill \\ \mathrm{\Delta }\mathit{Q}(\mathit{\theta },\mathit{V},\mathit{U},P_{\mathrm{ref}})=\mathrm{0}\hfill \\ \mathrm{\Delta }\mathit{G}(\mathit{\theta },\mathit{V},\mathit{U},P_{\mathrm{ref}})=\mathrm{0}\hfill \\ P_{ij}(\mathit{\theta },\mathit{V},\mathit{U},P_{\mathrm{ref}})=\mathrm{0}\hfill \end{array}\Rightarrow f\left(\mathit{x}\right)=\mathrm{0},

\left[\begin{array}{l}\mathrm{\Delta }\mathit{P}\hfill \\ \mathrm{\Delta }\mathit{Q}\hfill \\ \mathrm{\Delta }\mathit{G}\hfill \\ P_{ij}\hfill \end{array}\right]+\left[\begin{array}{llll}{\displaystyle \frac{\partial \mathrm{\Delta }\mathit{P}}{\partial \mathit{\theta }}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{P}}{\partial \mathit{V}}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{P}}{\partial \mathit{U}}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{P}}{\partial P_{\mathrm{ref}}}}\hfill \\ \hfill \\ {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{Q}}{\partial \mathit{\theta }}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{Q}}{\partial \mathit{V}}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{Q}}{\partial \mathit{U}}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{Q}}{\partial P_{\mathrm{ref}}}}\hfill \\ \hfill \\ {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{G}}{\partial \mathit{\theta }}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{G}}{\partial \mathit{V}}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{G}}{\partial \mathit{U}}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{G}}{\partial P_{\mathrm{ref}}}}\hfill \\ \hfill \\ {\displaystyle \frac{\partial P_{ij}}{\partial \mathit{\theta }}}\hfill & {\displaystyle \frac{\partial P_{ij}}{\partial \mathit{V}}}\hfill & {\displaystyle \frac{\partial P_{ij}}{\partial \mathit{U}}}\hfill & {\displaystyle \frac{\partial P_{ij}}{\partial P_{\mathrm{ref}}}}\hfill \end{array}\right]

(9)·\left[\begin{array}{l}\mathrm{\Delta }\mathit{\theta }\hfill \\ \mathrm{\Delta }\mathit{V}\hfill \\ \mathrm{\Delta }\mathit{U}\hfill \\ \mathrm{\Delta }{P}_{\mathrm{ref}}\hfill \end{array}\right]=\left[\begin{array}{l}\mathbf{0}\hfill \\ \mathbf{0}\hfill \\ \mathbf{0}\hfill \\ \mathrm{0}\hfill \end{array}\right],

where D denotes the increment; P and Q denote the active and reactive bus powers; G denotes the constraints of the UPFC; q and V denote the voltage angle and magnitude; U denotes the variables of the UPFC, i.e., q_se, q_sh, V_se, V_sh; and x = [q, V, U, P_ref]^T. When the integrated power flow converges, the critical setting to the UPFC, i.e., P_crit, is found.

4 UPFC setting to avoid APLFs based on IPF and considering wind power uncertainty

In the practical power system, the bus load and wind power changes with time, but the setting to the UPFC is relatively fixed and cannot change frequently, thus should be compromised with the wind power to avoid the loop flow. Therefore, the interval power flow (IPF) in Refs. [23,24] is introduced to the proposed integrated power flow to decide the range of the UPFC setting corresponding to the range of the wind power.

With the power factor fixed, the active and reactive wind powers are given by the interval variables, i.e., {\widehat{P}}_{\mathrm{wt}}=\left[P_{\mathrm{wt}}^{-},P_{\mathrm{wt}}^{+}\right], {\widehat{Q}}_{\mathrm{wt}}=\left[Q_{\mathrm{wt}}^{-},Q_{\mathrm{wt}}^{+}\right], where ^ denotes the interval variable, and the subscripts – and+ denote the infimum and supremum respectively.

Then Eq. (8) is rewritten to the IPF in Eq. (10), where {\widehat{\mathit{x}}}^{\left(t\right)} denotes the interval variables except for {\widehat{P}}_{\mathrm{wt}}.

\{\begin{array}{l}\mathrm{\Delta }\mathit{P}(\mathit{\theta },\mathit{V},\mathit{U},{\widehat{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}},{\widehat{P}}_{\mathrm{w}\mathrm{t}})=\mathrm{0}\hfill \\ \mathrm{\Delta }\mathit{Q}(\mathit{\theta },\mathit{V},\mathit{U},{\widehat{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}},{\widehat{P}}_{\mathrm{w}\mathrm{t}})=\mathrm{0}\hfill \\ \mathrm{\Delta }\mathit{G}(\mathit{\theta },\mathit{V},\mathit{U},{\widehat{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}},{\widehat{P}}_{\mathrm{w}\mathrm{t}})=\mathrm{0}\hfill \\ P_{ij}(\mathit{\theta },\mathit{V},\mathit{U},{\widehat{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}},{\widehat{P}}_{\mathrm{w}\mathrm{t}})=\mathrm{0}\hfill \end{array}

(10)\Rightarrow F(\widehat{\mathit{x}},{\widehat{P}}_{\mathrm{w}\mathrm{t}})=\mathrm{0}\mathrm{.}

To solve Eq. (10), the Krawczyk iteration is applied,

(11)\mathit{K}\left({\widehat{\mathit{x}}}^{\left(t\right)}\right)={\mathit{x}}_{\mathrm{M}}-{\mathit{J}}_{\mathrm{M}}^{-\mathrm{1}}f\left({\mathit{x}}_{\mathrm{M}}\right)+(\mathit{C}-{\mathit{J}}_{\mathrm{M}}^{-\mathrm{1}}\widehat{\mathit{J}}({\widehat{\mathit{x}}}^{\left(t\right)}\left)\right)({\widehat{\mathit{x}}}^{\left(t\right)}-{\mathit{x}}_{\mathrm{M}}),

where \mathit{K}\left({\widehat{\mathit{x}}}^{\left(t\right)}\right) is the Krawczyk operator, x_M is the midpoint of {\widehat{\mathit{x}}}^{\left(t\right)}, J_M is the midpoint matrix of the interval Jacobian matrix \widehat{\mathit{J}}\left({\widehat{\mathit{x}}}^{\left(t\right)}\right), C denotes the identity matrix, and t is the iteration number.

To describe the correlations between the interval variables, the affine arithmetic is applied. \widehat{\mathit{x}} is given by \widehat{\mathit{x}}={\mathit{x}}_{\mathrm{0}}+{\mathit{x}}_{\mathrm{1}}{ϵ}_{\mathrm{1}}+\mathrm{\cdots }+{\mathit{x}}_n{ϵ}_n, where x_i is the perturbation coefficient, ε_i

\in

[–1, 1] denotes the ith noise. To show the impact the wind power, only the fluctuation of the wind power is considered. Therefore, the affine form of the interval variables is given by \widehat{\mathit{x}}={\mathit{x}}_{\mathrm{0}}+{\mathit{x}}_{\mathrm{1}}ϵ. Then, \widehat{\mathit{J}}\left({\widehat{\mathit{x}}}^{\left(t\right)}\right) is calculated, and \mathit{K}\left({\widehat{\mathit{x}}}^{\left(t\right)}\right) with the affine form is converted to the interval form to solve {\widehat{\mathit{x}}}^{(t+\mathrm{1})},

(12){\widehat{\mathit{x}}}^{(t+\mathrm{1})}={\widehat{\mathit{x}}}^{\left(t\right)}\cap \mathit{K}\left({\widehat{\mathit{x}}}^{\left(t\right)}\right)\mathrm{.}

The initial interval is obtained by linearization at the midpoint of the interval, and the affine form of the initial interval is given by

(13)\left[\begin{array}{l}{\mathit{\theta }}^{\left(\mathrm{0}\right)}\hfill \\ {\mathit{V}}^{\left(\mathrm{0}\right)}\hfill \\ {\mathit{U}}^{\left(\mathrm{0}\right)}\hfill \\ P_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\left(\mathrm{0}\right)}\hfill \end{array}\right]=\left[\begin{array}{l}{\mathit{\theta }}_{\mathrm{0}}\hfill \\ {\mathit{V}}_{\mathrm{0}}\hfill \\ {\mathit{U}}_{\mathrm{0}}\hfill \\ P_{\mathrm{r}\mathrm{e}\mathrm{f},\mathrm{0}}\hfill \end{array}\right]+{[{\displaystyle \frac{\mathrm{d}\mathit{\theta }}{\begin{array}{l}{\mathrm{d}P}_{\mathrm{w}\mathrm{t}}\hfill \end{array}}}{\displaystyle \frac{\mathrm{d}\mathit{V}}{\begin{array}{l}{\mathrm{d}P}_{\mathrm{w}\mathrm{t}}\hfill \end{array}}}{\displaystyle \frac{\mathrm{d}\mathit{U}}{\begin{array}{l}{\mathrm{d}P}_{\mathrm{w}\mathrm{t}}\hfill \end{array}}}{\displaystyle \frac{\begin{array}{l}{\mathrm{d}P}_{\mathrm{r}\mathrm{e}\mathrm{f}}\hfill \end{array}}{\begin{array}{l}{\mathrm{d}P}_{\mathrm{w}\mathrm{t}}\hfill \end{array}}}]}^{\mathrm{T}}\mathrm{\Delta }\begin{array}{l}{P}_{\mathrm{w}\mathrm{t}}\hfill \end{array}ϵ,

where subscript 0 denotes the deterministic power flow solution at the midpoint P_wtM of the interval {\widehat{P}}_{\mathrm{wt}}, i.e., P_{\mathrm{wtM}}=\left(P_{\mathrm{wt}}^{-}+P_{\mathrm{wt}}^{+}\right)/\mathrm{2}; the superscript T denotes the transpose; and ∆P_wt = (P_{\mathrm{wt}}^{+}-P_{\mathrm{wt}}^{-})/\mathrm{2} is the radius of the interval {\widehat{P}}_{\mathrm{wt}}.

The sensitivities of the variables to the active power output is obtained by

{\left[\begin{array}{llll}{\displaystyle \frac{\mathrm{d}\mathit{\theta }}{\mathrm{d}{P}_{\mathrm{wt}}}}\hfill & {\displaystyle \frac{\mathrm{d}\mathit{V}}{\mathrm{d}{P}_{\mathrm{wt}}}}\hfill & {\displaystyle \frac{\mathrm{d}\mathit{U}}{\mathrm{d}{P}_{\mathrm{wt}}}}\hfill & {\displaystyle \frac{\mathrm{d}{P}_{\mathrm{ref}}}{\mathrm{d}{P}_{\mathrm{wt}}}}\hfill \end{array}\right]}^{\mathrm{T}}

(14)=-{\mathit{J}}_{\mathrm{0}}^{-\mathrm{1}}{\left[\begin{array}{llll}{\displaystyle \frac{\partial \mathrm{\Delta }\mathit{P}}{\partial P_{\mathrm{wt}}}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{Q}}{\partial P_{\mathrm{wt}}}}\hfill & {\displaystyle \frac{\partial \mathrm{\Delta }\mathit{G}}{\partial P_{\mathrm{wt}}}}\hfill & {\displaystyle \frac{\partial P_{ij}}{\partial P_{\mathrm{wt}}}}\hfill \end{array}\right]}^{\mathrm{T}},

where J₀ is the Jacobian matrix of the power flow solution at P_{wt, max}; DP/∂P_wt and ∂DQ/∂P_wt are the derivative of the mismatch bus powers; ∂DG/∂P_wt is the derivatives of DG to P_wt; and ∂P_ij/∂P_wt is the derivative of P_ij to P_wt.

Assuming P_ref<0, the solution to the active power setting of the UPFC with wind power fluctuation is demonstrated in Fig. 3, where P_lim is the power flow limit of the branch.

To find the critical branches whose active power directions are sensitive to the active power setting of the UPFC or the active power output of the wind farm, the power flow is performed in 3 steps,

Step 1: P_wt = 0, without UPFC,

Step 2: P_wt = 0, P_ref = P_lim, and

Step 3: P_wt = P_wtN, P_ref = P_lim.

In each step, the loop flow is found and the critical branch along each loop whose active power direction is sensitive to the UPFC is decided. By comparing the active power directions in Steps 1 and 2, the loop branches with change of active power direction are found.

The critical branch along each loop whose active power direction is sensitive to the wind power is decided. By comparing the active power directions in Steps 2 and 3, the branches along the loop with change of active power direction are found. The minimum marginal active power along the loop is the critical branch.

The IPF is performed and the interval of the active power setting corresponding to the critical branches that are sensitive to the UPFC and the wind farm, e.g., \left[P_{\mathrm{ref,U}}^{-},P_{\mathrm{ref,U}}^{+}\right] and \left[P_{\mathrm{ref,wt}}^{-},P_{\mathrm{ref,wt}}^{+}\right], where the subscript U denotes the UPFC, is obtained. The lower range of the active power setting to the UPFC is given by Eq. (15), where | | denotes the absolute value.

(15){\widehat{P}}_{\mathrm{ref},\mathrm{lower}}=\left[\min \left\{\left|P_{\mathrm{ref},\mathrm{U}}^{-}\right|,\left|P_{\mathrm{ref},\mathrm{wt}}^{-}\right|\right\},\min \left\{\left|P_{\mathrm{ref},\mathrm{U}}^{+}\right|,\left|P_{\mathrm{ref},\mathrm{wt}}^{+}\right|\right\}\right].

ƒIf P_ref>0, the upper range of the active power setting is given by

(16){\widehat{P}}_{\mathrm{ref},\mathrm{upper}}=\left[\max \left\{\left|P_{\mathrm{ref},\mathrm{U}}^{-}\right|,\left|P_{\mathrm{ref},\mathrm{wt}}^{-}\right|\right\},\max \left\{\left|P_{\mathrm{ref},\mathrm{U}}^{+}\right|,\left|P_{\mathrm{ref},\mathrm{wt}}^{+}\right|\right\}\right].

Some comments are given to the proposed methods,

(1)The system configuration and location and reference values of the UPFC have an obvious impact on the loop flow. The proposed methods, i.e., finding the loop flow, the integrated power flow with critical existence condition, and setting the UPFC to avoid loop flow, are based on theoretical derivation, with no specified precondition to the locations of the UPFC and the wind farms.

(2)The proposed methods are based on the theoretical derivation, thus not limited to fixed type of WTGs. The configuration and the control strategies of the PMSG, the photovoltaic (PV) generation, or the storage cell, are easier than those of the DFIGs, thus may be easily applied.

(3)The proposed integrated power flow and interval power flow models do not limit the number of the UPFCs and the wind farms, thus may be used for large power systems. The only burden may be the increasing calculation time with more UPFCs and wind farms.

(4)The control of the proposed model may be limited by the capacities of the series and shunt converters of the UPFC. It is possible that the recommended references exceed the capacities, thus the loop flows are partially instead of wholly eliminated. If so, the power output of the synchronous generators and other var controllers may be adjusted.

(5)The interval algorithm decides the range of the controlled parameters based on the range of the input parameters. However, it cannot give the desirable probabilistic distribution of the controlled parameters within the interval.

5 Numerical analysis

The proposed models are implemented by the MATLAB language compiled by the authors, and applied to the New England test system as depicted in Fig. 4 [25]. A wind farm with the rated capacity of 100 2-MW DFIGs is connected to Bus 3 through the step-up transformer. The power factor of the wind farm is fixed at 0.95. The UPFC is installed at the sending end of Branches 3–4. To study its impact on the loop flow, set Q_ref = –1.0712 p.u. and V_ref = 1.0302 p.u. The active power setting of the UPFC is adjusted within the limit of Branches 3–4 (5 p.u.).

5.1

At P_wt = 0, and the active power setting of the UPFC of –5 p.u., the shortest path matrix is given by

.

With the active power setting of the UPFC of 5 p.u., the shortest path matrix is given by

The numbers on the upper and right sides of the sub-matrix are the buses along the APLFs. It is found that M₁₂ = {1, 2} and M₂₁ = {2, 3, 41, 4, 5, 8, 9, 39, 1}, so the loop L1-1 of {1, 2, 3, 41, 4, 5, 8, 9, 39, 1} is found. For detailed procedure to find the loops, Ref. [11] may be referred.

Finally there are two loops L1-1 and L1-2 with the controlled power flow by the UPFC from Bus 3 to Bus 4, and one loop L2 with the controlled flow from Bus 4 to Bus 3 (Fig. 5).

Without or with the UPFC, the active power flows of the branches with change of active power directions are given in Table 1. The minimum active power flows are Branch 1–2 along L1-1, 4–14 along L1-2, and 17–18 along L2, which are the critical branches whose active powers directions are sensitive to the UPFC.

To quantify the efficiency of the integrated power flow with the APLF, the marginal active power settings yielding zero active power flows of the critical branches are found, and compared with the method proposed in Ref. [11]. The ideas of both methods are similar, i.e., iteratively approaching to the exact solution. The method is executed in two iterative processes, one for the power flow with the UPFC and the other for the critical condition of the loop flow. The latter is executed with one iterative process, thus theoretically is quicker.

The marginal power settings and the collation efficiencies of two methods are listed in Table 2. The method in Ref. [11] needs 2 or 3 power flow calculations converging with 4 iterations, thus totally 8–12 iterations are needed to solve the inverse of the Jacobian matrix. The proposed model needs only 5 iterations. The total calculation time is reduced remarkably, and the efficiency is improved with the integrated power flow model with the critical condition of the loop flow.

It is found that the APLF L1-1 is eliminated if P_ref is larger than –4.6564 p.u., but L1-2 still exists unless P_ref is larger than –4.3438 p.u. Therefore, the critical branch is 4–14 when the direction of the controlled power flowing from Bus 3 to Bus 4.

5.2 Impact of location of wind farm and UPFC on interval {{\widehat{P}}^{\prime }}_{\mathrm{wt}}=\left[P_{\mathrm{wt}}^{-},\hspace{0.17em}{P}_{\mathrm{wt}}^{+}\right]

To determine the interval {{\widehat{P}}^{\prime }}_{\mathrm{wt}} involved in the IPF calculation, it is necessary to judge whether there are APLFs when the active power output is set at its rated value.

(1) Wind farm integrated into the shunt side of UPFC

L1-1 and L1-2 are eliminated when the UPFC controls the power flow from Bus 3 to Bus 4, but L2 still exists when the UPFC changes the direction of the controlled power flow (Fig. 6). Compared with Fig. 5, the branches whose active power directions are sensitive to wind power are found, i.e., 1–2 along L1-1, 4–14 and 15–16 along L1-2.

The active power directions of the branches along L2 do not change. The active power outputs of the wind farm yielding zero active power flows of the branches along L2 are calculated based on the integrated power flow model, as given in Table 3.

When the direction of the controlled power flow from Bus 3 to Bus 4, five iterations are needed. The APLFs L1-1 and L1-2 exist when P_wt is less than 0.3125 p.u. If P_wt is increased to 0.9437 p.u., L1-1 and L1-2 are eliminated. Thus, the marginal active power output P_wt,crit is 0.9437 p.u., i.e., the upper limit of

{\widehat{P}}_{\mathrm{wt}}^{\text{'}}

is 0.9437 p.u., and 15–16 is the critical branch whose power direction is sensitive to the wind power.

When the UPFC changes the direction of the controlled power flow, i.e., P_ref = 5 p.u., Q_ref = 1.0712 p.u., the power flow converges to the solutions that are less than 0, or the Jacobian matrix is singular or even the power flow diverges. L2 cannot be eliminated even if the rated capacity of the wind farm is increased, because the wind power cannot flow directly from Bus 4 to Bus 3. Therefore, the upper limit of {\widehat{P}}_{\mathrm{wt}}^{\text{'}} is P_{\mathrm{wt}}^{+}.

(2) Wind farm integrated into the series side of UPFC

The APLFs are determined when P_wt = 0 and P_wt = 2 p.u. respectively, as exhibited in Fig. 7. They exist although the wind power has reached its rated value. The active power outputs yielding zero active power flows of all branches along each loop are tabulated in Table 4.

The solution to branch 1–2 along L3-1 converges with 6 iterations, while the active power flows of the branches along L3-2 converge to an impractical solution (negative active power output of wind farm). In other words, if the rated capacity of the wind farm is larger than 3.3574 p.u., L3-1 is eliminated, but L3-2 still exists, because the active power output cannot flow directly from Bus 3 to Bus 4, i.e., the upper limit of P_{\mathrm{wt}}^{\text{'}} is P_{\mathrm{wt}}^{+}. However, the Jacobian matrix is singular in solving all branches along L4, which shows that the APLFs may not be eliminated although the active power output flows directly from Bus 4 to Bus 3. If the rated capacity of the wind farm is further increased, it will not alleviate the APLF and may result in other problems.

Overall, if the controlled power flow flows to the wind farm bus, the upper limit of {P^{\prime }}_{\mathrm{wt}} is the upper limit of the given interval {\widehat{P}}_{\mathrm{wt}}. Otherwise, it is necessary to verify whether there is an APLF when P_wt = P_wtN.

6 Adjustment range of UPFC with wind speed range

When the wind farm is connected to the shunt side of the UPFC, the ranges of the upper and lower limits of the active power setting is analyzed. The given wind speed range of [5,15] m/s is converted to wind power. P_wt is a function of the cut-in speed (4 m/s), rated speed (15 m/s), cut-out speed (25 m/s), and rated active power output P_wtN, where the subscript N denotes the nominal value. The Monte Carlo simulation with 3000 sample is performed to validate the accuracy of the IPF model.

According to Sections 5.1 and 5.2, when the direction of the controlled power flowing from Bus 3 to Bus 4, the lower limit of the active power setting of the UPFC is determined by the critical branches 4–14 and 15–16, and the actual range of active power output is {\widehat{P^{\prime }}}_{\mathrm{wt}} = [0.1818, 0.9437] p.u., i.e., the corresponding range of wind speed is 5 to 9.19 m/s. Similarly, the upper limit of P_ref is only decided by the critical branch 17–18, and {\widehat{P^{\prime }}}_{\mathrm{wt}} = [0.1818, 2] p.u., i.e., the range of wind speed is 5 to 15 m/s. The interval results of the active power settings yielding zero active power flows of three critical branches are calculated by the Monte Carlo simulation and the IPF, as presented in Table 5. The absolute values of the relative errors of the IPF are displayed in Fig. 8. The largest error is less than 0.25%, thus the affine Krawczyk algorithm is accurate. In addition, the IPF method converges with 2 iterations and takes less time than the Monte Carlo simulation.

Within the range of the wind speed, the lower limit of the active power setting of the interval {\widehat{P}}_{\mathrm{ref},\mathrm{lower}} found from the IPF is [–5.0003, –4.2736] p.u., and the upper limit {\widehat{P}}_{\mathrm{ref},\mathrm{upper}} is [1.0221, 2.7688] p.u. Compared with the adjustment range of the active power setting of [–4.3438, 2.9461] p.u. when P_wt = 0, the absolute value of the lower limit is increased and the upper limit is decreased.

However, for the IPF results, the lower limit of {\widehat{P}}_{\mathrm{ref},\mathrm{upper}} may not correspond to the lower or upper limit of the wind speed. For example, when the wind speed is 5 m/s, the lower limit may not be –5.0003 p.u. or –4.4397 p.u., but within [–5.0003, –4.4397] p.u. Hence to avoid the loop flow caused by the UPFC within the range of wind speed, the adjustment range of the active power setting is [–4.4397, 1.0221] p.u., i.e., the active power setting is less than 4.4397 p.u. when the direction of the controlled power is from Bus 3 to Bus 4, and is less than 1.0221 p.u. if the UPFC changes the direction.

With different wind speeds and active power settings, the APLFs are plotted in Fig. 9(a), where 1 and –1 along the z-axis denote the operation conditions with and without the APLF. When the active power setting is adjusted within the range of [−4.4397, 1.0221] p.u. and the wind speed fluctuates within the range of [5,15] m/s, there is no APLF, as shown in Fig. 9(b). However, when the power setting is increased to 1.0767 p.u which is greater than 1.0221 p.u., the APLF occurs, thus the accuracy of the above interval setting to the UPFC is validated.

7 Conclusions

In this paper, an integrated power flow with the critical existence condition of the active power loop flow is proposed, extended to the IPF to include wind power fluctuation, and solved with the affine Krawcyzk iteration, based on which the adjustment range of the active power setting of the UPFC not yielding active power loop flow with wind power uncertainty is quantified. The following conclusions are reached:

The integrated power flow model with the critical existing condition of the loop flow is more straightforward than the sensitivity-based iterative solution.

The interval power flow model including uncertainty of wind power is more efficient to find the range of UPFC setting than the Monte Carlo simulation.

References

[1] Sadjad G, Mehrdad T H, Mohamad B B S. Unified power flow controller impact on power system predictability. IET Generation, Transmission & Distribution, 2013, 8(5): 819–827

[2] Golshannavaz S, Aminifar F, Nazarpour D. Application of UPFC to enhancing oscillatory response of series-compensated wind farm integrations. IEEE Transactions on Smart Grid, 2014, 5(4): 1961–1968

[3] Lin W M, Lu K H, Ou T C. Design of a novel intelligent damping controller for unified power flow controller in power system connected offshore power applications. IET Generation, Transmission & Distribution, 2015, 9(13): 1708–1717

[4] Wang L, Li H W, Wu C T. Stability analysis of an integrated offshore wind and seashore wave farm fed to a power grid using a unified power flow controller. IEEE Transactions on Power Systems, 2013, 28(3): 2211–2221

[5] Wei P, Ni Y X, Wu F F. Load flow tracing in power systems with circulating power. International Journal of Electrical Power & Energy Systems, 2002, 24(10): 807–813

[6] Marinakis A, Glavic M, Van Cutsem T. Minimal reduction of unscheduled flows for security restoration: application to phase shifter control. IEEE Transactions on Power Systems, 2010, 25(1): 506–515

[7] Lauria S, Palone F. Maximum undergrounding degree of HV sub transmission networks as dictated by unscheduled power flows. IET Generation, Transmission & Distribution, 2013, 7(11): 1202–1209

[8] Cvijić S, Ilić M D. Part II: PAR flow control based on the framework for modelling and tracing of bilateral transactions and corresponding loop flows. IEEE Transactions on Power Systems, 2014, 29(6): 2715–2722

[9] Sayed M A, Takeshita T. All nodes voltage regulation and line loss minimization in loop distribution systems using UPFC. IEEE Transactions on Power Electronics, 2011, 26(6): 1694–1703

[10] Miller J M, Ballamat B M, Morris K N, Malinowski J H, Pastermack B M, Eilts L E. Operating problems with parallel flow. IEEE Transactions on Power Systems, 1991, 6(3): 1024–1034

[11] Li S, Wang T, Zhang H, Wang L, Jiang Y, Xue J. Sensitivity-based coordination to controllable ranges of UPFCs to avoid active power loop flows. International Journal of Electrical Power & Energy Systems, 2020, 114: 105383

[12] Hajian M, Rosehart W D, Zareipour H. Probabilistic power flow by Monte Carlo simulation with Latin supercube sampling. IEEE Transactions on Power Systems, 2013, 28(2): 1550–1559

[13] Wang Z A, Alvarado F L. Interval arithmetic in power flow analysis. IEEE Transactions on Power Systems, 1992, 7(3): 1341–1349

[14] Moore R E, Cloud M J, Kearfott R B. Introduction to Interval Analysis. Philadelphia: Society for Industrial and Applied Mathematics, 2009

[15] Duan C, Jiang L, Fang W L, Liu J. Moment-SOS approach to interval power flow. IEEE Transactions on Power Systems, 2017, 32(1): 522–530

[16] Ding T, Bo R, Li F X, Guo Q, Sun H, Gu W, Zhou G. Interval power flow analysis using linear relaxation and optimality-based bounds tightening (OBBT) methods. IEEE Transactions on Power Systems, 2015, 30(1): 177–188

[17] Ding T, Li X, Li F, Bo R, Sun H. Interval radial power flow using extended DistFlow formulation and Krawcyzk iteration method with sparse approximate inverse preconditioner. IET Generation, Transmission & Distribution, 2015, 9(14): 1998–2006

[18] Wang Y, Wu Z, Dou X, Hu M, Xu Y. Interval power flow analysis via multi-stage affine arithmetic for unbalanced distribution network. Electric Power Systems Research, 2017, 142: 1–8

[19] Vaccaro A, Canizares C A. An affine arithmetic-based framework for uncertain power flow and optimal power flow studies. IEEE Transactions on Power Systems, 2017, 32(1): 274–288

[20] Pereira L E S, da Costa V M. Interval analysis applied to the maximum loading point of electric power systems considering load data uncertainties. International Journal of Electrical Power & Energy Systems, 2014, 54: 334–340

[21] Pereira L E S, da Costa V M, Rosa A L S. Interval arithmetic in current injection power flow analysis. International Journal of Electrical Power & Energy Systems, 2012, 43(1): 1106–1113

[22] Fuerte-Esquivel C R, Acha E, Ambriz-Perez H. A comprehensive Newton-Raphson UPFC model for the quadratic power flow solution of practical power networks. IEEE Transactions on Power Systems, 2000, 15(1): 102–109

[23] Ding T, Cui H T, Gu W, Wan Q L. An uncertainty power flow algorithm based on interval and affine arithmetic. Automation of Electric Power Systems, 2012, 36(13): 51–55

[24] Wolfram M, Schlegel S, Westermann D. Closed loop flow detection in power system based on Floyd-Warshall algorithm. In: IEEE Manchester PowerTech, Manchester, UK, 2017, 18–22

[25] Hiskens I. IEEE PES task force on benchmark systems for stability controls report on the 39-bus system (New England Reduced Model). 2020–6-9, available at website of Universidade de Sao Paulo

RIGHTS & PERMISSIONS

Higher Education Press