Acceleration strategies of Benders decomposition for the security constraints power system expansion planning

Abstract

Nowadays the power generation and transmission are substantial elements for the society, and will definitely play a more important role in the future. In this paper a modeling framework is presented to analyze security constrained composite generation and transmission expansion planning problem in power systems. Despite of the traditional expansion planning approaches where only supply side options are considered, the proposed approach accounts for both supply and demand side management (DSM) options simultaneously. DSM options are incorporated to correct the shape of the load duration curve in terms of peak clipping and load shifting programs. A mixed integer non-linear programming model is developed to find the optimal location and timing of electricity generation/transmission as well as DSM options while the effect of transmission losses are also taken into account. Nonlinearity of the transmission loss terms is eliminated using the piecewise linearization. Motivating from the structure of the model, Benders decomposition (BD) algorithm is devised. Three effective strategies named: valid inequalities, multiple generation cuts and strong high density cut are also employed to improve the convergence of the proposed BD algorithm. The performance of the accelerated BD algorithm is validated via applying it to the modified 6, 21, 48, 57, 118 and 300 bus IEEE reliability test systems. The computational experiments ensure the practicality of the proposed BD algorithm in terms of decreasing the number of iterations and CPU times.

Appendices

Appendix 1

Not with standing extensive literature about expansion planning of power systems (specifically on TEP), most of the time, transmission losses and its effect on expansion planning has been disregarded due to non-linear relationship between the transmission loss and the power flow. Usually, this non-linear term is approximated by a linear function. Alguacil et al. (2003) proposed a large-scale, mixed-integer, nonlinear and non-convex model for long-term transmission expansion planning problem. They derived a mixed-integer linear formulation through applying piecewise linear function to linearize nonlinear term corresponding to transmission losses.

This appendix describes how transmission losses are dealt with in the proposed mathematical programming model presented in Sect. 2. Similar with (Alguacil et al. 2003), the fraction of power lost on a type l or $l'$ lines connecting nodes i and j at stag t is considered to be a function of line type and power flow. Therefore, loss-adjusted power inflow for candidate,$\mathcal {L}\left( {f_{\ell ,j,i,m}^{t,{l}'} } \right)$, and existing,$\mathcal {L}\left( {f_{j,i,m}^{t,l} } \right)$, lines are presented in the Eqs. (65) and (66), respectively.

$$\begin{aligned}&\mathcal {L}\left( {f_{j,i,m}^{t,l} } \right) = f_{j,i,m}^{t,l} -\mathcal {E}_{i,j}^l .N_{i,j}^l .\left( {\theta _{m,j}^t -\theta _{m,i}^t } \right) ^{2} \quad \hbox {For existing lines} \end{aligned}$$

(65)

$$\begin{aligned}&\mathcal {L}\left( {f_{\ell ,j,i,m}^{t,{l}'} } \right) = f_{\ell ,j,i,m}^{t,{l}'} -\mathcal {E}_{i,j}^{{l}'} .\left( {\theta _{m,j}^t -\theta _{m,i}^t } \right) ^{2} \quad \hbox {For candidate lines} \end{aligned}$$

(66)

Substituting $\theta _{m,j}^t -\theta _{m,i}^t$ from Eqs. (4) and (5), the loss terms can be rewritten as:

$$\begin{aligned}&\mathcal {L}\left( {f_{j,i,m}^{t,l} } \right) = f_{j,i,m}^{t,l} -\frac{\mathcal {E}_{i,j}^l }{N_{i,j}^l .\left( {\Gamma _{i,j}^l } \right) ^{2}}.\left( {f_{j,i,m}^{t,l} } \right) ^{2} \quad \hbox {For existing lines}\end{aligned}$$

(67)

$$\begin{aligned}&\mathcal {L}\left( {f_{\ell ,j,i,m}^{t,{l}'} } \right) = f_{\ell ,j,i,m}^{t,{l}'} -\frac{\mathcal {E}_{i,j}^{{l}'} }{\left( {\Gamma _{i,j}^{{l}'} } \right) ^{2}}.\left( {f_{\ell ,j,i,m}^{t,{l}'} } \right) ^{2} \quad \hbox {For candidate lines} \end{aligned}$$

(68)

Fig. 3

Piecewise linearization technique per existing line loss function

Appendix 2

The quadratic terms in the Eq. (2) and product of discrete and continuous variables in Eq. (5) make the proposed model to be nonlinear. In this appendix we describe how these nonlinear terms can be converted to linear ones.

Since the flow on lines of type l and $l'$ are bounded within the intervals $\left[ {-N_{i,j}^l .\bar{{f}}_{i,j}^l }, \right. \left. {N_{i,j}^l .\bar{{f}}_{i,j}^l } \right]$ and $\left[ {\left. {-\bar{{f}}_{\ell ,i,j}^{{l}'} ,\bar{{f}}_{\ell ,i,j}^{{l}'} } \right] } \right.$, a linear approximation of quadratic terms $\left( {f_{j,i,m}^{t,l} } \right) ^{2}$ and $\left( {f_{\ell ,j,i,m}^{t,{l}'} } \right) ^{2}$ can be obtained by partitioning these intervals into the 2s smaller intervals $(s = 1, 2, \ldots , 2L')$ as shown in Fig. 3.

Using this technique, 2s new variables should be introduced to linearize the model. The same result may be obtained using only $s+ 2$ variables if we only focus on the positive area. Therefore, the quadratic terms in Eq. (2) can be approximated as follows:

$$\begin{aligned} \begin{array}{ll@{\quad }l} \hbox {For existing lines}&{} \hbox {For new lines} &{}\\ \left| {f_{i,j,m}^{t.l} } \right| = \sum _{s=1,\ldots ,{L}'} {pf_{i,j,m}^{t,l,s} } &{} \left| {f_{\ell ,i,j,m}^{t,{l}'} } \right| = \sum _{s=1,\ldots ,{L}'} {pf_{\ell ,i,j,m}^{t,{l}',s} } &{} (69)\\ \left( {f_{i,j,m}^{t,l} } \right) ^{2}\approx \sum _{s=1,\ldots ,{L}'} {m^{l,s}.pf_{i,j,m}^{t,l,s} }&{}\left( {f_{\ell ,i,j,m}^{t,{l}'} } \right) ^{2}\approx \sum _{s=1,\ldots ,{L}'} {m^{{l}',s}.pf_{\ell ,i,j,m}^{t,{l}',s} }&{} (70)\\ pf_{i,j,m}^{t,l,s} \le \left( {a_s -a_{s-1} } \right) .N_{i,j}^l .\bar{{f}}_{i,j}^l ; s=1,\ldots ,{L}' &{} pf_{\ell ,i,j,m}^{t,{l}',s} \le \left( {a_s -a_{s-1} } \right) .Y_{\ell ,i,j}^{t,{l}'} .\bar{{f}}_{i,j}^{{l}'} ; s=1,\ldots ,{L}'&{} (71)\\ pf_{i,j,m}^{t,l,s} \ge 0; s=1,\ldots ,{L}' &{} pf_{\ell ,i,j,m}^{t,{l}',s} \ge 0; s=1,\ldots ,{L}' &{} (72)\\ m^{l,s}= \frac{a_s^2 -a_{s-1}^2 }{a_s -a_{s-1} }.N_{i,j}^l .\bar{{f}}_{i,j}^l &{} m^{{l}',s}= \frac{a_s^2 -a_{s-1}^2 }{a_s -a_{s-1} }.\bar{{f}}_{i,j}^{{l}'} &{} (73) \end{array} \end{aligned}$$

where, $a_{s}$, $m^{l,s}$, $m^{{l}',s}$, $pf_{i,j,m}^{t,l,s}$ and $pf_{\ell ,i,j,m}^{t,{l}',s}$ represent coefficients (%), slops and flow magnitudes of sth interval for the existing and new transmission lines, respectively. Linear expression of the absolute value in the Eq. (69) is required which is obtained through the following substitution (Castillo et al. 2001):

$$\begin{aligned} \begin{array}{lll} \hbox {For existing lines}&{}\hbox {For new lines}&{}\\ \left| {f_{\ell ,j,i,m}^{{l}'} } \right| = f_{\ell ,j,i,m}^{+{l}'} +f_{\ell ,j,i,m}^{-{l}'}&{} \left| {f_{j,i,m}^l } \right| = f_{j,i,m}^{+l} +f_{j,i,m}^{-l}&{}\qquad (74)\\ f_{j,i,m}^l = f_{j,i,m}^{+l} -f_{j,i,m}^{-l} &{} f_{\ell ,j,i,m}^{{l}'} = f_{\ell ,j,i,m}^{+{l}'} -f_{\ell ,j,i,m}^{-{l}'}&{}\qquad (75)\\ \end{array} \end{aligned}$$

The modified form of Eqs. (2) and (4)–(7) are presented in Eqs. (76)–(83). These modified constraints are replaced with corresponding ones in the constraint sets of (2)–(20), also Eqs. (72) and (73) are added to the constraint sets of (2)–(20).

$$\begin{aligned}&\mathop \sum \limits _{ \mathop {j|\left( {i,j} \right) \in \textit{NN}_T ,} \limits ^ {l^{\prime }\in \textit{LN},l} }\left[ {f_{\ell ,j,i,m}^{+t,{l}'} -f_{\ell ,j,i,m}^{-t,{l}'} -\sum _s {r_{j,i}^{{l}',s} .pf_{\ell ,j,i,m}^{t,{l}',s} } } \right] \nonumber \\&\qquad +\mathop \sum \limits _{ \mathop {j|\left( {i,j} \right) \in \textit{NE}_T ,} \limits ^ {l\in \textit{LE}} } \left[ {f_{j,i,m}^{+t,l} -f_{j,i,m}^{-t,l} -\sum _s {r_{j,i}^{l,s} .pf_{j,i,m}^{t,l,s} } } \right] \nonumber \\&\qquad +\mathop \sum \limits _{n,nt\in NT,n} \left( {1-lg_{nt}^t } \right) .G_{n,nt,m,i}^t +\mathop \sum \limits _{et\in \textit{ET}} \left( {1-lg_{et}^t } \right) .G_{et,m,i}^t \nonumber \\&\qquad +\mathop \sum \limits _{d\in \mathcal {D}} \textit{SDSM}_{d,m,i}^t .\left( {\mathop \sum \limits _{v\le t} \textit{DSM}_{d,i}^v } \right) =D_{i,m}^t {\quad } \begin{array}{l} \forall m\in \mathcal {M}, \\ i\in N, \\ t=1,\ldots ,T \\ \end{array}\end{aligned}$$

(76)

$$\begin{aligned}&f_{i,j,m}^{+t,l} -f_{i,j,m}^{-t,l} -\Gamma _{i,j}^l .N_{i,j}^l .\left( {\theta _{m,i}^t -\theta _{m,j}^t } \right) =0\nonumber \\&\qquad \forall \left( {i,j} \right) \in \textit{NE}_T , m\in \mathcal {M}, t=1,\ldots ,T, l\in \textit{LE} \end{aligned}$$

(77)

$$\begin{aligned}&f_{\ell ,i,j,m}^{+t,{l}'} -f_{\ell ,i,j,m}^{-t,{l}'} -\Gamma _{i,j}^{{l}'} .Y_{\ell ,i,j}^{t,{l}'} .\left( {\theta _{m,i}^t -\theta _{m,j}^t } \right) =0\nonumber \\&\qquad \forall \left( {i,j} \right) \in \textit{NN}_T , m\in \mathcal {M}, t=1,\ldots ,T, {l}'\in \textit{LN},\ell \end{aligned}$$

(78)

$$\begin{aligned}&\sum _s {pf_{i,j,m}^{t,l,s} \le N_{i,j}^l .\bar{{f}}_{i,j}^l }\qquad \forall \left( {i,j} \right) \in \textit{NE}_T , m\in \mathcal {M}, t=1,\ldots ,T, l\in \textit{LE} \end{aligned}$$

(79)

$$\begin{aligned}&\sum _s {pf_{\ell ,i,j,m}^{t,{l}',s} \le Y_{\ell ,i,j}^{t,{l}'} .\bar{{f}}_{i,j}^{{l}'} }\qquad \forall \left( {i,j} \right) \in \textit{NN}_T , m\in \mathcal {M}, t=1,\ldots ,T, {l}'\in \textit{LN},\ell \qquad \end{aligned}$$

(80)

$$\begin{aligned}&f_{i,j,m}^{+t,l} +f_{i,j,m}^{-t,l} =\sum _s {pf_{i,j,m}^{t,l,s} }\qquad \forall \left( {i,j} \right) \in \textit{NE}_T , m\in \mathcal {M},l\in \textit{LE} \end{aligned}$$

(81)

$$\begin{aligned}&f_{\ell ,i,j,m}^{+t,{l}'} +f_{\ell ,i,j,m}^{-t,{l}'} =\sum _s {pf_{i,j,m}^{t,{l}',s} }\qquad \forall \left( {i,j} \right) \in \textit{NN}_T , m\in \mathcal {M},{l}'\in \textit{LN},\ell \end{aligned}$$

(82)

where $r_{j,i}^{{l}',s}$ and $r_{j,i}^{l,s}$ are equal to:

$$\begin{aligned} r_{i,j}^{{l}',s} =\frac{\mathcal {E}_{i,j}^{{l}'} }{\left( {\Gamma _{i,j}^{{l}'} } \right) ^{2}}. \frac{a_s^2 -a_{s-1}^2 }{a_s -a_{s-1} }.\bar{{f}}_{i,j}^{{l}'} \qquad r_{i,j}^{l,s}= & {} \frac{\mathcal {E}_{i,j}^l }{\left( {\Gamma _{i,j}^l } \right) ^{2}}. \frac{a_s^2 -a_{s-1}^2 }{a_s -a_{s-1} }.\bar{{f}}_{i,j}^l \end{aligned}$$

(83)

It is clear that constraint sets (71) ensure the constraint sets (79) and (80); therefore they are eliminated from the constraints sets. Finally, nonlinearity due to the products of discrete and continuous variables in Eq. (5) is eliminated via substituting this equation with Eqs. (84) and (85).

$$\begin{aligned}&f_{\ell ,j,i,m}^{+t,{l}'} -f_{\ell ,j,i,m}^{-t,{l}'} -\Gamma _{i,j}^{{l}'} . \left( {\theta _{m,i}^t -\theta _{m,j}^t } \right) \le M_{ij} \left( {1-Y_{\ell ,i,j}^{t,{l}'} } \right) \nonumber \\&\qquad \begin{array}{l} \forall \left( {i,j} \right) \in \textit{NN}_T , m\in \mathcal {M}, t=1,\ldots ,T, \\ {l}'\in \textit{LN},\ell \\ \end{array}\end{aligned}$$

(84)

$$\begin{aligned}&-f_{\ell ,j,i,m}^{+t,{l}'} +f_{\ell ,j,i,m}^{-t,{l}'} +\Gamma _{i,j}^{{l}'} .\left( {\theta _{m,i}^t -\theta _{m,j}^t } \right) \le M_{ij} \left( {1-Y_{\ell ,i,j}^{t,{l}'} } \right) \nonumber \\&\qquad \begin{array}{l} \forall \left( {i,j} \right) \in \textit{NN}_T , m\in \mathcal {M}, t=1,\ldots ,T, \\ {l}'\in \textit{LN},\ell \\ \end{array} \end{aligned}$$

(85)

where, $M_{ij}$ indicates a large enough number.

Appendix 3

In this appendix, we briefly describe the concept of $\alpha$-covering and $\eta$ parameter used in the CCB method proposed by Saharidis et al. (2010). Without loss of generality the following gives linear problem:

where, $c\in \mathfrak {R}^{n},d\in \mathfrak {R}^{q},b\in \mathfrak {R}^{m}$, A and B are $m \times n$ and $m \times q$ matrices.

In Benders decomposition we decompose the IP into PSP, which is a restriction of IP and provides an upper bound (UB) in the case of minimization, and the following relaxation of IP, which is called the restricted master problem (RMP), provides a lower bound (LB). In practice it is not the PSP which is solved in each iteration but the dual of PSP which has the following form:

A low (high) density cut is a cut covering a small (large) number of decision variables of RMP. A decision variable is covered in a cut if its coefficient is not equal to zero (or near to zero relative to the other coefficients).

Definition

In a feasibility cut, a variable $y_{k}$ is said to be $\alpha$-covered in the cut of the form $\sum _k\left( {v^{T}B} \right) _k y_k \ge v^{T}b$ if the kth row of the matrix $(v^{T}B)$ is greater than or equal to $\alpha \%$ of the coefficient with the maximum absolute value in the current cut: ${\vert }(v^{T}B)_{k}{\vert } \ge 10^{-2} \alpha \mathrm{Max}_{\forall k}\{{\vert }(v^{T}B)_{k}{\vert }\}$ where $\alpha$ is a given parameter chosen in [0, 100 %].

The above definition is used in the case of feasibility cut. In the case of optimality cut the vector $u \in U$ is used instead of the vector $v \in V$. Also, the parameter $\eta$ is the average of the coefficients of $\alpha$-covered decision variables in the classical Benders cut and takes the following value in each iteration.

$$\begin{aligned} \eta =\left( \frac{1}{k} \right) \sum _j \left| {\left( {v^{T}B} \right) _j } \right| ,\quad j\in \left\{ j:\left| \left( {v^{T}B} \right) _j \right| \ge \alpha \mathop {Max}\limits _{\forall j} \left\{ \left| \left( {v^{T}B} \right) _j \right| \right\} \right\} \end{aligned}$$

Appendix 4

See Figs. 4 and 5.

Fig. 4

6 and 21 Bus power systems

Fig. 5

48 Bus power system