Computationally Efficient Adjustment of FACTS Set ...

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Computationally Efficient Adjustment of FACTS Set Points in DC Optimal Power Flow with Shift Factor Structure M. Sahraei-Ardakani, Member, IEEE, and K. W. Hedman, Member, IEEE

Abstract—Enhanced utilization of the existing transmission network is a cheaper and paramount alternative to building new transmission lines. Flexible AC transmission system (FACTS) devices are advanced technologies that offer transfer capability improvements via power flow control. Although many FACTS devices exist in power systems, their set points are not frequently changed for power flow control purposes, which is mainly due to the computational complexity of incorporating FACTS flexibility within the market problem. This paper proposes a computationally efficient method for adjustment of variable impedance based FACTS set points, which is also compatible with existing market solvers. Thus, the method can be employed by the existing solvers with minimal modification efforts. This paper models FACTS reactance control as injections to keep the initial shift factors unchanged. Next, the paper formulates a DC optimal power flow that co-optimizes FACTS set points alongside generation dispatch. The resulting problem, which is in a nonlinear program, is then reformulated to a mixed-integer linear program. Finally, an engineering insight is leveraged to further reduce the computational complexity to a linear program. Simulation studies on IEEE 118-bus and Polish 2383-bus test cases show that the method is extremely effective in finding quality solutions and being very fast. Index Terms—FACTS devices, linear programming, optimal power flow, power system economics, power system operation, power transfer distribution factors.

I. NOMENCLATURE Matrices are represented by upper case bold, vectors by lower case bold, and scalars by lower case characters. Superscript D is used for the diagonal operator, while superscript T represents transpose of a matrix. is used to describe reduced matrices and vectors where the elements related to the reference bus are removed. Finally, is used to show the flow on the lines equipped with FACTS without the FACTS impact. A. Parameters 𝐹 Set of transmission elements equipped with FACTS devices. 𝑁! Number of transmission elements. Mostafa Sahraei-Ardakani is with the Department of Electrical and Computer Engineering at the University of Utah, Salt Lake City, UT 84112 USA (e-mail: [email protected]). Kory W. Hedman is with the School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287 USA (e-mail: [email protected]).

𝑘 𝒑 𝒑

Index for transmission elements. Upper limit vector of power generation. Lower limit vector of power generation.

𝒇 𝒇𝑬

Vector of transmission elements’ thermal limit. Vector of transmission elements’ emergency thermal limit. Vector of generators’ marginal costs. Vector of power consumptions. Upper limit vector of power injections representing FACTS devices. Lower limit vector of power injections representing FACTS devices. Upper limit vector of FACTS relative susceptance adjustment. Lower limit vector of FACTS relative susceptance adjustment. Upper limit for FACTS relative susceptance adjustment on transmission element k. Lower limit for FACTS relative susceptance adjustment on transmission element k. Generation locator matrix. Adjacency matrix. Reduced Adjacency matrix. Branch B matrix. Nodal B matrix. Base susceptance of transmission element k. PTDF matrix. LODF, representing the change on line l’s flow with respect to the outage of line k. Vector of ones. Vector of zeros. Identity matrix. A very large positive number.

𝒄 𝒅 𝝌 𝝌 𝜹 𝜹 𝛿! 𝛿! 𝜞 𝑨 𝑨 𝑩!" 𝑩 𝑏! 𝜱 𝜉!! 𝟏 𝟎 𝑰 𝑀

B. Variables 𝒑 𝒇 𝝍 𝝍 𝝌 𝝌 𝜹

Vector of power generations. Vector of power flows on transmission elements. Vector of power injections. Reduced vector of power injections. Vector of power injections representing FACTS devices. Reduced vector of power injections representing FACTS devices. Vector of FACTS relative susceptance adjustments.

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𝒛

Vector of binary variables identifying the direction of power flows. 𝑓! Power flow on transmission element k. 𝜃!,!" Voltage angle at the “to” bus of transmission element k. 𝜃!,!"#$ Voltage angle at the “from” bus of transmission element k. Δ𝑏! Changes in the susceptance of transmission element k by FACTS.

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II. INTRODUCTION

N many parts of the United States, the transmission network is under stress and needs to be upgraded [1]-[2]. High congestion costs, e.g. $670 million in 2013 for PJM [3], is one indicator of the scarcity of transfer capability over the existing transmission network. A trivial but unattractive solution to this problem would be to build new transmission lines. New transmission projects are extremely costly, take many years to complete, and are not preferred by the people living near the new lines. NIMBY (Not In My Back Yard) and BANANA (Build Absolutely Nothing Anywhere Near Anything) are two commonly used terms to refer to this problem. A cheaper, faster, and environmentally more appealing alternative to building new transmission lines is more efficient utilization of the existing system. This objective can be achieved with power flow control technologies, including transmission switching (TS) [4]-[6] and flexible AC transmission system (FACTS) devices [7]-[10]. With power flow control, the transfer capability over the existing network can be significantly improved [11]. Transfer capability improvement can reduce operational cost [12]-[15], improve system reliability [16]-[21], and enhance integration of renewable energy resources [22]-[23], thereby delaying the need for new transmission lines. Specific types of FACTS devices can provide significant power flow control [24]-[25] through adjustment of transmission elements’ reactances. This paper loosely employs the term “FACTS” to refer to those specific types of FACTS devices, which are known as variable impedance based FACTS: such as thyristor protected series compensator (TPSC) / thyristor controlled series compensator (TCSC) and unified power flow controller (UPFC). Despite the ability of the existing FACTS devices to provide power flow control, day-ahead security-constrained unit commitment (SCUC) or real-time security-constrained economic dispatch (SCED) rarely capture their full flexibility. This is mainly due to the lack of economic incentive [26]-[29] and computational complexity of modeling the flexibility of FACTS devices in SCUC and SCED [20], [30]. Solving the optimal power flow problem in its original nonlinear form is beyond today’s computational capabilities. The industry, thus, employs a linearized form of optimal power flow problem within the SCUC and SCED solvers. It is referred to as DC optimal power flow (DCOPF), which in its original form is a linear program (LP). Inclusion of FACTS flexibility would make the DCOPF a non-linear program (NLP), which is much more computationally burdensome than

the initial LP formulation of the problem. A recent work by the authors has investigated reformulation of the resulting NLP to a mixed-integer linear program (MILP) [30]. However, the formulation presented in [30] is developed using the 𝐵𝜃 DCOPF structure, which is not compatible with the industry practices. With a 𝐵𝜃 formulation, all the bus voltage angles need to be calculated in order to compute the line flows. The formulation, thus, does not scale well with the size of the power system [31]. Note that real power systems have tens of thousands of buses and transmission elements. Therefore, a reduced formulation with power transfer distribution factors (PTDF) is often used in SCUC and SCED solvers. With the use of PTDFs, the need for calculation of bus voltage angles vanishes. Moreover, only transmission elements, suspected to become overloaded, are monitored for thermal limit violations. DCOPF formulation with PTDF structure is explained in more details in the next section. Industrial implementations of SCED and SCUC employ a variety of methods to improve the speed and accuracy of their linearized PTDF-based optimal power flow model [32]-[34]. SCUC and SCED solvers become significantly faster as the number of transmission elements, whose flows are monitored for thermal limit violations, decreases. For instance, California Independent System Operator employs an adjustable threshold for flows to select the lines that are needed to be monitored. Any transmission element that is loaded above this threshold is included in SCED as a constraint [34]. Since the operator has information on the current flows, this threshold is often picked at levels very close to the thermal limit of the element for the next round of SCED. Therefore, the subset of precontingency and post-contingency line flows that are monitored is significantly smaller than the full set (e.g., hundreds to thousands in comparison to millions). PTDFs are calculated offline based on the topology and susceptance information. FACTS devices change the susceptance of the transmission elements and make the original PTDFs invalid. Therefore, the PTDF matrix needs to be recalculated every time the transmission network is changed either through transmission switching or adjustment of FACTS set point. Moreover, inclusion of FACTS adjustments makes DCOPF, an NLP. This paper contributes to the literature by addressing these two problems. First, the impact of FACTS devices is isolated form the initial PTDF matrix. Second, the resulting NLP is reformulated to an MILP and then to an LP. In order to avoid variable PTDFs, the impact of FACTS adjustments are modeled as additional injections at the two ends of the elements equipped with FACTS [22], [35]-[36]. Using this technique, the PTDF matrix remains constant with any possible adjustment of FACTS devices. The paper further shows that even with the constant PTDF matrix, the resulting problem remains an NLP. Building upon authors’ prior work [30], this NLP is reformulated to an MILP. An engineering insight is then introduced to further reduce the computational complexity of the problem to an LP. The method developed in this paper employs a similar structure to what is used in industry implementations of SCUC and SCED: PTDF-based power flow. Thus, the existing mar-

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ket solvers can be modified with minimal effort to incorporate this method and exploit the flexibility of FACTS devices. Through the co-optimization of FACTS set points alongside generation dispatch, cost savings [30] as well as reliability gains can be achieved [20]. This framework will also enable the system operators to effectively reduce spillage of intermittent generation from renewable energy resources [22]. To show the effectiveness of the method, simulation studies on IEEE 118-bus system, as well as the Polish system are conducted. The results are compared with those obtained by the 𝐵𝜃 formulation [30]. The results suggest that the algorithm developed in this paper is extremely effective in reducing the computational complexity of the problem and achieving quality solutions. The algorithm is significantly faster than [30] because it employs a PTDF structure and has better scaling behavior with respect to the size of the power system. Although optimality is not guaranteed, the results show that the method almost always find the optimal solution. Even when the optimal solution is not achieved, the solution obtained by the LP method is very close to the optimal solution found by the MILP algorithm. The rest of this paper is organized as follows: Section 3 presents the methodology and derives the formulation. Simulation studies on IEEE 118-bus system and the large-scale Polish system are presented in Section 4. Conclusions are drawn in Section 5. III. FORMULATION A DCOPF in 𝐵𝜃 form is shown in (1)-(5). Note that all the comparison operators for the vectors, throughout this paper, are meant to operate on an element to element basis. In this formulation, the line flows are calculated individually through their susceptance and voltage angle different. The formulation also creates inter-dependence between the flows of the transmission elements through bus voltage angles. Therefore, all the bus voltage angles need to be calculated before the flows can be compared with the transmission elements’ thermal limit (4). Such calculations become increasingly challenging with the size of meshed power systems. min 𝒄! 𝒑 𝒑≤𝒑≤𝒑

(1) (2)

𝒇 = 𝑩𝑩𝒓 𝑨! 𝜽 −𝒇 ≤ 𝒇 ≤ 𝒇 𝜞𝒑 − 𝒅 + 𝑨𝒇 = 𝟎

(3) (4) (5)

Given the size of real power systems, energy management systems (EMS) and market management system (MMS) software prevents such burdensome calculations by employing an alternative formulation based on shift factors. Using shift factors known as PTDFs, the need for calculation of bus voltage angles disappears and transmission flow calculations can be limited to only a subset of transmission elements. System operators are only concerned about a rather limited subset of transmission assets for thermal limit violations. Therefore, the market solver can only include those transmission elements to improve its computational capabilities. A PTDF-based OPF can be obtained by replacing (3) with:

𝒇 = 𝜱𝝍 𝝍 = 𝜞𝒑 − 𝒅

(6) (7)

PTDFs are calculated offline through network topology information as well as susceptance data: 𝜱 = 𝑩𝑩𝒓 𝑨𝑩!𝟏 𝑩 = 𝑨𝑩𝑩𝒓 𝑨!

(8) (9)

Series variable impedance FACTS devices offer power flow control through adjustment of the transmission element’s impedance. Such adjustment translates into a range for the transmission element’s susceptance in the DCOPF formulation. Therefore, the branch susceptance elements in (3) would no longer be constants for the transmission assets equipped with FACTS devices. The susceptance for those assets can be anywhere in the acceptable range of the FACTS device: 𝑏!!"# ≤ 𝑏! ≤ 𝑏!!"#

∀𝑘 ∈ 𝐹

(10)

The changes to the susceptance would propagate through PTDF calculations in (7)-(8). Since the PTDF calculations are performed offline, it is not suitable to model the flexibility of FACTS devices through their direct impact on the PTDF matrix. This problem has been discussed in the literature, for different problems, with the straightforward solution of modeling FACTS devices as injections [33]-[34]. The power flow on a transmission element equipped with FACTS can be calculated using DC power flow equation, with the impact of the FACTS adjustment separated from the line’s initial susceptance: 𝑓! = 𝑏! 𝜃!,!" −   𝜃!,!"#$ + Δ𝑏! 𝜃!,!" −   𝜃!,!"#$ = 𝑓! +

!!! !!

𝑓!

(11)

Thus, impact of FACTS adjustment can be modeled with injections at the two ends of the transmission element. This procedure is shown in Fig. 1. Since this modeling technique keeps the line’s initial susceptances unchanged, the PTDF matrix will remain unchanged as well. Each FACTS device will be modeled by two additional injections at the two ends of the transmission element as defined in (12). from

to 𝑏𝑘 + 𝛥𝑏𝑘

from

to 𝑏𝑘 𝛥𝑏! !   𝑓! ! 𝑏!

𝛥𝑏! !   𝑓! ! 𝑏!

Fig. 1. Conversion of FACTS adjustments to additional injections, while keeping the line’s susceptance unchanged.

In order to include the FACTS injections in the PTDFbased DCOPF formulation, they can be modeled as a vector:

4 !!! !!

𝜹=

⋮

!!!! ! !! !

𝝌 = 𝜹 𝒇     𝜹≤𝜹≤𝜹

(12) (13) (14)

where superscript D is the diagonal operator, creating a diagonal matrix from the elements of a vector. Note that the elements of 𝜹 vector, which correspond to transmission elements with no FACTS device, will be zero since the numerator is zero for those terms. Using (13), (6) can be rewritten as: 𝒇 = 𝜱 𝝍 + 𝑨𝝌 = 𝜱 𝝍 + 𝑨𝜹! 𝒇     ,

(15)

which can, then, be rearranged: 𝑰 − 𝜱𝑨𝜹! 𝒇 = 𝜱𝝍.

(16)

It is clear from (16) that the resulting DCOPF is a nonlinear program (NLP), since it includes production of FACTS adjustment vector, 𝜹, and power flow, 𝒇. NLPs are computationally expensive and are, thus, avoided in SCED and SCUC formulations. The nonlinearity that FACTS devices create in the DCOPF problem is a major barrier for enhanced operation of these devices. To avoid the product term in (16), FACTS injection can be represented as a range with inequality constraints instead of the equality constraint presented in (13). To do so, (14) can be reformulated to: 𝝌≤𝝌≤𝝌

(17)

The upper and lower limits of FACTS injection can be identified based on the direction of the flow and the limits on FACTS adjustment: for  𝑓! ≥ 0:                  𝛿! 𝑓! ≤ 𝜒! ≤ 𝛿! 𝑓!

(18)

for  𝑓! < 0:                𝛿! 𝑓! ≤ 𝜒! ≤ 𝛿! 𝑓!

(19)

These conditional inequalities can be modeled with binary variables: 1 − 𝑧! 𝛿! 𝑓! + 𝑧! 𝛿! 𝑓! ≤ 𝜒! ≤ 1 − 𝑧! 𝛿! 𝑓! + 𝑧! 𝛿! 𝑓! (20) 1 − 𝑧! 𝑓! ≥ 𝑧! 𝑓!

(21)

where 𝑧! takes a value of 0 for positive power flows and a value of 1 for negative power flows. Equations (20)-(21) include products of the binary variable 𝑧! and the power flow 𝑓! , making the resulting DCOPF a mixed integer nonlinear program (MINLP). However, using the big M reformulation [37] technique, (20)-(21) can be reformulated as a set of linear constraints: 𝛿! 𝑓! − 𝑧! 𝑀 ≤ 𝜒! ≤ 𝛿! 𝑓! + 𝑧! 𝑀 1 − 𝑧! −𝑀 + 𝛿! 𝑓! ≤ 𝜒! ≤ 1 − 𝑧! 𝑀 + 𝛿! 𝑓! 𝑓! ≥ −𝑧! 𝑀 1 − 𝑧! 𝑀 ≥ 𝑓!

(22) (23) (24) (25)

Thus, the DCOPF can be reformulated as a mixed integer linear program (MILP) with no loss of precision:

min 𝒄! 𝒑 𝑝≤𝑝≤𝑝

(26) (27)

𝒇 = 𝜱(𝝍 + 𝑨𝝌) 𝝍 = 𝜞𝒑 − 𝒅 𝜹! 𝒇 − 𝑀𝒛 ≤ 𝝌 ≤ 𝜹! 𝒇 + 𝑀𝒛 𝜹! 𝒇 − 𝑀 𝟏 − 𝒛 ≤ 𝝌 ≤ 𝜹! 𝒇 + 𝑀 𝟏 − 𝒛 𝒇 ≥ −𝑀𝒛 𝑀 𝟏−𝒛 ≥𝒇 −𝒇 ≤ 𝒇 + 𝝌 ≤ 𝒇 𝜞𝒑 − 𝒅 + 𝑨(𝒇 + 𝝌) = 𝟎 𝒛 ∈ 0,1 !!

(28) (29) (30) (31) (32) (33) (34) (35) (36)

MILPs, though less challenging than MINLPs, are still computationally expensive problems and are, thus, not preferred. As mentioned before, the binary variables in (26)-(36) only identify the direction of the power flow on the lines equipped with FACTS devices. Assuming that those directions are known, the binary variables in (26)-(27) can be fixed and the problem will become an LP. FACTS devices are often installed on major lines, where the power flow direction is relatively predictable. For instance, it is trivial to predict the flow direction on key corridors like the California-Oregon intertie (COI). Even for the cases when power flow directions are not known, the operator can use information from previous rounds of SCUC or SCED to estimate, with a high accuracy, this direction. The operator can also solve an initial DCOPF without considering FACTS devices and assign the same power flow direction to (26)-(36). It is not likely that the power flow directions change after FACTS adjustments on the lines equipped with FACTS. On the contrary, it is expected that the flows increase, in the same direction, on the paths parallel to the congested lines. Using this engineering insight, the MILP presented in (26)-(36) can be solved as a LP. Note that, although it is highly unlikely that the power flow directions on the lines equipped with FACTS change in the optimal solution, it is not impossible. Therefore, there is no guarantee that the solution to the LP version of the problem will be optimal. However, the significance of the proposed method does not vanish in absence of optimality; the algorithm will still very quickly find a solution with significantly lower cost. The results in the next section show that the algorithm almost always finds the optimal solution, and does so significantly faster than the original MILP formulation. As mentioned before, not all power flows are monitored for thermal limit violation in PTDF-based DCOPF problems. Therefore, (34) is only considered for a subset of transmission assets. However, all the transmission elements equipped with FACTS should be monitored in (34) to ensure the FACTS injections remain within the acceptable range. A. Contingency Constraints Modern SCUC and SCED solvers include explicit representation of a limited subset of contingency constraints. The impact of each transmission outage is represented through linear sensitivities known as line outage distribution factors (LODF). LODFs can be calculated through PTDFs and, thus, are readily available [31]. A contingency constraint representing line l’s flow after the outage of line k is shown in (37).

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−𝑓!! ≤ 𝑓! + 𝜉!! 𝑓! ≤ 𝑓!!

(37)

Note that, both 𝑓! and 𝑓! are represented through linear expansions using the PTDF structure (28). Thus, inclusion of contingency constraints like (37) would not add to the complexity of the problem, and the same logic explained before can be used to turn the problem into an LP. In the case that either, or both, of the lines k and l are equipped with FACTS devices, the linear technique presented in (26)-(36) still holds and can be used in contingency constraint modeling similar to the normal operations.

range, for 𝐵𝜃 and PTDF based methods presented here and in [30]. The figure shows that the PTDF-based DCOPF is faster than the 𝐵𝜃-based model both in the MILP and LP forms. Note that, monitoring fewer lines can further reduce the computational time. This is tested with simulations on a largescale Polish 2383-bus system.

IV. SIMULATION STUDIES To evaluate the effectiveness of the proposed method, simulation studies on IEEE 118-bus system as well as the Polish 2383-bus system are conducted. The results obtained here are compared to those presented in [30]. A. IEEE 118-Bus System The data is taken from [38], modified, and supplemented with cost data from [39]. The full dataset matches [4], [13][14] and in particular [30] Fig. 2 shows the savings for IEEE 118 bus system, for the case that FACTS devices are installed on the more heavily utilized lines. The total system cost is $2074/h when there is no FACTS device in the system. The results presented in Fig. 2 show the savings for the case where up to 20 FACTS devices with different range of reactance controls are used. These results are consistent with those obtained in [30].

Fig. 3. Average computational time (over reactance control range) for 𝐵𝜃 and PTDF-based models with different number of lines equipped with FACTS devices.

B. Polish 2383-Bus System The computational gains from PTDF-based formulation of the DCOPF problem become more significant and important when the size of the system increases. To test the effectiveness of the proposed method on large-scale systems, a number of simulation studies were conducted on a Polish system. The data is taken from [40]-[41], which represents the Polish electric grid in a peak winter hour in year 2000. The system includes 2383 buses, 2896 transmission lines, and 327 generators. The difference between the cost obtained by DCOPF and transportation model is $30,886, which is the upper limit for savings that can be achieved with power flow control technologies such as FACTS devices. Fig. 4 shows the savings achieved with up to 20 FACTS devices with different reactance control ranges, when FACTS devices are installed on the lines that carry larger flows. This is similar to simulation studies presented in [30] for the Polish system.

Fig. 2. Savings for IEEE 118-bus system with different number of FACTS devices and different reactance control ranges. The FACTS devices are installed on lines that are more heavily utilized and the reactance control range is shown on the figure for each set of the results.

As previously mentioned, the major advantage of PTDFbased formulation of SCED and SCUC is the ability to only monitor the flow of the lines that are of concern. For the results shown in Fig. 2, only line flows that are more than 70% of their capacity in the base case (no FACTS) are monitored. For all the simulations conducted in Fig. 2, the computational time was stored. Analysis of the results showed that the computational time heavily depends on the number of FACTS devices. The correlation between FACTS reactance range and the computational time was very weak. Fig. 3 shows the computational time, averaged over FACTS reactance control

Fig. 4. Savings for the Polish system with different number of FACTS devices and different reactance control range. The FACTS devices are installed on lines that carry larger power and the reactance control range is shown on the

6 figure for each set of the results.

The computational time for the results shown in Fig. 4 is presented in Fig. 5. For the PTDF-based models, only lines whose flow is larger than 70% of the line’s capacity in the base case (with no FACTS) are monitored. The computational time is averaged over the reactance control range of the FACTS devices, as the computational time heavily depends on the number of lines equipped with FACTS, and not so much on the reactance control range of the devices. The results show that the PTDF-based formulation is consistently faster than the 𝐵𝜃 model. The results also show that unlike the MILP models that get computationally more expensive with additional number of FACTS devices, the LP-based formulations show a robust behavior with respect to the computational time.

Fig. 5. Average computational time (over reactance control range) for 𝐵𝜃 and PTDF-based models with different number of lines equipped with FACTS devices. The FACTS devices are installed on lines that carry larger power.

Fig. 6 shows the savings when up to 50 FACTS devices are installed on the lines that have larger flows relative to their thermal capacity. The reactance control range of the devices is shown on the figure. Similar to the results shown previously, significant savings can be achieved with a limited number of FACTS devices.

Fig. 6. Savings for the Polish system with different number of FACTS devices and different reactance control ranges. The FACTS devices are installed on lines with larger flows relative to their thermal capacity and the reactance control range is shown on the figure for each set of the results.

The computational time for the results shown in Fig. 6 is presented in Fig. 7. Similar to the two sets of results presented

before, the PTDF-based methods outperform the 𝐵𝜃-based DCOPF models. The results shown in Fig. 7 were obtained by only monitoring the lines with flows more than 70% of their thermal capacity in the base case. Since the maximum number of lines with FACTS devices is 50 in Fig. 7, as opposed to 20 for Fig. 3 and 5, Fig. 7 better shows how the computational time grows in MILP-based implementation of FACTS adjustments. The linear methods, however, maintain the same level of complexity for different number of lines with FACTS. It should be noted that the difference between the computational time needed for the 𝐵𝜃-based and the PTDF-based models becomes significantly larger with the size of the power system. The difference will also become larger in computationally more difficult problems such as SCUC, which is an MILP with many binary variables.

Fig. 7. Average computational time (over reactance control range) for 𝐵𝜃 and PTDF-based models with different number of lines equipped with FACTS devices. The FACTS devices are installed on lines with larger flows relative to their thermal capacity.

As discussed before, with PTDF-based methods the number of monitored lines can be reduced. This subset of critical lines can be chosen either with the operator’s knowledge or the information present at the time of operation. For instance, real-time SCED can use the state estimation information including the current line flows. Thus, SCED can use such information to identify the lines that need to be monitored for thermal flow violation. As an example, only the lines with flows larger than 95% of their thermal capacity can be fed into SCED [34]. The computational time decreases as this subset of lines becomes more exclusive. To examine the impact of having fewer lines monitored in DCOPF, different thresholds on the line flow are studied. In particular, lines at 70% and 80% of capacity or above in the base case are picked as fixed thresholds. Three levels of adaptive thresholds are also tested. The difference between adaptive and fixed thresholds is that the information fed into the adaptive threshold includes the most recent run, i.e. the power flow in the previous simulation with one less number of FACTS devices. Therefore, the power flow information used for the adaptive thresholds is more accurate than the fixed thresholds. With the adaptive threshold, less number of lines can be monitored without violating the thermal limits. Note that, the lines equipped with FACTS devices should always be monitored to ensure the FACTS injections remain within the feasible range.

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Fig. 8 and Fig. 9 show the computational time with fixed and adaptive thresholds for the results shown in Fig. 4 and Fig. 6 respectively. Both figures show that employment of a method to reduce the number of lines, whose flows are monitored in DCOPF, significantly reduces the computational complexity of the FACTS adjustment problem. In particular, the method presented in this paper gained a speedup factor of around 2 by using an adaptive threshold instead of the fixed 70% threshold. Note that lines equipped with FACTS should always be monitored. Thus, monitoring fewer lines would lead to a significant improvement of the computational efficiency with fewer FACTS devices. Increasing the number of FACTS devices, and consequently increasing the number of lines needed to be monitored, would increase the computational time. The candidate lines for FACTS are likely already a part of the lines identified by the fixed 70% thresholds. However, many of those lines are not included in the set specified by tighter thresholds. Therefore, the computational time saving reduces as the number of lines equipped with FACTS devices increases.

and compatible with the industry implementations. Therefore, it is straightforward to add the proposed FACTS set point adjustment routine to the existing SCED and SCUC solvers. The results presented in this paper show that: 1. A few number of FACTS devices with mediocre reactance control range can bring significant savings. 2. The PTFD-based models outperform the 𝐵𝜃-based methods in terms of computational time. 3. The engineering insight that was used to reformulate the MILP-based DCOPF to a LP model is extremely effective. It is significantly faster than the MILP model and almost always finds the optimal solution. In fact, only 3 out of the 520 cases simulated in the paper, converged to a suboptimal solution using the LP-based method. For this 0.6% of the cases, which occurred for the Polish 2383-bus system, the difference between the suboptimal solution achieved by the LP-based model, and the optimal solution reached by the MILP-based method, was below $3. Note that the potential savings that can be achieved with FACTS devices is more than $30,000 for the Polish 2383bus system. Therefore, even when the LP-based method does not find the optimal solution, it finds a solution that is very close to the optimal solution and does that very quickly relative to the MILP-based method. V. CONCLUSIONS

Fig. 8. Average computational time (over reactance control range) for the PTDF-based models with different thresholds for the results shown in Fig. 4.

Fig. 9. Average computational time (over reactance control range) for the PTDF-based models with different thresholds for the results shown in Fig. 6.

C. Discussion The method proposed in this paper models the flexibility of variable impedance FACTS devices within PTDF-based DCOPF. Since SCUC and SCED solvers in the industry employ the same structure, the model presented here is consistent

Variable impedance FACTS devices can provide significant power flow control and, thus, reduce the cost, improve the system reliability, and enhance the integration of renewable resources. Despite the benefits of FACTS devices, their utilization has been limited due to a variety of factors including the computational complexity of FACTS set point adjustment within the DCOPF problem. Inclusion of FACTS flexibility in the DCOPF, originally an LP, would introduce nonlinearities with significant computational burden. FACTS set point adjustment would also change the initial PTDF matrix. This paper first models the FACTS reactance control as additional injections and develops an NLP with a constant PTDF matrix. The paper, then, reformulates this nonlinear problem to an MILP. Moreover, the MILP is reformulated to an LP using an engineering insight on the power flow directions. Since the industry employs PTDF-based linear market solvers, existing SCUC and SCED solvers can be straightforwardly modified using the method developed in this paper. To include the flexibility of FACTS devices, additional injections should be added to the model, which does not impose significant additional computational efforts. Additionally, all the lines equipped with FACTS should be monitored for thermal flow violation to ensure FACTS injections remain within the device’s reactance control range. The simulation studies on IEEE-118 bus as well as the Polish 2383-bus system show that a limited number of FACTS devices can provide significant cost savings. The method presented in this paper was able to effectively identify quality solutions with minimal computational effort. In fact, the LP-based method almost always (more than 99% of the time) found the optimal solution. For the very few cases, where the optimal solution was not found,

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