Alvarez et al 2023a - Revision history

Rimni at 09:33, 31 January 2023

2023-01-31T09:33:13Z

Rimni at 09:32, 31 January 2023

2023-01-31T09:32:34Z

Rimni: /* 3.1 SCOPF formulation */

2023-01-31T09:31:52Z

‎3.1 SCOPF formulation

Rimni at 09:25, 31 January 2023

2023-01-31T09:25:32Z

Rimni at 09:23, 31 January 2023

2023-01-31T09:23:00Z

Rimni at 09:22, 31 January 2023

2023-01-31T09:22:05Z

Rimni at 09:20, 31 January 2023

2023-01-31T09:20:55Z

Rimni at 09:19, 31 January 2023

2023-01-31T09:19:37Z

Srriverar: Srriverar moved page Alvarez et al 2022b to Alvarez et al 2023a

2023-01-30T15:35:44Z

Srriverar moved page Alvarez et al 2022b to Alvarez et al 2023a

Rimni: /* 2. Background */

2023-01-26T15:45:56Z

‎2. Background

@@ Line 415: / Line 415: @@
 ===4.2 Fixed demand and generation===
-To evaluate the influence of demand response and dispatchable generators on the benefit,  a sensitivity analysis is performed executing the SCOPF with different levels of dispatchable load and generation. For that, a share of generation and demand are modeled as fixed elements injecting or consuming power. The  02_Demand_Load shows the results obtained for different ratios of fixed demand and generation. A fixed value of 1 means that all  loads or generators (<math>1\times 10^2</math>) are non-responsive or non-dispatchable. On the contrary, a fixed value of 0 means that all generators are dispatchable or all demands are responsive. As expected, for this case, the higher benefit was obtained to fix on  1   of the total elements of demand and generation. The lowest benefit was to fix a non-demand response of <math>1\times 10^2</math>  of the total loads and fixing the non-dispatchable generator at <math>6\times 10^1</math>. Overall, for the case model C2T2N00617-185 the demand response costs present a stronger influence on the benefit than the generation costs, demonstrating the potential benefits of the proposed SCOPF algorithm ([[#img-3|Figure 3]] ).
+To evaluate the influence of demand response and dispatchable generators on the benefit,  a sensitivity analysis is performed executing the SCOPF with different levels of dispatchable load and generation. For that, a share of generation and demand are modeled as fixed elements injecting or consuming power. The  02_Demand_Load shows the results obtained for different ratios of fixed demand and generation. A fixed value of 1 means that all  loads or generators (<math>1\times 10^2</math>) are non-responsive or non-dispatchable. On the contrary, a fixed value of 0 means that all generators are dispatchable or all demands are responsive. As expected, for this case, the higher benefit was obtained to fix on  1   of the total elements of demand and generation. The lowest benefit was to fix a non-demand response of <math>1\times 10^2</math>  of the total loads and fixing the non-dispatchable generator at <math>6\times 10^1</math>. Overall, for the case model C2T2N00617-185 the demand response costs present a stronger influence on the benefit than the generation costs, demonstrating the potential benefits of the proposed SCOPF algorithm ([[#img-3|Figure 3]]).
 <div id='img-3'></div>
@@ Line 424: / Line 424: @@
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 3'''. Influence of fixed loads and generators on the benefit (<math>1\times 10^6</math> USD)for the case model C2T2N00617-185
 |}
 ===4.3 Large size systems===

← Older revision		Revision as of 09:32, 31 January 2023
Line 374:		Line 374:

	</ol>		</ol>
		+

	In step 5 is exploited the capabilities of parallel computing running each contingency as an OPF without cost variables. With this novel approach, it is guaranteed that during a contingency the power network operates fulfilling all security constraints (power balance, limits of loads, generators, voltage, branches, among others) ([[#img-1\|Figure 1]] ).		In step 5 is exploited the capabilities of parallel computing running each contingency as an OPF without cost variables. With this novel approach, it is guaranteed that during a contingency the power network operates fulfilling all security constraints (power balance, limits of loads, generators, voltage, branches, among others) ([[#img-1\|Figure 1]] ).

@@ Line 212: / Line 212: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{ f_0\left(P_{g_i},u,P_{l_i} \right)+\sum_{n=1}^{N_{gen}+N_{l}} y_n+\sum_{k=1}^{K} \left( f_k\left(P_{g_i},u,P_{l_i}\right)+\sum_{n=1}^{N_{gen}+N_{l}} y_n\right)}{}{} </math>
+| style="text-align: center;" | <math>{ f_0\left(P_{g_i},u,P_{l_i} \right)+\sum_{n=1}^{N_{gen}+N_{l}} y_n+\sum_{k=1}^{K} \left( f_k\left(P_{g_i},u,P_{l_i}\right)+\sum_{n=1}^{N_{gen}+N_{l}} y_n\right)} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (15)
@@ Line 224: / Line 224: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{P_i-\operatorname{Re}\left(E_i \sum_{k=1}^{N_{bus}} Y_{ij} E_j \right)}{=0}{i=1,\ldots,N_{bus}}</math>
+| style="text-align: center;" | <math>{P_i-\operatorname{Re}\left(E_i \sum_{k=1}^{N_{bus}} Y_{ij} E_j \right)}{=0}, \qquad {i=1,\ldots,N_{bus}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (16)
@@ Line 234: / Line 234: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{Q_i-\operatorname{Im}\left(E_i \sum_{k=1}^{N_{bus}} Y_{ij} E_j \right)}{=0}{i=1,\ldots,N_{bus}}</math>
+| style="text-align: center;" | <math>{Q_i-\operatorname{Im}\left(E_i \sum_{k=1}^{N_{bus}} Y_{ij} E_j \right)}{=0}, \qquad {i=1,\ldots,N_{bus}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (17)
@@ Line 244: / Line 244: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     { P_{l_i}^{0} t_{min}\leq P_{l_i}}{\leq P_{l_i}^{0} t_{max} \quad}{i=1,\ldots,N_{l}}</math>
+| style="text-align: center;" | <math>     { P_{l_i}^{0} t_{min}\leq P_{l_i}}{\leq P_{l_i}^{0} t_{max}}, \qquad {i=1,\ldots,N_{l}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (18)
@@ Line 254: / Line 254: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     {P_{l_i}^0-\Delta P_{l_i}^d \leq  P_{l_i} }{\leq P_{l_i}^0+\Delta P_{l_i}^u}{i=1,\ldots,N_{l}} </math>
+| style="text-align: center;" | <math>     {P_{l_i}^0-\Delta P_{l_i}^d \leq  P_{l_i} }{\leq P_{l_i}^0+\Delta P_{l_i}^u}, \qquad {i=1,\ldots,N_{l}} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (19)
@@ Line 264: / Line 264: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     { \cfrac{Q_{l_{i}^0}}{P_{l_{i}^0}}  P_{l_i}-Q_{l_i} }{= 0}{i=1,\ldots,N_{l}}
+| style="text-align: center;" | <math>     { \cfrac{Q_{l_{i}^0}}{P_{l_{i}^0}}  P_{l_i}-Q_{l_i} }{= 0}, \qquad  {i=1,\ldots,N_{l}}
   </math>
 |}
@@ Line 275: / Line 275: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     {P_{g_i}^{min}\leq P_{g_i}}{\leq P_{g_i}^{max}\quad}{i=1,\ldots,N_{gen}}
+| style="text-align: center;" | <math>     {P_{g_i}^{min}\leq P_{g_i}}{\leq P_{g_i}^{max}\quad}, \qquad  {i=1,\ldots,N_{gen}}
   </math>
 |}
@@ Line 286: / Line 286: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     {P_{g_i}^0-\Delta P_{g_i}^d \leq  P_{g_i}}{\leq P_{g_i}^0+\Delta P_{g_i}^u}{i=1,\ldots,N_{gen}}
+| style="text-align: center;" | <math>     {P_{g_i}^0-\Delta P_{g_i}^d \leq  P_{g_i}}{\leq P_{g_i}^0+\Delta P_{g_i}^u}, \qquad  {i=1,\ldots,N_{gen}}
   </math>
 |}
@@ Line 297: / Line 297: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     {Q_{g_i}^{min}\leq Q_{g_i}}{\leq Q_{g_i}^{max}\quad}{i=1,\ldots,N_{gen}}
+| style="text-align: center;" | <math>     {Q_{g_i}^{min}\leq Q_{g_i}}{\leq Q_{g_i}^{max}\quad}, \qquad  {i=1,\ldots,N_{gen}}
   </math>
 |}
@@ Line 308: / Line 308: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     {Q_{s_i}^{min}\leq Q_{s_i}}{\leq Q_{s_i}^{max}\quad}{i=1,\ldots,N_{shunt}}
+| style="text-align: center;" | <math>     {Q_{s_i}^{min}\leq Q_{s_i}}{\leq Q_{s_i}^{max}}, \qquad  {i=1,\ldots,N_{shunt}}
   </math>
 |}
@@ Line 319: / Line 319: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     { P_{s_i} }{= 0}{i=1,\ldots,N_{shunt}}
+| style="text-align: center;" | <math>     { P_{s_i} }{= 0}, \qquad {i=1,\ldots,N_{shunt}}
   </math>
 |}
@@ Line 330: / Line 330: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     {E_{i}^{min}\leq E_i}{\leq E_{i}^{max}\quad}{i=1,\ldots,N_{bus}}
+| style="text-align: center;" | <math>     {E_{i}^{min}\leq E_i}{\leq E_{i}^{max}}, \qquad {i=1,\ldots,N_{bus}}
   </math>
 |}
@@ Line 341: / Line 341: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     {S_{ij}}{\leq S_{ij}^{max}\quad}{i,j=1,\ldots,N_{branch}}
+| style="text-align: center;" | <math>     {S_{ij}}{\leq S_{ij}^{max}}, \qquad {i,j=1,\ldots,N_{branch}}
   </math>
 |}
@@ Line 352: / Line 352: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>     {m_j \left(x_i-x_j \right)+c_j}{\leq y_i\quad}{i=1,\ldots,N_{gen}+N_{l}}
+| style="text-align: center;" | <math>     {m_j \left(x_i-x_j \right)+c_j}{\leq y_i} , \qquad  {i=1,\ldots,N_{gen}+N_{l}}
   </math>
 |}

@@ Line 68: / Line 68: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>P_{g_i}^{min}\leq P_{g_i}\leq P_{g_i}^{max}, \qquad i=1,\ldots,N_{gen}</math>
+| style="text-align: center;" | <math>P_{g_i}^{\min}\leq P_{g_i}\leq P_{g_i}^{\max}, \qquad i=1,\ldots,N_{gen}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (4)
@@ Line 88: / Line 88: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>Q_{g_i}^{min}\leq Q_{g_i}\leq  Q_{g_i}^{max},\qquad i=1,\ldots,N_{gen}</math>
+| style="text-align: center;" | <math>Q_{g_i}^{\min}\leq Q_{g_i}\leq  Q_{g_i}^{\max},\qquad i=1,\ldots,N_{gen}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (6)
@@ Line 98: / Line 98: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>E_{i}^{min}\leq E_i\leq E_{i}^{max},\qquad i=1,\ldots,N_{bus}</math>
+| style="text-align: center;" | <math>E_{i}^{\min}\leq E_i\leq E_{i}^{\max},\qquad i=1,\ldots,N_{bus}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (7)
@@ Line 108: / Line 108: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>S_{ij}\leq S_{ij}^{max},\qquad i,j=1,\ldots,N_{branch} </math>
+| style="text-align: center;" | <math>S_{ij}\leq S_{ij}^{\max},\qquad i,j=1,\ldots,N_{branch} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (8)
 |}
-where <math display="inline">P_{g_i}</math> is the active power generated by the generator <math display="inline">i</math>, <math display="inline">f</math> is the cost function, <math display="inline">N_{gen}</math> is the number of generators, <math display="inline">P_i</math> is the injected power at bus <math display="inline">i</math>, <math display="inline">E</math> is the absolute value of bus voltage, <math display="inline">P_{g_i}^0</math> is the generating power at prior condition, <math display="inline">\Delta P_{g_i}^d</math>,<math display="inline">\Delta P_{g_i}^d</math> are the down and up generator ramps, <math display="inline">N_{bus}</math> is the number of buses, <math display="inline">S_{ij}</math> is the power flow by branch, <math display="inline">Y_{ik}</math> is the branch admittance between bus <math display="inline">i</math> and <math display="inline">k</math>, <math display="inline">P_{g}^{min}</math> and <math display="inline">P_{g}^{max}</math>, and <math display="inline">Q_{g}^{min}</math> and <math display="inline">Q_{g}^{max}</math> are active and reactive generators limits, <math display="inline">E_{i}^{min}</math> and <math display="inline">E_{i}^{max}</math> are bus voltage limits, <math display="inline">S_{ij}^{max}</math> is the branch thermal limit. Finally, <math display="inline">u</math> is the vector of decision and state variables, bus angles <math display="inline">\left(\theta \right)</math> and voltages, and active and reactive generator power.
+where <math display="inline">P_{g_i}</math> is the active power generated by the generator <math display="inline">i</math>, <math display="inline">f</math> is the cost function, <math display="inline">N_{gen}</math> is the number of generators, <math display="inline">P_i</math> is the injected power at bus <math display="inline">i</math>, <math display="inline">E</math> is the absolute value of bus voltage, <math display="inline">P_{g_i}^0</math> is the generating power at prior condition, <math display="inline">\Delta P_{g_i}^d</math>,<math display="inline">\Delta P_{g_i}^d</math> are the down and up generator ramps, <math display="inline">N_{bus}</math> is the number of buses, <math display="inline">S_{ij}</math> is the power flow by branch, <math display="inline">Y_{ik}</math> is the branch admittance between bus <math display="inline">i</math> and <math display="inline">k</math>, <math display="inline">P_{g}^{\min}</math> and <math display="inline">P_{g}^{\max}</math>, and <math display="inline">Q_{g}^{\min}</math> and <math display="inline">Q_{g}^{\max}</math> are active and reactive generators limits, <math display="inline">E_{i}^{\min}</math> and <math display="inline">E_{i}^{\max}</math> are bus voltage limits, <math display="inline">S_{ij}^{\max}</math> is the branch thermal limit. Finally, <math display="inline">u</math> is the vector of decision and state variables, bus angles <math display="inline">\left(\theta \right)</math> and voltages, and active and reactive generator power.
 ===2.1 Switched shunt capacitors===
@@ Line 140: / Line 140: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>t_{min} \leq t \leq t_{max} </math>
+| style="text-align: center;" | <math>t_{\min} \leq t \leq t_{\max} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (10)
 |}
-where <math display="inline">t</math> is the fraction of current load and <math display="inline">t_{min}</math> and <math display="inline">t_{max}</math> are the maximum and minimum fraction of load, respectively. Hence, the resulting active and reactive parts of the load are:
+where <math display="inline">t</math> is the fraction of current load and <math display="inline">t_{\min}</math> and <math display="inline">t_{\max}</math> are the maximum and minimum fraction of load, respectively. Hence, the resulting active and reactive parts of the load are:
 {| class="formulaSCP" style="width: 100%; text-align: left;"

@@ Line 23: / Line 23: @@
 * For the SCOPF algorithm, the fixed switchable shunts are modeled as generators with an active power equal to zero and no cost, and the responsive demand is modeled as generators injecting negative power with constant power factor. This novel modeling can improve the optimal operation of modern power systems
 * The SCOPF algorithm can run on a multi-node cluster platform. The ARPA-GO platform was used to eval the effectiveness  of the SCOPF algorithm using large and realistic networks
 This paper is organized as follows: in Background the theoretical background of OPF is introduced. Section  [[#3 SC-OPF Proposal |3]]  presents the proposed methodology to compute SCOPF using parallel computing, followed by  Results, where the algorithm is assessed for different study cases.  Finally, in Conclusion, conclusions are addressed.

@@ Line 16: / Line 16: @@
 To increase the performance of the SCOPF algorithm many advantages  are obtained. However, in order to employ SCOPF during real-time operation, the solution of this optimization problem needs to be computationally efficient given the well-known issues of OPF such as non-convexity  <span id='citeF-4'></span>[[#cite-4|[4]]], network size, integer-mixed variable decisions, among others. Therefore, to reduce the solution time of SCOPF algorithms, two approaches have been proposed. For one hand, to select a set of critical contingencies  <span id='citeF-5'></span><span id='citeF-6'></span>[[#cite-5|[5,6]]]. For other hand,  to simplify the problem using a linear approximation [Mithun2010].
-Given the uncertainty in reducing the number of contingencies or using linear approximations during the planning and operation of the next-generation energy systems, a robust SCOPF assessment is required  <span id='citeF-7'></span>[[#cite-7|[7]]], However, this kind of algorithm presents a high computational cost. New approaches emerge to face this challenge, such as the use of reinforced learning (artificial intelligence algorithms) to operate networks during contingencies  <span id='citeF-8'></span>[[#cite-8|[8]]] or paralleling the Computing of the SCOPF by employing High-Performance Computing (HPC), both to reduce the computation cost. The issue with reinforced learning is that an artificial agent needs to be trained for each network with known states, therefore HPC is commonly used for different applications for power systems. For instance, in  Rodriguez et al. <span id='citeF-9'></span>[[#cite-9|[9]]] an embedded computer is used to partially solve power flows paralleling the Newton-Raphson algorithm. In  <span id='citeF-10'></span><span id='citeF-11'></span>[[#cite-10|[10,11]]] a hybrid CPU-GPU approach to solving power flows using Parallel Computing is employed. Finally, in <span id='citeF-12'></span>[[#cite-12|[12]]] a Multi-CPU/Core Computing Environment is used for contingency analysis showing an acceleration ratio of 2.69x10<sup>1</sup> compared with sequential processing. However, the OPF is not considered.
+Given the uncertainty in reducing the number of contingencies or using linear approximations during the planning and operation of the next-generation energy systems, a robust SCOPF assessment is required  <span id='citeF-7'></span>[[#cite-7|[7]]], However, this kind of algorithm presents a high computational cost. New approaches emerge to face this challenge, such as the use of reinforced learning (artificial intelligence algorithms) to operate networks during contingencies  <span id='citeF-8'></span>[[#cite-8|[8]]] or paralleling the Computing of the SCOPF by employing High-Performance Computing (HPC), both to reduce the computation cost. The issue with reinforced learning is that an artificial agent needs to be trained for each network with known states, therefore HPC is commonly used for different applications for power systems. For instance, in  Rodriguez et al. <span id='citeF-9'></span>[[#cite-9|[9]]] an embedded computer is used to partially solve power flows paralleling the Newton-Raphson algorithm. In  <span id='citeF-10'></span><span id='citeF-11'></span>[[#cite-10|[10,11]]] a hybrid CPU-GPU approach to solving power flows using Parallel Computing is employed. Finally, in  Zhou and Feng <span id='citeF-12'></span>[[#cite-12|[12]]] a Multi-CPU/Core Computing Environment is used for contingency analysis showing an acceleration ratio of 2.69x10<sup>1</sup> compared with sequential processing. However, the OPF is not considered.
 Based on the reported benefits of employing HPC in power systems applications  <span id='citeF-13'></span><span id='citeF-14'></span>[[#cite-13|[13,14]]], in this paper a novel SCOPF analysis using parallel computing is addressed. This approach maximizes the market surplus during regular operation and contingencies, considering network constraints of power generation and load ramps. Also, generators’ active and reactive capacity, lines and transformers thermal limits, switchable shunts, voltage security limits, and price responsive demand are considered. The main contributions of this paper are summarized as follows:

@@ Line 12: / Line 12: @@
 ==1. Introduction==
-Power systems must operate in energy equilibrium in a secure, reliable, economical, and sustainable way; commonly contrary issues. Hence, tools to optimize the planning, maintenance, and operation of electrical networks are required. The optimal power flow (OPF) is a well-known problem  <span id='citeF-1'></span>[[#cite-1|[1]]] where the power balance and generation, thermal and voltage limits among other constraints are considered. However, the solution of this formulation could be non-optimal or insecure if contingencies occur in the grid. To securely operate considering the multiple contingencies that could arise, the OPF needs to be reformulated considering security constraints. This reformulation is known as Security Constraint Optimal Power Flow (SCOPF). According to  <span id='citeF-2'></span>[[#cite-2|[2]]],  an improvement of  5 % in the speed efficiency of the SCOPF algorithm can save between  619$   in the  U.S and it can also save  between 2687$  in the world. Illustrating, these savings are close to  1x10<sup>1</sup>  of the 2021 annual investment in the power sector  <span id='citeF-3'></span>[[#cite-3|[3]]].
+Power systems must operate in energy equilibrium in a secure, reliable, economical, and sustainable way; commonly contrary issues. Hence, tools to optimize the planning, maintenance, and operation of electrical networks are required. The optimal power flow (OPF) is a well-known problem  <span id='citeF-1'></span>[[#cite-1|[1]]] where the power balance and generation, thermal and voltage limits among other constraints are considered. However, the solution of this formulation could be non-optimal or insecure if contingencies occur in the grid. To securely operate considering the multiple contingencies that could arise, the OPF needs to be reformulated considering security constraints. This reformulation is known as Security Constraint Optimal Power Flow (SCOPF). According to  Cain et al. <span id='citeF-2'></span>[[#cite-2|[2]]],  an improvement of  5 % in the speed efficiency of the SCOPF algorithm can save between  619$   in the  U.S and it can also save  between 2687$  in the world. Illustrating, these savings are close to  1x10<sup>1</sup>  of the 2021 annual investment in the power sector  <span id='citeF-3'></span>[[#cite-3|[3]]].
 To increase the performance of the SCOPF algorithm many advantages  are obtained. However, in order to employ SCOPF during real-time operation, the solution of this optimization problem needs to be computationally efficient given the well-known issues of OPF such as non-convexity  <span id='citeF-4'></span>[[#cite-4|[4]]], network size, integer-mixed variable decisions, among others. Therefore, to reduce the solution time of SCOPF algorithms, two approaches have been proposed. For one hand, to select a set of critical contingencies  <span id='citeF-5'></span><span id='citeF-6'></span>[[#cite-5|[5,6]]]. For other hand,  to simplify the problem using a linear approximation [Mithun2010].

← Older revision	Revision as of 15:35, 30 January 2023
(No difference)

@@ Line 47: / Line 47: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> P_i-\operatorname{Re}\left(E_i \sum_{k=1}^{N_{bus}} Y_{ij} E_j \right)=}{0,} \qquad i=1,\ldots,N_{bus}</math>
+| style="text-align: center;" | <math> P_i-\operatorname{Re}\left(E_i \sum_{k=1}^{N_{bus}} Y_{ij} E_j \right)={0,} \qquad i=1,\ldots,N_{bus}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2)
@@ Line 67: / Line 67: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>P_{g_i}^{min}\leq P_{g_i}\leq}{ P_{g_i}^{max}, \qquad i=1,\ldots,N_{gen}</math>
+| style="text-align: center;" | <math>P_{g_i}^{min}\leq P_{g_i}\leq P_{g_i}^{max}, \qquad i=1,\ldots,N_{gen}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (4)
@@ Line 107: / Line 107: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>S_{ij}\leq S_{ij}^{max},\qquad {i,j=1,\ldots,N_{branch} </math>
+| style="text-align: center;" | <math>S_{ij}\leq S_{ij}^{max},\qquad i,j=1,\ldots,N_{branch} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (8)