He et al 2024a - Revision history

Rimni at 12:14, 13 May 2024

2024-05-13T12:14:32Z

Rimni at 12:13, 13 May 2024

2024-05-13T12:13:56Z

Rimni at 12:13, 13 May 2024

2024-05-13T12:13:35Z

Rimni at 12:12, 13 May 2024

2024-05-13T12:12:12Z

Rimni at 12:11, 13 May 2024

2024-05-13T12:11:17Z

Rimni: /* 2.2 Distributed scheduling model */

2024-05-13T11:43:13Z

‎2.2 Distributed scheduling model

Rimni at 11:41, 13 May 2024

2024-05-13T11:41:35Z

Rimni at 11:40, 13 May 2024

2024-05-13T11:40:07Z

Rimni at 11:34, 13 May 2024

2024-05-13T11:34:10Z

Rimni at 11:22, 13 May 2024

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@@ Line 230: / Line 230: @@
 To surmount this impediment, the present study introduces the augmented Lagrangian relaxation method (ALR), which incorporates a quadratic penalty term for the relaxed constraints within the objective function and adheres to a fixed step length for multiplier updates, thereby enhancing convergence. However, this incorporation of a quadratic penalty term compromises the factorability of the optimization problem, representing a significant limitation of the ALR approach. Two prevalent resolutions are the auxiliary problem principle method (APP) [24-27] and the alternating direction multiplier method (ADMM) [28-32]. Each methodology exhibits its own merits and demerits: the APP method facilitates parallel solutions albeit with moderate convergence velocity, whereas the ADMM approach necessitates serial resolution but boasts rapid convergence rates. Given the relatively modest number of regions within the extant power system architecture, the ADMM method is selected for resolving the multi-regional distributed power dispatching conundrum.
-====1) Traditional ADMM method====
+'''1) Traditional ADMM method'''
 Consider the following optimization problems:

@@ Line 230: / Line 230: @@
 To surmount this impediment, the present study introduces the augmented Lagrangian relaxation method (ALR), which incorporates a quadratic penalty term for the relaxed constraints within the objective function and adheres to a fixed step length for multiplier updates, thereby enhancing convergence. However, this incorporation of a quadratic penalty term compromises the factorability of the optimization problem, representing a significant limitation of the ALR approach. Two prevalent resolutions are the auxiliary problem principle method (APP) [24-27] and the alternating direction multiplier method (ADMM) [28-32]. Each methodology exhibits its own merits and demerits: the APP method facilitates parallel solutions albeit with moderate convergence velocity, whereas the ADMM approach necessitates serial resolution but boasts rapid convergence rates. Given the relatively modest number of regions within the extant power system architecture, the ADMM method is selected for resolving the multi-regional distributed power dispatching conundrum.
-=====1) Traditional ADMM method=====
+====1) Traditional ADMM method====
 Consider the following optimization problems:

@@ Line 230: / Line 230: @@
 To surmount this impediment, the present study introduces the augmented Lagrangian relaxation method (ALR), which incorporates a quadratic penalty term for the relaxed constraints within the objective function and adheres to a fixed step length for multiplier updates, thereby enhancing convergence. However, this incorporation of a quadratic penalty term compromises the factorability of the optimization problem, representing a significant limitation of the ALR approach. Two prevalent resolutions are the auxiliary problem principle method (APP) [24-27] and the alternating direction multiplier method (ADMM) [28-32]. Each methodology exhibits its own merits and demerits: the APP method facilitates parallel solutions albeit with moderate convergence velocity, whereas the ADMM approach necessitates serial resolution but boasts rapid convergence rates. Given the relatively modest number of regions within the extant power system architecture, the ADMM method is selected for resolving the multi-regional distributed power dispatching conundrum.
-'''1) Traditional ADMM method'''
+=====1) Traditional ADMM method=====
 Consider the following optimization problems:

@@ Line 490: / Line 490: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\begin{array}{c}
+| <math>P_{l,t}=H_i\cdot [(CP_{n,t}+C_\mbox{w}W_{n_\mbox{w},t}+
-P_{l,t}=H_i\cdot [(CP_{n,t}+C_\mbox{w}W_{n_\mbox{w},t}+\\
+C_\mbox{s}S_{n_\mbox{s},t}-D_{n_\mbox{B},t}),T_{ij,t}^i,T_{i-j+1,t}^i]</math>
 C_\mbox{s}S_{n_\mbox{s},t}-D_{n_\mbox{B},t}),T_{ij,t}^i,T_{i-j+1,t}^i]
-\end{array}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (33)

← Older revision		Revision as of 12:11, 13 May 2024
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−	[[#img-4\|Figure 4]] illustrates the transformation from traditional ADMM method to fully distributed ADMM method. The solid line denotes the power connection line, and the dashed line denotes the information connection line. The traditional ADMM method performs power transmission between regions, and each region transmits information to the upper data center. The fully distributed ADMM method eliminates the upper layer data center, and accomplishes the transmission of power and information directly between regions. The specific process is as follows: in the <math display="inline"> k </math>-th iteration, the problem of region <math display="inline"> i </math> is solved sequentially, and the corresponding power result of the tie line is obtained; Region <math display="inline"> i </math> sends this result to region <math display="inline"> i+1 </math> and participates in the problem solving of region <math display="inline"> i+1 </math> as the latest iteration result. After solving the problem for region <math display="inline"> i+1 </math>, region <math display="inline"> i+1 </math> already has the necessary information to update the line multiplier between region <math display="inline"> i </math> and region <math display="inline"> i+1 </math>, namely <math display="inline">T_{i-j+1,t}^{i(k)}</math> and <math display="inline">T_{i-j+1,t}^{i+1(k)}</math>, so the update of the corresponding multiplier can be done directly in the region <math display="inline"> i+1 </math>. Before the <math display="inline"> k+1 </math> iteration, region <math display="inline"> i+1 </math> passes the updated multiplier back to region <math display="inline"> i </math>, starting the next iteration. Similarly, region <math display="inline"> i+2 </math> can update the multipliers of region <math display="inline"> i </math> with region <math display="inline"> i+2 </math> and region <math display="inline"> i+1 </math> with region <math display="inline"> i+2 </math>. Ultimately, all multiplier updates are done by the regions, without the involvement of the upper data center. In addition, the stop condition of the iteration can also be determined by each region. The region responsible for updating the multiplier can further calculate the corresponding relaxed constrained error and compare it with the given error tolerance.	+	[[#img-4\|Figure 4]] illustrates the transformation from traditional ADMM method to fully distributed ADMM method. The solid line denotes the power connection line, and the dashed line denotes the information connection line. The traditional ADMM method performs power transmission between regions, and each region transmits information to the upper data center. The fully distributed ADMM method eliminates the upper layer data center, and accomplishes the transmission of power and information directly between regions. The specific process is as follows: in the <math display="inline"> k </math>-th iteration, the problem of region <math display="inline"> i </math> is solved sequentially, and the corresponding power result of the tie line is obtained. Region <math display="inline"> i </math> sends this result to region <math display="inline"> i+1 </math> and participates in the problem solving of region <math display="inline"> i+1 </math>, as the latest iteration result. After solving the problem for region <math display="inline"> i+1 </math>, region <math display="inline"> i+1 </math> already has the necessary information to update the line multiplier between region <math display="inline"> i </math> and region <math display="inline"> i+1 </math>, namely <math display="inline">T_{i-j+1,t}^{i(k)}</math> and <math display="inline">T_{i-j+1,t}^{i+1(k)}</math>, so the update of the corresponding multiplier can be done directly in the region <math display="inline"> i+1 </math>. Before the <math display="inline"> k+1 </math> iteration, region <math display="inline"> i+1 </math> passes the updated multiplier back to region <math display="inline"> i </math>, starting the next iteration. Similarly, region <math display="inline"> i+2 </math> can update the multipliers of region <math display="inline"> i </math> with region <math display="inline"> i+2 </math> and region <math display="inline"> i+1 </math> with region <math display="inline"> i+2 </math>. Ultimately, all multiplier updates are done by the regions, without the involvement of the upper data center. In addition, the stop condition of the iteration can also be determined by each region. The region responsible for updating the multiplier can further calculate the corresponding relaxed constrained error and compare it with the given error tolerance.

	<div id='img-4'></div>		<div id='img-4'></div>

← Older revision		Revision as of 11:43, 13 May 2024
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−	[[#img-4\|Figure 4]] illustrates the transformation from traditional ADMM method to fully distributed ADMM method. The solid line denotes the power connection line, and the dashed line denotes the information connection line. The traditional ADMM method performs power transmission between regions, and each region transmits information to the upper data center. The fully distributed ADMM method eliminates the upper layer data center, and accomplishes the transmission of power and information directly between regions. The specific process is as follows: in the <math display="inline"> k </math>-th iteration, the problem of region <math display="inline"> i </math> is solved sequentially, and the corresponding power result of the tie line is obtained; Region <math display="inline"> i </math> sends this result to region <math display="inline"> i+1 </math> and participates in the problem solving of region <math display="inline"> i+1 </math> as the latest iteration result. After solving the problem for region <math display="inline"> i+1 </math>, region <math display="inline"> i+1 </math> already has the necessary information to update the line multiplier between region <math display="inline"> i </math> and region <math display="inline"> i+1 </math>, namely <math display="inline">T_{i-j+1,t}^{i(k)}</math> and <math display="inline">T_{i-j+1,t}^{i+1(k)}</math>, so the update of the corresponding multiplier can be done directly in the region <math display="inline"> i+1 </math>; Before the <math display="inline"> k+1 </math> iteration, region <math display="inline"> i+1 </math> passes the updated multiplier back to region <math display="inline"> i </math>, starting the next iteration. Similarly, region <math display="inline"> i+2 </math> can update the multipliers of region <math display="inline"> i </math> with region <math display="inline"> i+2 </math> and region <math display="inline"> i+1 </math> with region <math display="inline"> i+2 </math>. Ultimately, all multiplier updates are done by the regions, without the involvement of the upper data center. In addition, the stop condition of the iteration can also be determined by each region. The region responsible for updating the multiplier can further calculate the corresponding relaxed constrained error and compare it with the given error tolerance.	+	[[#img-4\|Figure 4]] illustrates the transformation from traditional ADMM method to fully distributed ADMM method. The solid line denotes the power connection line, and the dashed line denotes the information connection line. The traditional ADMM method performs power transmission between regions, and each region transmits information to the upper data center. The fully distributed ADMM method eliminates the upper layer data center, and accomplishes the transmission of power and information directly between regions. The specific process is as follows: in the <math display="inline"> k </math>-th iteration, the problem of region <math display="inline"> i </math> is solved sequentially, and the corresponding power result of the tie line is obtained; Region <math display="inline"> i </math> sends this result to region <math display="inline"> i+1 </math> and participates in the problem solving of region <math display="inline"> i+1 </math> as the latest iteration result. After solving the problem for region <math display="inline"> i+1 </math>, region <math display="inline"> i+1 </math> already has the necessary information to update the line multiplier between region <math display="inline"> i </math> and region <math display="inline"> i+1 </math>, namely <math display="inline">T_{i-j+1,t}^{i(k)}</math> and <math display="inline">T_{i-j+1,t}^{i+1(k)}</math>, so the update of the corresponding multiplier can be done directly in the region <math display="inline"> i+1 </math>. Before the <math display="inline"> k+1 </math> iteration, region <math display="inline"> i+1 </math> passes the updated multiplier back to region <math display="inline"> i </math>, starting the next iteration. Similarly, region <math display="inline"> i+2 </math> can update the multipliers of region <math display="inline"> i </math> with region <math display="inline"> i+2 </math> and region <math display="inline"> i+1 </math> with region <math display="inline"> i+2 </math>. Ultimately, all multiplier updates are done by the regions, without the involvement of the upper data center. In addition, the stop condition of the iteration can also be determined by each region. The region responsible for updating the multiplier can further calculate the corresponding relaxed constrained error and compare it with the given error tolerance.

	<div id='img-4'></div>		<div id='img-4'></div>

← Older revision		Revision as of 11:41, 13 May 2024
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	Step 5: Compare the maximum error bound by the tie line with the specified allowable error. If the error requirement is met, the iteration ends and the result is output. Otherwise, <math display="inline">k=k+1</math>, <math display="inline"> i=1 </math>, go to Step 2.		Step 5: Compare the maximum error bound by the tie line with the specified allowable error. If the error requirement is met, the iteration ends and the result is output. Otherwise, <math display="inline">k=k+1</math>, <math display="inline"> i=1 </math>, go to Step 2.
−

	==3. Simulation result analysis==		==3. Simulation result analysis==

@@ Line 230: / Line 230: @@
 To surmount this impediment, the present study introduces the augmented Lagrangian relaxation method (ALR), which incorporates a quadratic penalty term for the relaxed constraints within the objective function and adheres to a fixed step length for multiplier updates, thereby enhancing convergence. However, this incorporation of a quadratic penalty term compromises the factorability of the optimization problem, representing a significant limitation of the ALR approach. Two prevalent resolutions are the auxiliary problem principle method (APP) [24-27] and the alternating direction multiplier method (ADMM) [28-32]. Each methodology exhibits its own merits and demerits: the APP method facilitates parallel solutions albeit with moderate convergence velocity, whereas the ADMM approach necessitates serial resolution but boasts rapid convergence rates. Given the relatively modest number of regions within the extant power system architecture, the ADMM method is selected for resolving the multi-regional distributed power dispatching conundrum.
-) Traditional ADMM method
+'''1) Traditional ADMM method'''
 Consider the following optimization problems:
@@ Line 304: / Line 304: @@
 This iterates over and over again until the iteration termination condition is met.
-) Multi-region distributed power scheduling based on traditional ADMM method.
+'''2) Multi-region distributed power scheduling based on traditional ADMM method'''
 A multi-region distributed power dispatching model is derived from the centralized dispatching model by applying the conventional ADMM method. The main process is as follows: The centralized scheduling model reveals that the coupling constraint between regions is manifested in the constraint related to the liaison line, that is, the maximum capacity constraint of the regional liaison line  <math display="inline">-T_{ij}^{max}\leqslant T_{ij,t}^{}\leqslant T_{ij}^{max}</math>. However, the  <math display="inline">T_{ij,t}^{}</math> variable is correlated with both region <math display="inline"> i </math> and region <math display="inline"> j </math>. If the constraint is directly relaxed, the decoupling in the true sense cannot be achieved. Therefore, the constraint should be first rewritten as follows:
@@ Line 510: / Line 510: @@
 Similarly, the corresponding PTDF matrix can be solved for other regions. The PTDF for all regions still only needs to be solved once and can be used in all iterations with great convenience.
-) Multi-region distributed power dispatching with fully distributed ADMM method.
+'''3) Multi-region distributed power dispatching with fully distributed ADMM method'''
 A limitation of traditional ADMM is that it still requires an upper level data center to collect data from each contact line, and perform the calculation and distribution of multipliers. Although each region does not need to share its own key power information, such upper-level data centers still have a certain authority in practical applications, which is hard to achieve. As shown in [[#tab-1|Table 1]], the only relaxed constraint in the whole iteration process of the traditional ADMM method is the contact line constraint, and the information on each contact line is only generated and used by the two areas of the contact, and is irrelevant to other areas.
@@ Line 562: / Line 562: @@
-. Simulation result analysis
+==3. Simulation result analysis==
 In this investigation, the efficacy of the proposed methodology was substantiated through the development of a tripartite interconnection test case, predicated on the IEEE 30-node, 39-node, and 57-node benchmark test systems. The schematic representation of the interconnected system’s architecture is illustrated in [[#img-5|Figure 5]]. Pertaining to each subsystem, the network’s topology and ancillary parameters were preserved in congruence with the original configuration, with modifications confined solely to the corresponding numerals to fulfill the prerequisites of the interconnected framework. In alignment with the archetypal diurnal load fluctuation pattern, characterized by “two peaks and one trough” [33], a 24-hour load dataset was synthesized. Subsequently, generators situated at nodes 13, 23, 61, 68, 72, and 81 were designated as contractual power entities, with the stipulated power being allocated at a quantum equating to half of the maximal power output. The tie line’s maximal capacity threshold was established at 500 MW, while the transactional power interlinking the 30-node and 39-node systems was ascertained at 2 GWh, and the exchange power schedule was constrained within a bandwidth spanning from −50 MW to 400 MW. Employing a parameterization of <math>\rho =0.1</math> and a maximal tie line constraint deviation of 0.1 MW, the model underwent resolution via both the centralized approach and the fully distributed ADMM technique, with the outcomes delineated in [[#tab-2|Table 2]].

@@ Line 28: / Line 28: @@
 In light of these challenges, the necessity for a distributed scheduling framework becomes critical, one that allocates scheduling responsibilities to regional dispatching centers. This paradigm shift paves the way for globally optimal power scheduling based on a distributed architecture that minimizes the need for extensive inter-regional information sharing. Consequently, Distributed Dynamic Scheduling (DDS) has attracted increasing academic attention, emerging from research on Distributed Optimal Power Flow (DOPF) optimization approaches.
-In the scholarly discourse on power systems, the concept of virtual nodes has been introduced as a means to balance power flow post-decoupling within subregions, thus enabling the formulation of a multi-regional decomposition algorithm. Significantly, the literature [12] endorses a parallel Distributed Optimal Power Flow (DOPF) model, designed for extensive regional interconnection systems, validated through simulations on systems of moderate size. Further, literature [13] mathematically implements the proposed distributed framework by amalgamating two distinct decomposition methods: the Predictor-Corrector Proximal Multiplier (PCPM) and the Alternating Direction Method (ADM). Additionally, literature [14] addresses the distributed processing of the multi-objective reactive power optimization issue within large-scale power systems, effectively resolving the centralized reactive power optimization challenge.
+In the scholarly discourse on power systems, the concept of virtual nodes has been introduced as a means to balance power flow post-decoupling within subregions, thus enabling the formulation of a multi-regional decomposition algorithm. Significantly, Kim and Baldick [12] endorse a parallel Distributed Optimal Power Flow (DOPF) model, designed for extensive regional interconnection systems, validated through simulations on systems of moderate size. Further, Kim and Baldick [13] mathematically implement the proposed distributed framework by amalgamating two distinct decomposition methods: the Predictor-Corrector Proximal Multiplier (PCPM) and the Alternating Direction Method (ADM). Additionally, Cheng et al. [14] addresse the distributed processing of the multi-objective reactive power optimization issue within large-scale power systems, effectively resolving the centralized reactive power optimization challenge.
-Furthermore, literature [15] incorporates network loss into the foundational DOPF model using cosine approximation. The Auxiliary Problem Principle (APP) is employed in literature [16-18] to manage regional coupling constraints, leading to the development of the DOPF algorithm. The Alternating Direction Multiplier Method (ADMM), thoroughly examined in literature [19-23], has been extensively refined for deployment within the Distributed Dynamic Scheduling (DDS) sphere. Notably, literature [19] converts the optimal power flow problem within microgrids into a convex issue via semi-definite programming relaxation techniques, thereafter seeking a distributed solution through the ADMM. Literature [20] utilizes the ADMM, predicated on consistency, for the distributed optimal control of reactive power in power systems, demonstrating its superiority to the dual-ascent method.
+Furthermore, Conejo and Aguado [15] incorporate network loss into the foundational DOPF model using cosine approximation. The Auxiliary Problem Principle (APP) is employed in literature [16-18] to manage regional coupling constraints, leading to the development of the DOPF algorithm. The Alternating Direction Multiplier Method (ADMM), thoroughly examined in literature [19-23], has been extensively refined for deployment within the Distributed Dynamic Scheduling (DDS) sphere. Notably, Dall'Anese et al. [19] convert the optimal power flow problem within microgrids into a convex issue via semi-definite programming relaxation techniques, thereafter seeking a distributed solution through the ADMM. Sulc et al. [20] utilize the ADMM, predicated on consistency, for the distributed optimal control of reactive power in power systems, demonstrating its superiority to the dual-ascent method.
 This research articulates a fully distributed algorithm to address the centralized dynamic scheduling issue within power systems featuring a multi-regional interconnection structure. Based on the inherent characteristics of power systems and employing the ADMM for the decoupling of inter-regional connections, this approach decomposes the central issue into individual sub-optimization tasks for each region. Simultaneously, the traditional role of the data center, responsible for multiplier updates, becomes redundant, resulting in a completely distributed scheduling framework. The achievement of an optimal system-wide solution is accomplished through the iterative resolution of economic scheduling sub-problems specific to each region. An illustrative analysis, utilizing a multi-regional interconnected system modeled on the IEEE standard test system, affirms the effectiveness and precision of the proposed model in managing the intricacies of multi-regional distributed dynamic scheduling in power systems.

@@ Line 677: / Line 677: @@
 [[#img-6|Figures 6]] and [[#img-7|7]] elucidate the trajectory of iterative errors, revealing that with <math>\rho =0.01</math>, the error curve is characterized by a smooth yet gradual decline, necessitating an increased number of iterations. Conversely, when <math>\rho =2</math> is employed, the error curve exhibits a precipitous descent accompanied by notable fluctuations, resulting in a heightened iteration count. Consequently, the parameter <math>\rho</math> serves a dual function: it not only modulates the multiplier correction step within the iterative sequence but also impacts the oscillatory nature of the process. These dual roles exert antithetical effects on the rate of convergence. In practical scenarios, the optimal relative value of <math>\rho</math> can be ascertained through empirical experimentation.
-=4. Conclusion=
+==4. Conclusion==
 The current investigation addresses the dynamic economic dispatch (DED) dilemma within a trans-regional power system context, advancing a model that leverages the alternating direction multiplier method (ADMM). This model innovatively obviates the traditional requirement for a superior data center to perform multiplier updates, thus fostering a fully distributed dynamic economic scheduling (DDES) framework. The model’s validation was conducted through simulation tests on a set of interconnected systems, based on the IEEE standard test system. The results from these simulations substantiate the model’s ability to generate highly precise solutions across a prudent range of iterations. The valuation strategy for the parameter <math display="inline">\rho  </math> was scrutinized, revealing its significant impact on the iteration count. It was observed that an overly large or small <math display="inline">\rho  </math> value could either amplify fluctuations in the iteration process or slow down the multiplier’s update speed, both of which are counterproductive to rapid iterative convergence. In practical scenarios, optimizing the parameter <math display="inline">\rho  </math> can be achieved through iterative experimental adjustments.