A fully distributed algorithm for dynamic economic dispatch of cross regional power systems based on alternating direction multiplier method

Abstract: In the present study, a novel approach to dynamic economic dispatch for power systems spanning multiple regions is introduced, utilizing the alternating direction multiplier method (ADMM) as its computational foundation. The proposed economic scheduling model is designed to optimize the operational costs of the system comprehensively, while adhering to a spectrum of operational constraints. Employing the ADMM, the study achieves a distributed resolution of the model by severing the inter-regional interconnections, thereby partitioning the overarching optimization challenge into manageable sub-problems specific to each region. This methodology facilitates the attainment of the system’s global optimum through the iterative resolution of these regional sub-problems. Furthermore, the algorithm obviates the necessity for a centralized data repository for multiplier updates, thereby endorsing a fully distributed scheduling paradigm. Concurrently, the study incorporates a multi-period optimization technique within the economic scheduling model to accommodate the inherent temporal dependencies of the power system. The culmination of this research is the empirical analysis of a tri-regional interconnected system, predicated on the IEEE standard test system, which substantiates the efficacy of the proposed economic dispatch strategy.

Key words: Microgrid; Distribution side power market; Intermediation; Supply function equilibrium; Distributed optimization

1. Introduction

In the realm of power systems, Economic Dispatch (ED) is delineated as an optimization quandary that endeavors to minimize network losses or generation costs, whilst adhering to the constraints imposed by power flow and operational parameters [1-8]. The deployment of renewable energy sources, such as photovoltaic and wind power, has engendered an increased intertemporal coupling within power systems. This development has rendered static ED, which is predicated on a singular temporal analysis, as inadequate for the nuanced scenarios presented by contemporary power systems. Consequently, the focus has shifted towards dynamic ED. The burgeoning power market, with its rapid expansion, necessitates an enhanced interconnectivity and an escalated frequency of power exchanges across various regions within power systems. This objective is attainable only through the centralized dispatching executed by a cross-regional dispatching center wielding superior authority or through the coordination of regional dispatching centers. However, centralized scheduling (CS), the traditional scheduling archetype that mandates a central hub to amass intricate data from each scheduling domain and to centrally arbitrate their operational modalities, encounters multifarious impediments. A plethora of scholars, both domestic and international, have engaged with the quandary of centralized dispatching within power systems, deploying classical optimization algorithms such as the interior point method [9], Lambda iteration method [10], and Lagrange method [11], all of which are inherently centralized. A salient merit of the centralized scheduling algorithm is its obviation of iterative processes, thereby enabling direct global optimization. However, this approach becomes untenable when confronted with large-scale power scheduling issues, as the sheer volume and complexity of the variables necessitate resolution escalate dramatically. Furthermore, the intensification of research on the global energy Internet and the consequent establishment of transnational regional UHV power grids are poised to facilitate power complementarity, load peak shifting, and power support. China has already realized a modicum of power interconnection with neighboring nations such as Russia, Mongolia, Vietnam, Laos, and Myanmar, cumulatively amounting to an interconnection capacity of approximately 2.6 million kilowatts. Yet, the reticence of countries to divulge intricate details pertaining to their power systems, owing to strategic confidentiality imperatives, imposes limitations on the feasibility of higher-level dispatch centers with global dispatch authority.

Given the aforementioned exigencies, the imperative to adopt a distributed scheduling framework becomes paramount, one that delegates scheduling authority to regional dispatching centers. This shift heralds the potential for globally optimal power scheduling, predicated on a distributed architecture that necessitates minimal inter-regional information exchange. In response to these dynamics, Distributed Dynamic Scheduling (DDS) has garnered escalating scholarly interest. DDS algorithms are an outgrowth of research into Distributed Optimal Power Flow (DOPF) optimization strategies.

The literature delineates the introduction of virtual nodes as a mechanism to equilibrate power flow post-decoupling within subregions, thereby facilitating the construction of a multi-regional decomposition algorithm. Notably, literature [12] advocates for a parallel DOPF model tailored for expansive regional interconnection systems, a model whose efficacy is corroborated through simulations of moderately sized systems. Literature [13] mathematically operationalizes the proposed distributed framework by integrating two distinct decomposition methodologies: the Predictor-Corrector Proximal Multiplier (PCPM) and the Alternating Direction Method (ADM). Literature [14] undertakes the distributed processing of the multi-objective reactive power optimization quandary within large-scale power systems, thereby resolving the centralized reactive power optimization dilemma.

Moreover, literature [15] integrates network loss into the foundational DOPF model via cosine approximation. The Auxiliary Problem Principle (APP) is applied in literature [16-18] to navigate regional coupling constraints, culminating in the realization of the DOPF algorithm. The Alternating Direction Multiplier Method (ADMM), extensively documented in literature [19-23], has undergone significant refinement for application within the DDS domain. Literature [19] transmutes the optimal power flow conundrum within microgrids into a convex problem through the application of semi-definite programming relaxation techniques, subsequently pursuing a distributed resolution via the ADMM. Literature [20] harnesses the ADMM predicated on consistency for the distributed optimal control of reactive power within power systems, validating its preeminence over the dual-ascent method.

In this investigation, a fully distributed algorithm is articulated for addressing the centralized dynamic scheduling conundrum within power systems characterized by a multi-regional interconnection topology. Predicated upon the intrinsic attributes of power systems and leveraging the Alternating Direction Multiplier Method (ADMM) for the decoupling of inter-regional linkages, this model facilitates the decomposition of the central problem into discrete sub-optimization tasks specific to each region. Concomitantly, the conventional role of the data center, tasked with multiplier updates, is rendered obsolete, culminating in an entirely distributed scheduling schema.The attainment of the system-wide optimal solution is realized through the iterative resolution of the economic scheduling sub-problems inherent to each region. An exemplar analysis, employing a multi-regional interconnected system modeled after the IEEE standard test system, corroborates the proficiency and accuracy of the proposed model in navigating the complexities of multi-regional distributed dynamic scheduling within power systems.

2. Multi-region dynamic scheduling model

The multi-regional power scheduling model, as depicted in Figure 1, encapsulates the intricate scheduling dynamics extant among various regions within a power system. It is constituted by an array of generator sets and loads, each interlinked via transmission lines. The transfer of power and information is facilitated through contact lines bridging regions, thereby enabling their interconnectivity. Owing to the autonomous dispatching attributes inherent to each region, the real-time sharing of comprehensive power dispatching data presents a formidable challenge. Consequently, the adoption of a distributed scheduling approach is necessitated, one that eschews the traditional dispatching center, and through iterative processes, converges upon the globally optimal power scheduling, all while minimizing the exchange of information between regions.

Figure 1 Fig. Multi-region economic dispatch model

2.1 Traditional centralized scheduling model

The traditional centralized power dispatching model can be established as follows:

1) objective function.

The objective function is to minimize the total operating cost of the system. Among them, the main consideration of the operating cost of the generator set. The expression of the objective function is as follows:

File:Draft He 943252806-image2.png

(1)

Where: ${\textstyle F}$ is the total operating cost of the system; ${\textstyle t}$ is the number of each scheduling period; ${\textstyle T}$ represents the total number of scheduling periods; ${\textstyle n}$ and ${\textstyle n_{\mbox{B}}}$ are the number of the generator set and the system node respectively; ${\textstyle N}$ and ${\textstyle N_{\mbox{B}}}$ are the total number of generator sets and system nodes respectively; ${\textstyle P_{n,t}}$ represents the actual output of generator set ${\textstyle n}$ during the period ${\textstyle t}$ ; ${\textstyle a_{n}}$ , ${\textstyle b_{n}}$ and ${\textstyle c_{n}}$ are the coefficient of output characteristic of generator set ${\textstyle n}$ .

2) Constraints.

a) System power balance constraint, that is, the total generator output at any time is equal to the load demand:

${\textstyle \sum _{n=1}^{N}P_{n,t}=\sum _{n_{\mbox{B}}=1}^{N_{\mbox{B}}}D_{n_{\mbox{B}},t}}$

(2)

Where, ${\textstyle D_{n_{\mbox{B}},t}}$ is the load demand of node ${\textstyle n}$ during the period ${\textstyle t}$ .

b) Constraints on the upper and lower limits of generator set output:

${\textstyle P_{n}^{min}\leqslant P_{n,t}\leqslant P_{n}^{max}}$

(3)

Where, ${\textstyle P_{n}^{min}}$ and ${\textstyle P_{n}^{max}}$ are respectively the lower limit and upper limit of output of generator set ${\textstyle n}$ in any period.

c) The upper and lower limits of the generator set climbing:

${\textstyle {\mbox{Δ}}P_{n}^{min}\leqslant P_{n,t}-P_{n,t-1}\leqslant {\mbox{Δ}}P_{n}^{max}}$

(4)

Where ${\textstyle {\mbox{Δ}}P_{n}^{min}}$ and ${\textstyle {\mbox{Δ}}P_{n}^{max}}$ indicate the minimum climb rate (i.e. maximum downward climb rate) and maximum climb rate (i.e. maximum upward climb rate) of generator set ${\textstyle n}$ , respectively.

d) The power constraint of the generator set contract, that is, the total power generation of the generator set within a given period of time must comply with the agreed contract:

${\textstyle {\sum }_{t=1}^{T}P_{n,t}=E_{n},n\in \phi }$

(5)

Where: ${\textstyle E_{n}}$ is the contracted electricity quantity of generator set ${\textstyle n}$ in a given period of time; ${\textstyle \phi }$ is the set of generator sets for which the contracted electricity quantity is agreed.

e) The contract unit can be decomposed value deviation constraint, that is, the contract electricity decomposed to the day needs to be within a certain deviation range:

${\textstyle E_{n}^{{\mbox{day}},{\mbox{min}}}\leqslant E_{n,d}^{\mbox{day}}\leqslant E_{n}^{{\mbox{day}},{\mbox{max}}}}$

(6)

Where: ${\textstyle d}$ is the number of the dispatch date; ${\textstyle E_{n,d}^{\mbox{day}}}$ is the generation capacity of generator set ${\textstyle n}$ in date ${\textstyle d}$ ; ${\textstyle E_{n}^{{\mbox{day}},{\mbox{min}}}}$ and ${\textstyle E_{n}^{{\mbox{day}},{\mbox{max}}}}$ are the planned allowable minimum and maximum of the energy decomposition value of the generator set ${\textstyle n}$ , respectively.

f) Transmission line maximum capacity constraints:

${\textstyle -P_{l}^{max}\leqslant P_{l,t}\leqslant P_{l}^{max}}$

(7)

Where: ${\textstyle P_{l,t}}$ is the transmission power of line ${\textstyle l}$ in the time period ${\textstyle t}$ ; ${\textstyle P_{l}^{max}}$ is the maximum transmission power of line ${\textstyle l}$ . The maximum capacity of the two-way transmission of the line is equal.

g) Maximum capacity constraint of regional liaison line:

${\textstyle -T_{ij}^{max}\leqslant T_{ij,t}\leqslant T_{ij}^{max}}$

(8)

Among them: ${\textstyle T_{ij,t}}$ is a vector variable, representing the transmission power of each contact line between region ${\textstyle i}$ and region ${\textstyle j}$ in the time period ${\textstyle t}$ ; ${\textstyle T_{ij}^{max}}$ is a vector constant that represents the maximum transmission power of the contact lines between region ${\textstyle i}$ and region ${\textstyle j}$ . This paper assumes that the maximum two-way transmission capacity of each link line is equal.

h) Constraint on the power of the regional contact line transaction, that is, the power of the contact line transaction within a given period must comply with the agreement between regions:

${\sum }_{t=1}^{T}{\boldsymbol {1}}^{\mbox{T}}T_{ij,t}=E_{ij},ij\in {\mbox{Γ}}$

(9)

Where: ${\textstyle {\boldsymbol {1}}={\left(1\cdot \cdot \cdot 1\right)}^{\mbox{T}}}$ ; ${\textstyle E_{ij}}$ is the total amount of agreed transaction power between region ${\textstyle i}$ and region ${\textstyle j}$ in a given period of time; ${\mbox{Γ}}$ is the set of region pairs for the agreed transaction of electricity.

i) Regional liaison line switching power plan constraints, that is, the regional liaison line switching power of each period should be within the allowable range of assessment deviation:

${\textstyle T_{ij}^{kh,min}\leqslant {\boldsymbol {1}}^{\mbox{T}}T_{ij,t}\leqslant T_{ij}^{kh,max}}$

(10)

Where, ${\textstyle T_{ij}^{kh,min}}$ and ${\textstyle T_{ij}^{kh,max}}$ are the minimum and maximum value allowed by the switching power plan between region ${\textstyle i}$ and region ${\textstyle j}$ during the ${\textstyle t}$ period, respectively.

The power transmission distribution factor (PTDF) matrix can be used to obtain the power transmission lines. The PTDF matrix directly links the net injected power of each node to the power transmission of each branch, and its main advantage is that it avoids introducing redundant voltage phase angle variables, and only depends on the grid structure and line parameters, and is independent of the variation of the injected power of each node, that is, for a given network, only one PTDF matrix is needed to be reused. The basic format of its application is:

${\textstyle P_{\mbox{flow}}=HP_{{\mbox{in}}j}}$

(11)

Where: ${\textstyle P_{\mbox{flow}}}$ is the vector composed of the power of each branch; ${\textstyle H}$ is the corresponding PTDF matrix; ${\textstyle P_{{\mbox{in}}j}}$ is the vector composed of the net injected power of each node.

For this centralized scheduling problem:

${\textstyle P_{l,t}=H(CP_{n,t}+C_{\mbox{w}}W_{n_{\mbox{w}},t}+C_{\mbox{s}}S_{n_{\mbox{s}},t}-}$$D_{n_{\mbox{B}},t})$

(12)

Where ${\textstyle C}$ is the sorting matrix, the generator sets are rearranged in the order of node numbers.

2.2 Distributed scheduling model

In the discourse of distributed scheduling (DS), the concept is predicated on the decoupling of inter-regional connections, thereby enabling the segmentation of the aggregate power scheduling optimization quandary into distinct regional sub-problems. These sub-problems are iteratively addressed until a pre-defined error threshold is attained. Within the mathematical frameworks employed for DS, Lagrange relaxation (LR) is commonly invoked. This method mitigates the constraints, amalgamates them into the objective function, and iteratively refines the Lagrange multipliers until the cessation criteria of the iteration are fulfilled. Nonetheless, the LR method grapples with the challenge of selecting an appropriate step size for updating the multipliers, which impinges upon convergence rates.

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.

Consider the following optimization problems:

${\textstyle minf(x)+g(x)}$

(13)

${\textstyle {\mbox{s}}.{\mbox{t}}.Ax+Bz=c}$

(14)

Applying augmented Lagrange relaxation to equality constraints, the unconstrained optimization problem is

${\textstyle {\begin{array}{c}{\mbox{min}}f(x)+g(z)+{\lambda }^{\mbox{T}}(Ax+Bz-c)+\\{\frac {\rho }{2}}Ax+Bz-c_{2}^{2}\end{array}}}$

(15)

Where λ is the Lagrange multiplier and ρ is the positive quadratic penalty term coefficient.

The quintessence of the Alternating Direction Method of Multipliers (ADMM) approach resides in the premise that, during the resolution of a variable, concomitant variables are held constant, utilizing the outcomes from the antecedent iteration. This stratagem effectively “decouples” the inter-variable nexus, thereby adroitly circumventing the complexities associated with the quadratic penalty term. The fundamental iterative schema of ADMM unfolds in the ensuing manner:

${\textstyle x^{k+1}:=arg{\underset {x}{min}}L_{\rho }(x,z^{k},{\lambda }^{k})}$

(16)

${\textstyle z^{k+1}:=arg{\underset {z}{min}}L_{\rho }(x^{k+1},z,{\lambda }^{k})}$

(17)

${\textstyle {\lambda }^{k+1}={\lambda }^{k}+\rho (Ax^{k+1}+Bz^{k+1}-}$$c)$

(18)

This iterates over and over again until the iteration termination condition is met.

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 ${\textstyle -T_{ij}^{max}\leqslant T_{ij,t}^{}\leqslant T_{ij}^{max}}$ . However, the ${\textstyle T_{ij,t}^{}}$ variable is correlated with both region i and region j. If the constraint is directly relaxed, the decoupling in the true sense cannot be achieved. Therefore, the constraint should be first rewritten as follows:

${\textstyle T_{ij,t}^{i}=T_{ij,t}^{j}}$

(19)

${\textstyle -T_{ij}^{max}\leqslant T_{ij,t}^{i}\leqslant T_{ij}^{max}}$

(20)

${\textstyle -T_{ij}^{max}\leqslant T_{ij,t}^{j}\leqslant T_{ij}^{max}}$

(21)

Among them: ${\textstyle T_{ij,t}^{i}}$ represents the transmission power of the contact lines between region i and region j in the time period t when solving the sub-problem of region i; ${\textstyle T_{ij,t}^{j}}$ represents the transmission power of the contact lines between region i and region j in the time period t when solving the subproblem of region j. Figure 2 illustrates this process by treating the contact lines between region i and region j as two, each belonging to its own region and satisfying the same maximum capacity constraint, and finally ensuring that the power of the two contact lines is equal at all times. After processing, the relationship between the two regions is clearly reflected in the constraint ${\textstyle T_{ij,t}^{i}=T_{ij,t}^{j}}$ .

FIG. 2 Schematic diagram of constraint rewriting of the contact line.

For constraint ${\textstyle T_{ij,t}^{i}=T_{ij,t}^{j}}$ , applying augmented Lagrange relaxation, the objective function is:

${\textstyle {\begin{array}{c}F_{\mbox{L}}=\sum _{t=1}^{T}\sum _{n=1}^{N}\left(a_{n}P_{n,t}^{2}+b_{n}P_{n,t}+c_{n}\right)+\\\sum _{j\in i,j\not =i}{\lambda }_{ij,t}^{\mbox{T}}(T_{ij,t}^{i}-T_{ij,t}^{j})+\\\sum _{j\in i,j\not =i}\rho \parallel T_{ij,t}^{i}-T_{ij,t}^{j}{\parallel }_{2}^{2}\end{array}}}$

(22)

Where: ${\textstyle {\lambda }_{ij,t}^{\mbox{T}}}$ is the Lagrangian vector multiplier of the relaxed constraint ${\textstyle T_{ij,t}^{i}=T_{ij,t}^{j}}$ ; ρ is the positive coefficient of the corresponding quadratic penalty term. Notice that the quadratic penalty term ${\textstyle \sum _{j\in i,j\not =i}\rho \parallel T_{ij,t}^{i}-T_{ij,t}^{j}{\parallel }_{2}^{2}}$ destroyed the decomposability of the problem, so the ADMM method was applied for distributed solution, that is, each region was solved separately, and the variables of other regions were regarded as constants in each solution, and the latest iteration results were used. The k-th iteration optimization problem for region i can be given as follows:

${\textstyle {\begin{array}{c}minF_{\mbox{L}}^{i(k)}=\sum _{t=1}^{T}\sum _{n\in A^{i}}\left(a_{n}P_{n,t}^{2}+b_{n}P_{n,t}+c_{n}\right)+\\\sum _{j\in i,j\not =i}{\lambda }_{ij,t}^{(k-1)T}T_{ij,t}^{i}+\\\sum _{j\in i,j\not =i}\rho \parallel T_{ij,t}^{i}-T_{ij,t}^{j({\mbox{newest}})}{\parallel }_{2}^{2}{\mbox{ }}\end{array}}}$

(23)

${\textstyle {\mbox{s}}.{\mbox{t}}.\sum _{n\in A^{i}}P_{n,t}+\sum _{j\in i,j\not =i}{\boldsymbol {1}}^{\mbox{T}}T_{ij,t}^{i}=}$$\sum _{n_{\mbox{B}}\in A^{i}}D_{n_{\mbox{B}},t}$

(24)

${\textstyle P_{n}^{min}\leqslant P_{n,t}\leqslant P_{n}^{max},n\in A^{i}}$

(25)

${\textstyle {\mbox{Δ}}P_{n}^{min}\leqslant P_{n,t}-P_{n,t-1}\leqslant {\mbox{Δ}}P_{n}^{max},n\in A^{i}}$

(26)

${\textstyle \sum _{t=1}^{T}P_{n,t}=E_{n},n\in {\mbox{Φ}},n\in A^{i}}$

(27)

${\textstyle -P_{l}^{max}\leqslant P_{l,t}\leqslant P_{l}^{max},l\in A^{i}}$

(28)

${\textstyle -T_{ij}^{max}\leqslant T_{ij,t}\leqslant T_{ij}^{max},j\in i,j\not =}$$i$

(29)

${\textstyle \sum _{t=1}^{T}x{\boldsymbol {1}}^{\mbox{T}}T_{ij,t}=E_{ij},ij\in {\mbox{Γ}},j\in i,j\not =}$$i$

(30)

Where: ${\textstyle A^{i}}$ represents the set of all elements in region i; ${\textstyle T_{ij,t}^{j({\mbox{newest}})}}$ is the value of the latest iteration of ${\textstyle T_{ij,t}^{j}}$ :

${\textstyle {\boldsymbol {T}}_{ij,t}^{j({\mbox{newest}})}=\lbrace {\begin{array}{ll}{\boldsymbol {T}}_{ij,t}^{j(k)},&j<i\\{\boldsymbol {T}}_{ij,t}^{j(k-1)},&j>i\end{array}}}$

(31)

After each iteration, the update process of the multiplier is:

${\textstyle {\lambda }_{ij,t}^{\left(k\right)}={\lambda }_{ij,t}^{\left(k-1\right)}+}$$\rho (T_{ij,t}^{i(k)}-T_{ij,t}^{j(k)})$

(32)

The PTDF matrix can still be used to calculate the power transmission of the lines in region i. However, the whole network PTDF matrix derived from the centralized scheduling requires the information of all nodes, which obviously cannot be directly applied to the problem solving of region i. Therefore, the corresponding PTDF matrix needs to be found, which can utilize the information of region i and the information of the contact lines between region i and other regions to obtain the power transmission of the lines in region i.

Figure 3 illustrates the solution method of the target PTDF matrix. First, all regions connected to region i (i.e., regions j₁ and j₂ in the figure) are equivalent to nodes outside region i, whose injected power is the corresponding power transmitted by the liaison line. Then, the PTDF matrix is computed for this equivalent network. Using the PTDF matrix, the power transmission of the lines in region i can be acquired by using the information of region i and the information of the contact lines between region i and other regions. Its formula is as follows (all variables belong to region i):

${\textstyle {\begin{array}{c}P_{l,t}=H_{i}\cdot [(CP_{n,t}+C_{\mbox{w}}W_{n_{\mbox{w}},t}+\\C_{\mbox{s}}S_{n_{\mbox{s}},t}-D_{n_{\mbox{B}},t}),T_{ij,t}^{i},T_{i-j+1,t}^{i}]\end{array}}}$

(33)

FIG. 3 Schematic diagram of PTDF matrix construction corresponding to region i.

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.

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 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.

Hence, by exploiting the feature that the contact line is not a global constraint, the update process of the corresponding multiplier for each contact line can be completed between the two interconnection areas, without uploading to the data center. Thus, the multiplier calculation and assignment of each contact line can be done by the relevant subareas, and the upper data center can be eliminated, resulting in a fully distributed ADMM method. It should be noted that the mathematical formulas for solving subproblems and updating multipliers for each region are unchanged, so the method has the same mathematical properties as the traditional ADMM method.

Table 1 Comparison of relaxation constraints in ADMM methods

Relax constraints	Global constraints	Contact line constraints
Multiplier update features	Information for all regions is required	Only contact area information is required
Multiplier assignment features	Assignment to all regions is required	Only need to be assigned to the contact area
Upper-tier data center	need	No, the corresponding work can be transferred to the liaison area

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 k-th iteration, the problem of region i is solved sequentially, and the corresponding power result of the tie line is obtained; Region i sends this result to region i+1 and participates in the problem solving of region i+1 as the latest iteration result. After solving the problem for region i+1, region i+1 already has the necessary information to update the line multiplier between region i and region i+1, namely ${\textstyle T_{i-j+1,t}^{i(k)}}$ and ${\textstyle T_{i-j+1,t}^{i+1(k)}}$ , so the update of the corresponding multiplier can be done directly in the region i+1; Before the k+1 iteration, region i+1 passes the updated multiplier back to region i, starting the next iteration. Similarly, region i+2 can update the multipliers of region i with region i+2 and region i+1 with region i+2. 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.

FIG. 4 Schematic diagram of transformation from traditional ADMM method to fully distributed ADMM method

If the conditions are not met, this signal can be transmitted to each area through the entire information network, and the iteration continues. The solution steps of multi-region distributed power scheduling with fully distributed ADMM method are as follows:

Step 1: Give the initial values of all variables and parameters (number of iterations k=0, area number i=1).

Step 2: Solve the subproblem of region i based on the ADMM method.

Step 3: Compare the number of area i and the interconnection area. If i is greater than the number of any interconnect zone, zone i is responsible for updating the corresponding multiplier and transmitting the result to the corresponding interconnect zone.

Step 4: i=i+1. If i is greater than the maximum area number, go to Step 5. Otherwise, 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, k=k+1, i=1, go to Step 2.

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 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 ρ=0.1

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 Table 2.

FIG. 5 Schematic diagram of the three-system interconnection example

As delineated in Table 2, the methodology articulated herein is corroborated to engender precise solution outcomes, with the iteration count remaining well within an acceptable ambit. In pragmatic deployment, this approach is adept at navigating the complexities of inter-regional interconnected power dispatching quandaries, particularly in scenarios where centralized coordination proves to be infeasible.

Table 2 Comparison of solution results between centralized method and distributed method model

Scheduling method	P21,1/MW	1TT30−39,1 / MW	F/106 yuan	Number of iterations
centralized	314.8780	−131.0152	1.8167	—
distributed	314.9792	−130.8220	1.8167	23

The exhibit delineates the fluctuation in the iteration count consequent to the selection of disparate regional iteration sequences within the distributed modality, given the parameter ρ=0.1. It is discernible that the sequential order of iterations across the various regions exerts a negligible influence on the iteration quantity. This implies that, within the distributed resolution framework, the iteration of each region may proceed in an arbitrary sequence, yet still achieve expeditious convergence. The determination of the parameter ρ is subjected to additional scrutiny in the ensuing discourse.

Table 3 Influences of regional iteration order on iteration times

Region iteration sequence	30–39–57		30–57–39	39–30–57
Number of iterations	23		24	21
Region iteration sequence		39–57–30	57–30–39	57–39–30
Number of iterations		20	23	24

Table 4 elucidates the variability of solution outcomes contingent upon the selection of divergent values for the parameter ρ. It is observed that the objective function’s value, derived from varying ρ parameters, manifests with commendable precision. However, the iteration frequency exhibits discernible disparities: an excessively augmented or diminished ρ value precipitates an escalation in iteration quantity. Additionally, the error trajectory corresponding to ρ=0.01 and ρ=2 is graphically represented in Figures 6 and 7, respectively.Table 4 Influence of the selection of parameter ρ on the solution results.

ρ	0.01	0.05	0.1	0.5	2
F /106 yuan	1.8168	1.8166	1.8167	1.8167	1.8172
Number of iterations	79	21	23	24	34

Figure 6 Iterative error curve of parameter ρ=0.01

Figure 7. Iterative error curve of parameter ρ=2

Figures 6 and 7 elucidate the trajectory of iterative errors, revealing that with ρ = 0.01, the error curve is characterized by a smooth yet gradual decline, necessitating an increased number of iterations. Conversely, when ρ = 2 is employed, the error curve exhibits a precipitous descent accompanied by notable fluctuations, resulting in a heightened iteration count. Consequently, the parameter ρ 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 ρ can be ascertained through empirical experimentation.

4. Conclusion

The present study delves into the dynamic economic dispatch (DED) conundrum within a trans-regional power system framework, formulating a model that harnesses the alternating direction multiplier method (ADMM). This model innovatively circumvents the conventional necessity for an upper echelon data center to update the multiplier, thereby engendering a wholly distributed dynamic economic scheduling (DDES) paradigm. Validation of this model was executed via simulation tests on a triad of interconnected systems, anchored in the IEEE standard test system. The empirical evidence gleaned from these tests corroborates the method’s capacity to yield highly accurate solutions within a judicious iteration spectrum. The valuation strategy of the parameter ρ underwent meticulous examination within the exemplar, unveiling its tangible influence on the iteration count. An excessively large or minuscule ρ value was found to respectively intensify iteration process fluctuations or decelerate the multiplier’s update velocity, both scenarios being detrimental to swift iterative convergence. In practical applications, the parameter ρ’s optimization is achievable through a series of experimental trials.

References

[1] G. Hu, Z. Bie, X. Wang, Optimal dispatch in wind integrated system considering operation reliability, Trans. China Electrotech. Soc. 28(5) (2013) 58-65.

[2] D. C. Walters, G. B. Sheble, Genetic algorithm solution of economic dispatch with valve point loading, IEEE Trans. Power Syst. 8(3) (1993) 1325-1332.

[3] C. L. Chiang, Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels, IEEE Trans. Power Syst. 20(4) (2005) 1690-1699.

[4] J. Hetzer, C. Y. David, K. Bhattarai, An economic dispatch model incorporating wind power, IEEE Trans. Energy Convers. 23(2) (2008) 603-611.

[5] T. Ding, R. Bo, W. Gu, et al., Big-M based MIQP method for economic dispatch with disjoint prohibited zones, IEEE Trans. Power Syst. 29(2) (2014) 976-977.

[6] T. Ding, R. Bo, F. X. Li, et al., A bi-level branch and bound method for economic dispatch with disjoint prohibited zones considering network losses, IEEE Trans. Power Syst. 30(6) (2015) 2841-2855.

[7] P. Attaviriyanupap, H. Kita, E. Tanaka, et al., A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost function, IEEE Trans. Power Syst. 17(2) (2002) 411-416.

[8] H. Chen, Y. Wang, P. Xuan, et al., Robust economic dispatch method of microgrid containing high proportion of wind power, Control Theory & Appl. 34(8) (2017) 1104-1111.

[9] W. M. Lin, S. J. Chen, Bid-based dynamic economic dispatch with an efficient interior point algorithm, Int. J. Electr. Power Energy Syst. 24(1) (2002) 51-57.

[10] C. E. Lin, G. L. Viviani, Hierarchical economic dispatch for piecewise quadratic cost functions, IEEE Trans. Power Appar. Syst. 103(6) (1984) 1170-1175.

[11] T. Guo, M. I. Henwood, M. van Ooijen, An algorithm for combined heat and power economic dispatch, IEEE Trans. Power Syst. 11(4) (1996) 1778-1784.

[12] B. H. Kim, R. Baldick, Coarse-grained distributed optimal power flow, IEEE Trans. Power Syst. 12(2) (1997) 932-939.

[12] B. H. Kim, R. Baldick, Coarse-grained distributed optimal power flow, IEEE Trans. Power Syst. 12(2) (1997) 932-939.

[13] B. H. Kim, R. Baldick, A comparison of distributed optimal power flow algorithms, IEEE Trans. Power Syst. 15(2) (2000) 599-604.

[14] X. Cheng, J. Li, L. Cao, et al., Multi-objective distributed parallel reactive power optimization based on subarea division of the power systems, Proc. CSEE 23(10) (2003) 109-113.

[15] A. J. Conejo, J. A. Aguado, Multi-area coordinated decentralized DC optimal power flow, IEEE Trans. Power Syst. 13(4) (1998) 1272-1278.

[16] W. Zheng, W. Wu, B. Zhang, et al., Fully distributed multi-area economic dispatch method for active distribution networks, IET Gener. Transm. Distrib. 9(12) (2015) 1341-1351.

[17] X. Cheng, J. Li, L. Cao, et al., Distributed and parallel optimal power flow solution of electronic power systems, Autom. Electr. Power Syst. 27(24) (2003) 23-27.

[18] F. J. Nogales, F. J. Prieto, A. J. Conejo, A decomposition methodology applied to the multi-area optimal power flow problem, Ann. Oper. Res. 120(1/4) (2003) 99-116.

[19] E. Dall'Anese, H. Zhu, G. B. Giannakis, Distributed optimal power flow for smart microgrids, IEEE Trans. Smart Grid 4(3) (2013) 1464-1475.

[20] P. Sulc, S. Backhaus, M. Chertkov, Optimal distributed control of reactive power via the alternating direction method of multipliers, IEEE Trans. Energy Convers. 29(4) (2014) 968-977.

[21] T. Erseghe, Distributed optimal power flow using ADMM, IEEE Trans. Power Syst. 29(5) (2014) 2370-2380.

[22] H. Zhang, Y. Li, W. D. Gao, et al., Distributed optimal energy management for energy internet, IEEE Trans. Ind. Inform. 13(6) (2017) 3081-3097.

[23] Y. Wang, L. Wu, S. Wang, A fully-decentralized consensus-based ADMM approach for DC-OPF with demand response, IEEE Trans. Smart Grid 8(6) (2017) 2637-2647.

[24] G. Cohen, Auxiliary problem principle and decomposition of optimization problems, J. Optim. Theory Appl. 32(3) (1980) 277-305.

[25] G. Cohen, Auxiliary problem principle extended to variational inequalities, J. Optim. Theory Appl. 59(2) (1988) 325-333.

[26] D. Hur, J. K. Park, B. H. Kim, Evaluation of convergence rate in the auxiliary problem principle for distributed optimal power flow, IEE Proc.-Gener. Transm. Distrib. 149(5) (2002) 525-532.

[27] A. Losi, M. Russo, On the application of the auxiliary problem principle, J. Optim. Theory Appl. 117(2) (2003) 377-396.

[28] C. Chen, B. He, Y. Ye, et al., The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent, Math. Program. 155(1/2) (2016) 57-79.

[29] E. Ghadimi, A. Teixeira, I. Shames, et al., Optimal parameter selection for the alternating direction method of multipliers (ADMM): quadratic problems, IEEE Trans. Autom. Control 60(3) (2015) 644-658.

[30] M. Hong, Z. Q. Luo, On the linear convergence of the alternating direction method of multipliers, Math. Program. 162(1/2) (2017) 165-199.

[31] M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems, Comput. Optim. Appl. 1(1) (1992) 93-111.

[32] E. Wei, A. Ozdaglar, Distributed alternating direction method of multipliers, in: The 51st IEEE Conf. on Decision and Control, Maui, HI, USA: IEEE, 2012, pp. 5445-5450.

[33] D. Xia, Power System Analysis, 2nd ed., Beijing: China Electric Power Press, 2011.