Latest revision as of 14:14, 13 May 2024

Abstract

In this study, a novel approach to dynamic economic dispatch for multi-region power systems is introduced, leveraging the alternating direction multiplier method (ADMM) for computational efficiency. The method begins by dividing an interconnected power system into coherent groups, assigning local processors to each area to estimate black-box transfer function models using Lagrange multipliers. These processors then collaborate to form a global transfer function model through a consensus algorithm, leading to the derivation of a transfer function residue for the optimal wide-area control loop. A wide-area damping controller is designed from this residue, and its effectiveness is validated on two area and IEEE-39 bus test systems using RTDS/RSCAD and MATLAB co-simulation platforms. Simultaneously, the study proposes an economic scheduling model that aims to minimize operational costs while meeting various operational constraints. By severing inter-regional connections, the ADMM enables distributed resolution of the optimization problem, breaking it down into regional sub-problems that are iteratively solved to reach the global optimum. This process eliminates the need for a centralized data repository for multiplier updates, supporting a fully distributed scheduling approach. Additionally, a multi-period optimization technique is integrated to address the power system’s temporal dependencies. The approach's validity is demonstrated through empirical analysis of a tri-regional interconnected system based on the IEEE standard test system, confirming the strategy's effectiveness in economic dispatch.

Keywords: Microgrid, distribution side power market, intermediation, supply function equilibrium, distributed optimization

1. Introduction

Within the domain of power systems, Economic Dispatch (ED) is characterized as an optimization dilemma aimed at minimizing network losses or generation costs while conforming to the constraints of power flow and operational parameters [1-8]. The integration of renewable energy sources, notably photovoltaic and wind power, has introduced a heightened degree of intertemporal coupling in power systems. Such advancements have deemed static ED, reliant on single temporal analysis, insufficient for the complex scenarios of modern power systems, thereby necessitating a transition to dynamic ED. The burgeoning power market demands increased interconnectivity and more frequent power exchanges across regions, achievable through centralized dispatching by an authoritative cross-regional center or via coordination among regional centers. Nevertheless, centralized scheduling (CS), which requires a central entity to collect detailed data and centrally determine operational strategies, faces numerous challenges. Scholars globally have explored centralized dispatching, utilizing classical optimization algorithms like the interior point method [9], Lambda iteration method [10], and Lagrange method [11], which are intrinsically centralized. Centralized scheduling algorithms eliminate iterative processes, facilitating direct global optimization. However, this method is impractical for large-scale scheduling due to the exponential increase in variable volume and complexity. Moreover, research intensification on the global energy Internet and the development of transnational regional UHV power grids aim to enhance power complementarity, load peak shifting, and support. China has achieved some power interconnection with neighboring countries, including Russia, Mongolia, Vietnam, Laos, and Myanmar, totaling an interconnection capacity of roughly 2.6 million kilowatts. Nonetheless, the reluctance of nations to share detailed power system information, for strategic confidentiality reasons, limits the viability of higher-level dispatch centers with global authority.

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, 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, 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.

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. 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:

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:

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:

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:

where ${\textstyle \Delta P_{n}^{\min }}$ and ${\textstyle \Delta P_{n}^{\max }}$ 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:

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:

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 ${\textstyle d}$; ${\textstyle E_{n}^{{\mbox{day}},\min }}$ and ${\textstyle E_{n}^{{\mbox{day}},\max }}$ 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:

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:

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:

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; ${\displaystyle {\mbox{Γ}}}$ 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:

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:

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:

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:

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

where ${\displaystyle \lambda }$ is the Lagrange multiplier and ${\displaystyle \rho }$ 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:

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 ${\textstyle i}$ and region ${\textstyle j}$. If the constraint is directly relaxed, the decoupling in the true sense cannot be achieved. Therefore, the constraint should be first rewritten as follows:

Among them: ${\textstyle T_{ij,t}^{i}}$ represents the transmission power of the contact lines between region ${\textstyle i}$ and region ${\textstyle j}$ in the time period ${\textstyle t}$ when solving the sub-problem of region ${\textstyle i}$; ${\textstyle T_{ij,t}^{j}}$ represents the transmission power of the contact lines between region ${\textstyle i}$ and region ${\textstyle j}$ in the time period ${\textstyle t}$ when solving the subproblem of region ${\textstyle j}$. Figure 2 illustrates this process by treating the contact lines between region ${\textstyle i}$ and region ${\textstyle j}$ 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 ${\textstyle T_{ij,t}^{i}=T_{ij,t}^{j}}$.

Figure 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:

where ${\textstyle {\lambda }_{ij,t}^{\mbox{T}}}$ is the Lagrangian vector multiplier of the relaxed constraint ${\textstyle T_{ij,t}^{i}=T_{ij,t}^{j}}$; ${\displaystyle \rho }$ 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 ${\textstyle i}$ can be given as follows: