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Demand‐side management using a distributed initialisation‐free optimisation in a smart grid

2019; Institution of Engineering and Technology; Volume: 13; Issue: 9 Linguagem: Inglês

10.1049/iet-rpg.2018.5858

1752-1424

Yi Dong, Tianqiao Zhao, Zhengtao Ding,

Smart Grid Security and Resilience

IET Renewable Power GenerationVolume 13, Issue 9 p. 1533-1543 Research ArticleFree Access Demand-side management using a distributed initialisation-free optimisation in a smart grid Yi Dong, Yi Dong orcid.org/0000-0003-3047-7777 School of Electrical and Electronic Engineering, University of Manchester, M13 9PL Manchester, UKSearch for more papers by this authorTianqiao Zhao, Tianqiao Zhao orcid.org/0000-0001-8272-3103 School of Electrical and Electronic Engineering, University of Manchester, M13 9PL Manchester, UKSearch for more papers by this authorZhengtao Ding, Corresponding Author Zhengtao Ding zhengtao.ding@manchester.ac.uk School of Electrical and Electronic Engineering, University of Manchester, M13 9PL Manchester, UKSearch for more papers by this author Yi Dong, Yi Dong orcid.org/0000-0003-3047-7777 School of Electrical and Electronic Engineering, University of Manchester, M13 9PL Manchester, UKSearch for more papers by this authorTianqiao Zhao, Tianqiao Zhao orcid.org/0000-0001-8272-3103 School of Electrical and Electronic Engineering, University of Manchester, M13 9PL Manchester, UKSearch for more papers by this authorZhengtao Ding, Corresponding Author Zhengtao Ding zhengtao.ding@manchester.ac.uk School of Electrical and Electronic Engineering, University of Manchester, M13 9PL Manchester, UKSearch for more papers by this author First published: 18 April 2019 https://doi.org/10.1049/iet-rpg.2018.5858Citations: 4AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Due to the integration of the renewable generation and the distributed load that inherently uncertain and unpredictable, developing an efficient distributed management structure of such a complex system remains a challenging issue. Most of the existing works on the demand-side management concentrate on the centralised methods or need a proper initialisation process. This study proposed a demand-side management strategy that can solve the optimisation problem in a distributed manner without initialisation. The objective of the designed demand management system is to maximise the social welfare of a smart grid by controlling the active power economically. The proposed optimisation strategy that generates the optimal power references uses the neighbouring information while considering the local feasible constraints by using a projection operation. Furthermore, the optimisation algorithm is initialisation free, which avoids any initialisation process when plugging-in new customers or plugging-out power units, such as demand loads, battery energy storage systems and distributed generators. The proposed strategy only uses the neighbouring information, so that the proposed approach is scalable and potentially applicable to large-scale smart grids. The effectiveness and scalability of the proposed algorithm are established and verified through case studies. Nomenclature Indices and sets index of units set of projection area set of BESS units set of controllable load units set of demand units set of generator units set of renewable generator units Variables penalty function of ith BESS emission penalty function of ith generator operation and mainteinance penalty function of ith generator k numbers of power generator units control variables of consensus algorithm communicated information of ith power unit between its neighbours percentage of ith unit that knows the total power mismatch cost function of ith unit m numbers of power demand units n numbers of battery energy storage system units p electricity price welfare function of ith BESS welfare function of ith demand unit welfare function of ith generator Parameters cost parameters of ith controllable demand unit fixed operation cost, the charging or discharging cost and the charging efficiency of ith BESS cost parameters of ith generator active power mismatch between supply and demand in a smart grid active power output of ith power unit maximum active power output of ith power unit minimum active power output of ith power unit active power output of ith BESS active power output of ith controllable demand unit consumed active power of ith demand unit active power output of ith generator unit active power output of ith renewable generator 1 Introduction Due to the increasing energy demand, economic consideration and the emission concerns, the power grid is facing the challenges and opportunities of transforming the traditional power grid into a smart grid. With the integration of renewable energy resource, battery energy storage systems (BESSs) and controllable loads, the power grid becomes distributed and complex [1]. The operation situation of the power grid may change frequently, and therefore the reasonable energy management strategy of the power grid is an important aspect in the smart grid research which is designed to meet the demand requirements at different time intervals and to realise the efficient operation of the smart grid [2]. Due to increasing power demand, economic consideration and emission concern, the smart grid is facing the challenges and opportunities of integrating renewable energy [1]. The percentage of energy derived from renewable sources has risen from 6.7% in 2009 to 29.4% in the UK, 2017 [3]. Due to the intermittency and unpredictability of renewable energy, the future smart grid inevitably integrates more dynamic elements. Meanwhile, the smart grid must be able to maintain the balance between the supply and demand [4, 5]. Demand-side management (DSM) refers to all those strategies aiming at varying controllable load profiles to optimise the entire power system from the supply side to the demand side, optimising power allocation to obtain efficient and eco-friendly usage of electricity [6]. DSM can be implemented by additional equipment to reduce and shift consumption, such as BESS and smart control of EV. Unbalanced conditions resulting from the uncertain load changes and the renewable power generation affect the power quality, and may even damage customer equipment [7]. Therefore, it is crucial for DSM to have an effective and optimal strategy [8]. Typically, DSM focuses on centralised algorithms. For example, Tsikalakis and Hatziargyriou [9] propose a hierarchical control structure to maximise the economic benefits through a central controller, whereas it does not consider the inequality constraints, such as the maximum and the minimum power generation of distributed generation (DG) and the limitations of the charging/discharging power in BESSs. To keep the power units working in a feasible mode, Fu et al. [10] divide an inequality constraint into five intervals, which greatly increases the complexity of the algorithm. The penalty function is applied to eliminate the inequality constraints in [11], whereas the studied algorithm may still exceed the constraint area in some cases. In [12], a centralised second-order economic power dispatch system is utilised to minimise the consumption costs among the supply and demand profiles. Nazari-Heris et al. [13] solve an optimal power and heat scheduling problem, which minimises the operation cost of the smart grid with the uncertain power market prices. In [14], a centralised optimisation strategy is applied in a photovoltaic solar farm, which optimises the investment cost, operation and maintenance cost, fuel cost, emission cost and network losses cost. However, this centralised computation process is very complex, and it is time consuming to calculate the optimal results. The centralised algorithms require a powerful control centre and a large data server centre to collect the global information and process the massive data [15], which are not conducive to the development and the upgrade of smart grids. Besides, the complexity of the centralised demand management system grows exponentially with the increasing number of power units [16]. Due to the uncertainty and the variability of renewable energy resources and demands, the topology of a smart grid may change frequently and suddenly. Therefore, the centralised algorithms are not suitable for the requirement of plug-and-play. Therefore, different flexible distributed control strategies are studied in a significant number of papers, e.g. [17-27]. In [17], the authors work on a social maximal welfare problem in a smart grid in a distributed manner. Hug et al. [18] investigate a consensus-based distributed power management algorithm to solve the demand of a smart grid. In [19], the authors introduce a cooperative distributed demand management system based on Karush–Kuhn–Tucker (KKT) conditions. Deng et al. [20] present a distributed demand response problem and define sub-problems by decomposing the main optimisation problem and solving each sub-problem locally. In [21], a three-layer strategy is used to control a power grid, which includes supervision, optimisation and execution process. Meanwhile, Xu et al. [23] employ a cooperative control strategy of multiple BESSs to maintain the active power balance and minimise the total active power loss associated with BESSs' charging/discharging inefficiency. In [25], the authors propose an optimal distributed solution for economic dispatch to minimise the operation cost. However, most of them are sharing users' information, e.g. the output power or the incremental cost, which would cause privacy concerns. Furthermore, the optimal solutions of the algorithms in [21, 23, 24, 28] can be only obtained when certain initial conditions, i.e. the sum of initial power allocations should equal to the system active power mismatch, are satisfied. In other words, the network resource constraint can be ensured only if it satisfies the initial conditions. Hence, it is not compatible with the plug-and-play function. In this paper, a distributed initialisation-free optimisation strategy is proposed to handle the optimal demand management under the uncertain renewable generation. To deal with the physical constraints and the initialisation problem, we combine the proportional-integral consensus dynamics [29] and the projection algorithm [30], which can solve the constraints problem by local power units. This strategy considers the costs of demand units, BESSs and DGs. Different from the existing results, the emission costs and the battery degradation costs are also considered in our DSM model. In addition, the proposed algorithm can handle DSM problem with any initial errors so that it can adapt the plug-and-play operation. In order to achieve the control objectives of DSM, each power unit generates an optimal power reference through the proposed algorithm by coordinating information with its neighbours. At the meantime, each power unit will meet the power reference through a local controller. The effectiveness of the proposed distributed algorithm is validated through simulation studies in an IEEE 14-bus system and a complex power grid with 40 generators, 15 BESSs and 200 loads. The major contributions of the proposed distributed optimisation strategy for DSM are summarised as follows: The power outputs of all units including controllable loads, BESSs and DGs are optimised according to their welfare functions while maintaining the supply–demand balance in various conditions, i.e. different prices, communication failures and time-varying active power mismatches. The proposed strategy is distributed based on a distributed average estimator. Thus, the proposed strategy is scalable and could be applied to large-scale systems. Additionally, it only requires the exchange of one information variable that does not contain the information of the welfare function. The customer privacy could be protected during the information transmission process. The practical implementation of a distributed algorithm would have initial mismatches. To this end, the proposed algorithm in this paper is initialisation free, which has the advantage of solving the initial errors. Meanwhile, this feature enables the plug-and-play functions in DSM since it does not require any initialisation process. The rest of this paper is organised as follows. The mathematical preliminaries used in this paper are summarised in Section 2. The welfare functions of power units in a smart grid are designed in Section 3, including DGs, BESSs and local users. A distributed projection-based consensus protocol is proposed in Section 4. Simulation results and corresponding analysis are presented in Section 5. Finally, Section 6 concludes this paper. 2 Preliminary In this section, we recall some preliminaries related to the graph theory, the convex analysis and the projection. Let be the set of real matrices and the superscript means the transpose of real matrices. denotes the identity matrix of dimension N and represents a column vector with all entries being 1. denotes the positive real numbers. represents the 2-norm of the argument. 2.1 Graph theory Following [31], an undirected graph can be used to describe the communication topology among the power units, where is the vertex set and is the edge set. The adjacency matrix of is an matrix, such that if and otherwise. Define the degree matrix A graph is connected if and only if every pair of vertices can be connected by a path, namely, a sequence of edges. In this paper, we assume that the graph is connected and undirected. The Laplacian matrix related to is defined as , i.e. (1) When is a connected undirected graph, 0 is an eigenvalue of Laplacian with the eigenvector and all the other eigenvalues are positive. Then (2) 2.2 Convex analysis and projection By [32], a set is convex if for any and . For a closed convex , the projection map is defined as (3) Then the following inequalities hold, (4) For , the normal cone to is (5) A function is convex if for any and . 3 Problem formulation In this section, we formulate the welfare functions of the power units involved in a smart grid, consisting of demand units, BESSs and DGs. The objective of the proposed model is to maximise the total welfare associated with the demand units, BESSs and DGs, including operating costs [1, 5]. Also, we consider emission cost in our model. Therefore, we have (6) To ensure that all power units work in the normal mode, the formulated problem should be subject to the power balance constraint and local power constraints that will be discussed later. Note that the power output of the battery can be positive or negative, depending on discharging and charging states, respectively. 3.1 Controllable power units 3.1.1 Welfare on controllable demand units The welfare function of the ith load is formulated as the level of consumer satisfaction, which is related to the power consumption of applications. For the demand customers, consuming more power will bring more satisfaction. Therefore, similar to [5], the welfare function is defined as (7) where the power utility satisfies . Here, the satisfaction level of customer increases with the consumption of electrical power and will eventually get saturated. 3.1.2 Welfare on BESSs To save electricity and balance the uncertain power generation, some BESSs are also installed in the smart grid. Referring to [33], we formulate the welfare function of BESSs as (8) Due to the cost varies with the characteristics of BESSs, following the approximation in [33], both of the charging and discharging process will increase the DoD cost of BESSs. Hence, the cost function can be uniformly expressed as (9) Furthermore, the power output satisfies with the minimum and maximum power output . Here we assume that BESSs are eco-friendly [34], and therefore the emission cost of BESSs is zero. 3.1.3 Welfare on generators The welfare function for the power generators is usually formulated by the income minus the costs [5, 1]. The costs of DGs mainly include the operation and maintenance (O&M) costs [14, 35] which can be expressed as a quadratic function and a linear function of active power, respectively. Generally, the O&M cost of DGs is expressed as (10) where the power output satisfies with . For comparison purposes, the total emission cost of various pollutants is generally expressed as [36-38] (11) where is the total pollution emission cost for the ith power generator. Therefore, the welfare function is expressed as (12) Basically, the welfare for the generation unit represents the benefit of selling power minus the cost of operation and emission. The first term in (12) means the income by selling energy and the other terms denote the cost caused by maintenance and pollution. 3.2 Uncontrollable power units From [2, 14], renewable generators and users' loads are uncontrollable power units and their power output and consumption are related to the light intensity, illumination time, wind speed, customers' habits and so on. Therefore, these uncontrollable power units generate power by different conditions, and these uncontrollable power units are considered as undispatchable in this paper. In the smart grid, transmission losses are inevitable, accounting for around 5–7% of the total power load [39], which can be modelled by multiplying the load with this percentage. Overall, to maintain system stability, the active power balance between the supply and the demand side is described as (13) Since the renewable source is non-dispatchable and the load is related to the customers' habits which are partial controllable, we rewrite the above constraint as (14) Here, can be negative or positive. 3.3 Problem reformulation In this paper, our objective is to design a reliable DSM system that can maximise the social welfare while maintaining active power balance under various conditions. To this end, an objective function is formulated by integrating the above welfare functions and subjecting to physical constraints. For notation convenience, we denote the power vector as where m, n and k denote the numbers of demand units, BESSs and DGs with (15) where and denotes the welfare of the ith unit. Notice that the cost function of each unit is strictly convex and continuously differentiable. Traditionally, the constrained optimisation problem can be solved using centralised methods, but those algorithms require a powerful control centre to collect data from the subsystems and distribute control instruction to the units after calculation. In the following section, we design a distributed algorithm, where each subsystem is allocated with a low-price processor, by which the collection and calculation can be performed locally. 4 Distributed solution In this section, a projection-based gradient decent algorithm is developed, by which the inequality constraints can be tackled accordingly. We consider the Lagrangian function for each unit with the affine equality constraint, written as (16) where is the Lagrangian multiplier which is used to ensure the equality constraints are met during the optimisation process. The inequality constraints are not considered in the Lagrangian function since the inequality constraint can be solved by local units with projection algorithm. The optimal solution can be obtained by using a well-known centralised saddle-point dynamics as (17) Its equilibrium points (16) satisfy the KKT conditions (see, e.g. [32]). However, it collects the global information about the Lagrangian multiplier () of all the power units. In the modern smart grid, most loads are distributed, so that it is desired to design a distributed algorithm for DSM that solves the optimisation problem locally. To facilitate our design, a local copy of the global variable is used to estimate the global (). As a result, the problem is solved when the local copy variables converge to the . According to the KKT conditions, the following lemma is proposed. Lemma 1.The optimisation problem has an optimal solution if and only if there exists a vector of Lagrangian multiplier such that (18) where denotes cost function vector, and represents its gradient. 4.1 Algorithm design Inspired by the authors in [29, 30], the objective of the distributed algorithm is developed in order to achieve a consensus of the Lagrangian multiplier and reach the optimal power output. Let denote the feasible domain of the ith unit's power output, which is clearly a compact and convex set. The problem (15) can be solved by the following distributed algorithm, : (19a) (19b) (19c) (19d) where is the power output of the ith power unit and are two auxiliary variables of the ith unit. The convergence analysis of (19a)–(19d) is detailed in the Appendix. Note that the proposed algorithm can be implemented with respect to the local feasible constraints by projection operations. Furthermore, the algorithm (19a)–(19d) does not require any initialisation process, as proved in the Appendix. Therefore, the system constraints can be satisfied without specifying initial conditions. Due to free of any control centre and initialisation process, the algorithm can work under a 'plug-and-play' operation that improves the flexibility of future power systems. The algorithm (19a)–(19d) is distributed, in this sense that the ith unit only needs the local data and the information which is shared with the neighbouring units. Therefore, (19a)–(19d) does not require any centre to process the data or coordinate the units. Since the local data do not need to be uploaded and downloaded from the centre, each unit can respond to local data changes rapidly which can quickly adapt the local decisions. The algorithm can be understood concerning singular perturbation, where the third dynamic is on a faster scale than the second one. Hence, goes to , as t goes to infinity, and substituting this to the algorithm yields a saddle-point seeking algorithm. We will show the detailed proof in the next section. Different from the algorithm in [30], the proposed algorithm does not require each unit to know the load information, is not always equal to . Furthermore, the algorithm (19a)–(19d) introduced the estimation ability to detect the total mismatch, which can be selected according to [40] and . Therefore, this algorithm could be applied to a large grid. Notably, the proposed algorithm does not require the initialisation process so that it is more compatible with the operation of a smart grid. The initialisation for the network resource constraint is quite restrictive for a sizeable dynamical grid because it is related to the global coordination and has to be performed whenever the network data/configuration changes. However, the communication network may change frequently and rapidly in a smart grid, such as when plugging in an electric vehicle and routine maintenance. Furthermore, it is not trivial to achieve the initialisation coordination with both the local feasibility and the network resource constraints. 4.2 Algorithm implementation The proposed distributed algorithm adopts a distributed average consensus estimator to measure the global mismatch information locally. The step-by-step algorithm for all power units is shown in Algorithm 1 (see Fig. 1). To illustrate more clearly, a flow chart for solving distributed demand-side management by the proposed algorithm is shown in Fig. 2. As in [11], the communication structure can be designed independently of the power system in a cost-efficient way based on the location and the convenience of the smart grid. For example, a communication topology is designed as Fig. 3. DGs, BESSs and controllable loads are connected through a communication network as the power units in a smart grid. To obtain the optimal power outputs, each power unit only interacts with its neighbouring units to exchange the information through the communication network at the top level, and then each power unit performs the proposed algorithm accordingly. Fig. 1Open in figure viewerPowerPoint Algorithm 1: distributed optimal DSM Fig. 2Open in figure viewerPowerPoint Flowchart for ith power unit management Fig. 3Open in figure viewerPowerPoint Two-level structure of optimisation algorithm 5 Case study In this section, four cases are employed to validate the effectiveness and applicability of the proposed algorithm. At the beginning, two sub-cases in the first case study are used to test the algorithm in a modified IEEE-14 bus system. First sub-case considers a constant power mismatch as 20 MW, and the results obtained using the proposed algorithm are compared with the results based on previous work [28]. In the second sub-case, the optimisation algorithm is studied under different time-of-use (ToU) prices. Next, case 2 is studied to test the proposed algorithm under communication failures, which assumes that all communication links of the DG4 fail to exchange information with neighbours. The plug-and-play adaptability of our algorithm has been tested in case 3. Lastly, a large-scale power system is adopted to test the scalability of the proposed algorithm in case 4. The parameters are chosen from [41]. Results of the four cases are discussed in the following section. 5.1 Case 1 In this study, we test the algorithm on an IEEE 14-bus system which consists of four DGs, two BESSs and ten loads in Fig. 4, whose coefficients are shown in Table 1. Table 1. Cost coefficients for simulation studies , MW , MW DG1 0.008 33.83 2.3 × 10−3 −1.5 × 10−3 2.0 × 10−4 2.857 — — — 0 70 DG2 0.0062 34.03 2.1 × 10−3 −1.82 × 10−3 5.0 × 10−4 3.333 — — — 0 65 DG3 0.0075 33.93 2.2 × 10−3 −1.249 × 10−3 1.0 × 10−6 8.0 — — — 0 70 DG4 0.0072 33.97 2.3 × 10−3 −1.355 × 10−3 2.0 × 10−3 2.0 — — — 0 65 BESS1 — — — — — — 0.025 3.41 95% −40 40 BESS2 — — — — — — 0.025 3.41 95% −30 30 L1 — — 0.072 8.25 — — — — — 0 50 L2 — — 0.066 7.90 — — — — — 0 60 L3 — — 0.070 7.55 — — — — — 0 30 L4 — — 0.055 8.00 — — — — — 0 40 L5 — — 0.075 7.75 — — — — — 0 40 L6 — — 0.045 8.05 — — — — — 0 60 L7 — — 0.058 7.85 — — — — — 0 60 L8 — — 0.062 8.45 — — — — — 0 40 L9 — — 0.070 7.25 — — — — — 0 50 L10 — — 0.058 8.00 — — — — — 0 50 Fig. 4Open in figure viewerPowerPoint Modified IEEE 14-bus system 5.1.1 Constant demand and price As a baseline test, we first consider the situation with a constant power mismatch and electricity price, where all the power units are connected. To reveal the effectiveness of the proposed strategy, our algorithm is first compared with another algorithm in [28]. Here, the electrical price is set to 40 £/MWh, which is chosen from the UK electric price report [42], and the BESSs are distributed among the communication network. The electricity price and the supply–demand mismatch are assumed to be constant, and the system parameters are set to be the same to make the comparison study more convincible. As shown in Fig. 5, the power outputs of each power unit can converge to its optimal value with the proposed algorithm. Here, the negative/positive power value means the consumed/generated power. The BESSs can be charged or discharged depending on the electricity price and the power mismatch. Fig. 6 shows the power mismatch during the optimisation process. It indicates that the power is not balanced at the beginning and the power mismatch converges to zero within 5 s, which reflects that our algorithm is capable of maintaining the power balance. Thus, the proposed strategy can address the DSM problem from any initial error. Fig. 5Open in figure viewerPowerPoint Update of power outputs with the proposed algorithm 1 Fig. 6Open in figure viewerPowerPoint Power mismatch with the proposed algorithm In order to show the advantage of initialisation-free feature, we first compare the proposedalgorithm with an existing work in [28]. In the comparison study, we assumed both algorithms areinitialised randomly and do not satisfy the initialisation conditions in[28]. The results are shown inFig. 7. The power outputs using these algorithms can converge to stablevalues, but the results of the algorithm in [28] are not optimal since it needs the sum of all initial poweroutput is equal to the power mismatch during the initialisation process.Then, we further compare the proposed algorithm with the algorithm in [27]. To make the comparison moreclearly, the ideal results are obtained through a centralised solver inMatlab, and the comparison result is given in Table 2. Compared with the algorithms in [27, 28], the proposed algorithm can achieve the optimal active powervalues even under initial errors. Table 2. Comparison results Algorithm (19a)–(19d) Algorithm in[28] Algorithm in[27] Matlab(fmincon) DG1 30.6819 22.9948 26.7769 30.6820 DG2 55.3153 43.9412 50.2758 55.3155 DG3 39.3940 30.2443 35.2270 39.3941 DG4 43.8133 35.7782 39.4714 43.8134 BESS1 −22.1363 −24.7803 −24.2210 −22.1362 BESS2 −24.8522 −25.8211 −26.4163 −