Hierarchically Coordinated Energy Management for A Regional Multi-microgrid Community

$\Delta t$

Time interval.

$t$

Index of time.

$i,j$

Indices of MG.

$k$

Index of controllable appliances.

$n$

Index of special ordered set of type 2 (SOS-2).

$\boldsymbol{N_{G}}$

Set of MGs.

$\boldsymbol{{ES}^{i}}$

Set of Energy Storages (ESs) in MG $i$.

$\boldsymbol{{EV}^{i}}$

Set of Electric Vehicles (EVs) in MG $i$.

$\boldsymbol{L_{1}^{i}}$

Set of type 1 loads in MG $i$.

$\boldsymbol{L_{2}^{i}}$

Set of type 2 loads in MG $i$.

$\boldsymbol{N_{k}^{i}}$

SOS-2 for $k$th ES in MG $i$.

$p_{\!M\!,b}^{i,t},p_{\!M\!,s}^{i,t}$

Purchasing and selling power of upstream grid.

$p_{C,b}^{ij,t},p_{C,s}^{ij,t}$

Transmitted power between MG $i$ and $j$.

$p_{k,b}^{i,t},p_{k,s}^{i,t}$

Discharging and charging power of ES (EV).

$u_{M}^{i,t}$

Binary indicator for power from utility grid.

$u_{C}^{ij,t}$

Binary indicator for transmitted power between $i$th and $j$th MG.

$\delta_{k}^{i,t}$

Binary indicator for power of ES.

$\theta_{k}^{i,t}$

Binary indicator for power of EV.

$\lambda_{k}^{i,t}$

Binary indicator for type 1 load.

$\mu_{k}^{i,t}$

Binary indicator for type 2 load.

$\nu_{k,s}^{i,t},\nu_{k,e}^{i,t}$

Integral variables to indicate starting and ending time of $k$th type 2 load.

$\alpha_{k,n}^{i,t}$

Element in SOS-2 $\boldsymbol{N_{k}^{i}}$ of $k$th ES.

${N_{g}}$

Number of MGs.

${T}$

Length of time horizon.

$c_{b}^{t},c_{s}^{t}$

Electricity purchasing and selling price between individual MGs and the upstream grid.

$c_{C}^{t}$

Electricity transaction price between MGs in the community.

$\varepsilon_{ij}$

Loss factor between MG $i$ and $j$.

$\overline{P_{M}^{i}},\underline{P_{M}^{i}}$

Power limits of utility grid.

$p_{C}^{i,t}$

Aggregation of transmitted power in community.

$\overline{P_{C}^{ij}},\underline{P_{C}^{ij}}$

Transmitted power limits.

$\overline{P_{k}^{i,t}},\underline{P_{k}^{i,t}}$

ES (EV) power limits.

$\zeta_{k}^{i}$

ES (EV) charging/discharging efficiency.

$E_{k}^{i,t}$

ES (EV) energy level.

$\overline{E_{k}^{i}},\underline{E_{k}^{i}}$

ES (EV) energy limits.

$E_{k}^{i,dep}$

EV energy requirement at departure.

$\boldsymbol{T_{k}^{i}}$

Set of Parking time region.

$p_{L_{1}}^{i,t}$

Power of non-dispatchable loads.

$p_{PV}^{i,t}$

Power of PV.

$p_{k}^{i,t}$

Power of type 1 and 2 loads.

$P_{k}^{i}$

Total required energy of type 1 and 2 loads.

$H_{k}^{i}$

Operation durations of $k$th type 1 (or 2) load sets.

$c_{k,n}^{i}$

ES degradation cost coefficient.

$G_{k,n}^{i}$

Energy level in which $c_{k,n}^{i}\in\boldsymbol{N_{k}^{i}}$ .

$l_{i}^{x},l_{i}^{y}$

Coordinates of $i$th MG.

$W$

Weighting matrix of MG community.

$w_{ij}$

Weighting coefficient between MG $i$ and $j$.

$f_{k}(\bullet)$

ES degradation cost.

$F(\bullet)$

Linearized ES degradation cost.

Without loss of generality, the individual objective function can be formulated as follows:

Each local EMS aims to minimize the operational cost within the time $\boldsymbol{T}$, which depends on its own operational requirements and is not necessarily correspondent with the community-level EMS. Investment costs of controllable appliances such as EV and PV are not considered, since the proposed method is focused on operational optimization where investment planning is out of this work’s scope.

The first row in (4) represents the electricity cost with the upstream grid. The time-varying dynamic pricing scheme provides economic incentives by bilateral transaction, in which the purchasing price $c_{b}^{t}$ is usually higher than the selling price $c_{s}^{t}$ to prevent energy arbitrage. The second row represents the transaction cost by power exchanges in the community, in which the transaction price $c_{C}^{t}$ is predefined as the average of $c_{b}^{t}$ and $c_{s}^{t}$. The last row represents the degradation cost associated with ES. It is known that the degradation process of ES is nonlinearly related with its lifetime and operation mode [23]. To take the degradation cost in a practical fashion, the linearized model is defined from its nonlinear form in [24] by using inclining blocks, as shown in Fig. 2. The ES degradation cost $f_{k}$ with special ordered set of type 2 (SOS-2) constraints can be written as follows:

where $\alpha_{k,n}^{i,t}$ is the element of $k$th ES in $\boldsymbol{N_{k}^{i}}$ that only the adjacent elements are nonzero. $c_{k,n}^{i}$ is the degradation cost coefficient of $k$th ES, and $G_{k,n}^{i}$ is the power level at $c_{k,n}^{i}$.

The objective function (4) is subject to the various constraints on power balance, energy exchange, ES, EV and loads.

Each individual optimization problem $\boldsymbol{M_{i}}$ for MG $i$ can be described as follows:

$\boldsymbol{M_{i}}$ is formulated as a MILP which can be effectively solved by many open-source and commercial solvers.

The pairing algorithm identifies each distinct pair of MGs with minimal power transmission losses, so that the pattern of exchanging energy inside the MG community can be established. To achieve this target, appropriate weighting coefficients need to be addressed to determine the pairing priority. Since the MG community is particularly regional, electrical distances of MGs are considered to be so close that specific line resistance parameters may not be attained explicitly [25, 26]. Therefore, geographical distances are used instead as weighting coefficients to indirectly reflect transmission losses induced by power exchange among MGs, since they are approximately proportional to line resistances within small regions [27, 28]. The loss factor coefficient $\varepsilon_{ij}$ is specified as well to present the linear transmission loss in a practical fashion.

Accordingly, a 2-D Cartesian coordinate system is formulated where the location of MG $i$ is expressed by its geographical coordinates $(l_{i}^{x},l_{i}^{y})$. Hence, the weighting coefficient $w_{ij}$ of MGs $\{i,j\}$ representing the transmission loss can be expressed by their Euclidean distance as follows:

Note that $w_{ij}$ for MGs with no physical connection can be set as a large positive number $M$, and intuitively, $w_{ii}=M,~{}i\in\boldsymbol{N_{G}}$ as no self connection exists. The weighting matrix $W$ of the entire community is expressed as follows:

It can be easily recognized that $W$ is ${N_{g}}{\times}{N_{g}}$ symmetric. Each row ${r_{i}}=\{w_{ij}|j\in\boldsymbol{N_{G}}\}$ in $W$ comprises the weighting coefficients of MG $i$ to other MGs. Next, we prove that there always exists a pairing $(w_{ij},w_{ji})$ in $W$ for the MG community set $\boldsymbol{N_{G}}$, in which they are of minimal values in their corresponding rows ${r_{i}}$ and ${r_{j}}$.

By removing invalid pairings whose connections are not constructed and self-pairings in prior, it is straightforward to determine the MG pairings with minimal transmission losses in the community by using the pairing algorithm. Accordingly, after the minimal value of rows in $W$ is found, the pairing $x,y\in\boldsymbol{N_{G}}$ for $W$ can be determined by looking into the same values of these minimums as follows:

The pairing algorithm has demonstrated MG pairings with minimal transmission loss can be always located for the entire community. However, such a pairing is regarded valid between two MGs only with different power flow directions. To this extent, the coordination strategy recognizes all the valid pairings of MGs with power surplus and deficits, determines specific values for corresponding energy transactions and simultaneously minimizes the total transaction loss in the MG community. To reach this target, total transmitted power of individual MGs with respect to the community need to be firstly distinguished. An auxiliary variable $p_{C}^{i,t}$ for $i$th MG is introduced as follows:

where $p_{C}^{i,t}$ is the summed transmitted power of $i$th MG to all other MGs in $\boldsymbol{N_{G}}$ and the upstream grid.

It is recognized that $p_{C}^{i,t}$ is positive when the $i$th MG has power surplus and negative with power deficit. Based on the power flow directions, therefore, the corresponding elements in $W$ can be excluded by the signs of $p_{C}^{i,t}$ as follows:

where $M$ is a large positive number.

After elimination of irrelevant parameters, valid pairings of MGs with different power flow directions can be always established by using the modified coefficient matrix $W$ in (34), (39) and (53). Based on the optimized dispatch signals transmitted from local EMSs, the community-level EMS can determine the transactions among MGs explicitly.

The community-level EMS executes the pairing algorithm to find MGs in a pairing with minimal weighting coefficients, marked as $x,y\in\boldsymbol{N_{G}}$. Correspondingly, the variables related to exchanged power of MGs $x,y$ need to be updated in the MG level. For all possible scenarios, the transmitted power of MGs $x,y$ are defined by (58), and variables $p_{C}^{x,t}$ and $p_{C}^{y,t}$ indicating summed transmitted power of MGs $x,y$ respectively are updated by (63) as well.

At last, the weighting matrix $W$ is updated as follows so that the successfully paired MGs have been excluded:

(64) finishes the first MG pairing, and the community-level EMS starts searching next pairing for the rest of MGs, until any of the total energy surplus of deficit among $\boldsymbol{N_{G}}$ becomes zero. The stopping criterion can be expressed as follows:

The overall procedure of the proposed EMS is described in Fig. 3. In the beginning of each time period, PV and load forecasts are made locally and the day-ahead electricity price is obtained. For each MG $i\in\boldsymbol{N_{G}}$, dispatch decisions in the entire scheduling horizon are obtained by solving its own optimization problem in the local EMS individually, and $p_{C}^{i,t}$ is calculated by (52). Then for the community-level EMS, $W$ is re-established based on (34), (39) and (53). Afterwards, the community-level EMS finds the MG pairing for current $W$. Procedures of the coordination strategy in the community-level EMS are iterated by (52)–(64) until (65) is met.

It is worth mentioning that the proposed pairing algorithm is not to physically control the power flow from one MG to the other. Rather, it is to settle the energy transactions within different MGs in a hierarchical way. Specifically, the local EMS determines the optimal power dispatch for each individual MG, and all the excessive power will go to the PCC for power exchange. The pairings of power surplus and deficit are then established based on the in-prior local scheduling in the community-level EMS. In this way, each MG can transact with others rather than the upstream grid to save the operating cost. It is important to note that the community-level EMS is not to physically dispatch the power but to settle the energy transactions, and the power surplus and deficit of all the individual MGs are balanced with the upstream grid at the PCC. Also note in Fig. 3 that the community-level EMS has no iterative information exchange with the local EMS since $p_{C}^{i,t}$ is only collected once from the corresponding $i$th local EMS, and that the local EMS can solve the MILP optimization problem in parallel. Such the non-iterative interaction and parallel optimization will improve the computational speed of the proposed EMS over the existing techniques. The verification on computation speed will be provided in Section V.

On the selection of weighting coefficients, as the electrical distances of MGs are close due to the regionally small area, it is usually not feasible to obtain exact line parameters and thus hard to calculate the explicit power losses. It is similarly suggested in [29, 30] that the fixed R/X ratio can be employed approximately in the homogeneous distribution system to calculate the line parameters. On the other hand, since it is only required in Theorem 1 that the weighting coefficient matrix $W$ is symmetric, the specific line parameters can be readily employed as weighting coefficients for the proposed algorithm without affecting its effectiveness if they are explicitly known beforehand.

In this case study, a MG community with 4 different types of MGs (MG1-MG4) is investigated. MG1 and MG2 are considered as two individual houses with different load patterns. MG3 is an apartment building with 10 households. MG4 is a small-scale MG with high PV penetration, where its renewable output is much larger than the total load. The specifications of 4 MGs are detailed in Table I. For each households in individual houses and apartment, type 1 and type 2 loads are listed in Table II with predefined operation time ranges. The PV profile is based on the solar radiation data from [32], which is depicted in Fig. 4. The non-dispatchable load and electricity price data are based on the average household statistics in 2017 [33] and the hourly market data in Singapore [34], which are shown in Fig. 5 and Fig. 6, respectively. Specifically, to indicate different patterns, the load profile of MG2 is shifted so that it has the peak power consumption in the daytime. The loss factors in the community $\varepsilon_{ij},~{}i,j\in\boldsymbol{N_{G}}$ are set to be 0.05.

A regional MG community with 50 MGs in four types including 20 of MG1, 20 of MG2, 5 of MG3 and 5 of MG4, respectively, is further investigated to validate the scalability of the proposed EMS. Individual MG characteristics follow those of same types in Case 1.

The operational cost results are presented in Table V. It is observed the entire MG community has reduced the operational cost in 24 hours from $548.361 to $490.268, while the total transmission loss is $8.036. In total, the electricity expense has been improved by 9.129%. In addition, the proposed EMS benefits all types of individual MGs with average cost improvements ranging from 5.422% to 14.958%. As a result, the implementation of the proposed EMS may implicitly decrease the electricity price from the upstream grid in turn, since the stress to power congestion at peak hours in the distribution level would be alleviated.

Comparing the result with that in the 4-MG case, it is noted that the ratio of operational result reduction does not necessarily decrease with the increasing size of MG community, since it is closely related on the different MG types and their independent operational states, such as RES outputs, fixed and dispatchable load profiles, electricity prices and so on.

Three existing approaches in the literature are conducted for comparison to further evaluate the performance and advantages of the proposed EMS. Firstly, the single-level energy management framework in [35] is adopted as the centralized benchmark in which all the MGs in the community are combined just as one unity. A mathematical model similar in Section III-B is formulated and solved. Secondly, the optimization method proposed in [22] without stochastic processes is utilized in the community level as the two-level EMS benchmark, in which a Lagrangian relaxation based algorithm is used to eliminate power exchange imbalances. Finally, a purely decentralized framework by only using local EMSs is added in which each MG makes transaction directly with the upstream grid, without any communication inside the community.

Comparative results with existing methods in Case 1 and 2 are presented in Table VI and Table VII, respectively. It is seen in Table VI that the proposed EMS has achieved the optimized operational cost slightly higher by just 0.2% to the best result with [35], whereas the computation time is nearly 19 times faster than the second best result with [22]. It is noted that the operational cost in [35] is the minimum since all the information from individual MGs has been fully collected and gathered into the centralized EMS, which may nevertheless bring serious privacy concerns. Similar privacy problems exist in [22] that information of dispatch signals needs to be exchanged among MGs and with the community-level EMS. On the contrary, private information is well preserved in the proposed EMS, since no communication among local EMSs or with the community-level EMS is required by any means. Communications between the community-level EMS and local EMSs only involve total power exchanges of individual MGs that do not expose any detailed decision making inside MGs.

The benefits on computational speed of the proposed EMS are further revealed with the increasing number of MGs, as shown in Table VII. It is seen that the proposed EMS spends just 15 seconds to the optimized result with a degradation of 0.21%, however it takes nearly half an hour for the centralized method in [35] and 23 minutes for the decentralized approach in [22]. Such the discrepancy on computational efficiency would become more noticeable with size expansion of the MG community, therefore, the proposed EMS has outstanding advantages in view of trade-offs between computation efficiency and solution quality. With this respect, it is noted that the uncertainty related with renewables and loads are not modeled in the proposed method because of its superior performance on computation speed. The proposed EMS is able to be carried out more frequently if uncertain factors are taken into consideration.

In this paper, a hierarchically coordinated EMS model for multiple small-scale MGs in a regional community is proposed, accounting for minimization of the community-level operational cost and maximization of individual MG-level benefits simultaneously. The local EMSs aim to minimize the individual operational cost, while the community-level EMS determines specific energy transactions in the community by using the pairing algorithm to further reduce individual operational costs. The proposed EMS has been validated by two case studies with different scales. By comparing with existing approaches, simulation results have shown significant advantages of the proposed EMS on modeling generality, computational complexity and privacy security, that the optimization time has been reduced significantly by the non-iterative algorithm, and privacy issues have been eliminated by minimal information exchange.

There have been studies addressing uncertainties by implementing stochastic optimization techniques [36, 37, 38], in which uncertainties are usually handled by sampling reduction and uncertainty set in stochastic programming, and then the optimization problems can be solved in multiple stages. Nevertheless, most of the techniques and algorithms mentioned above have been implemented to the centralized energy management framework whereas little application has been reported regarding decentralized energy management. In our future research, it is planned to incorporate uncertainties into the proposed model with distributed stochastic optimization schemes, and develop computationally efficient algorithm for practical application.