It is said the DG and load are “matched” in a distribution network during a period T only if the following condition is satisfied: | P DG ge ( t ) | ≤ | P L ( t ) | ,     ∀ t ∈ [ 0 , T ] .                                 (1)

Otherwise, it is said the DG and load are “unmatched” if | P DG ge ( t ) | > | P L ( t ) | ,     ∃ t ∈ [ 0 , T ] .                                 (2)

In Eqs. 1, 2, T is the period of observation, and the unit is hour. | P DG ge ( t ) | is the sum of maximum available output power of all the DGs at the moment t. | P L ( t ) | is the sum of consumption power of all the load at the moment t. | P DG ge ( t ) | is usually influenced by the weather conditions, such as light intensity and wind speed, as well as by the installed capacity. Equations 1, 2 show that if the DG and load are unmatched, definitely, there is a moment t, the output power of DGs has to be limited.

If the DG and load are matched in a distribution network, that is, Eq. 1 is satisfied, the matching degree, represented by E _Ω, is defined as E Ω = ∫ 0 T | P DG ge ( t ) | d t ∫ 0 T | P L ( t ) | d t   × 100 %     ( “matched” ) ,                 (3) where, ∫ 0 T | P DG ge ( t ) | d t means the sum of maximum available output power energy of all the DGs during period T. ∫ 0 T | P L ( t ) | d t means the sum of consumption power energy of all the load during period T. The value range of E _Ω in Eq. 3 is [0%, 100%]. Particularly, when | P DG ge ( t ) | = | P L ( t ) | , ∀ t ∈ [ 0 , T ] , E _Ω = 100%. When | P DG ge ( t ) | = 0 , ∀ t ∈ [ 0 , T ] , E _Ω = 0%. The DG and load will be better matched if E _Ω is larger.

If the DG and load are unmatched in a distribution network, that is, Eq. 1 is not satisfied, the matching degree is defined as E Ω = −   ∫ 0 T | P DG ge ( t ) | d t − ∫ 0 T min ( | P DG ge ( t ) | , | P L ( t ) | ) d t ∫ 0 T | P DG max | d t − ∫ 0 T min ( | P DG ge ( t ) | , | P L ( t ) | ) d t × 100 %       ( “unmatched” ) , (4) where ∫ 0 T | P DG ge ( t ) | d t − ∫ 0 T min ( | P DG ge ( t ) | , | P L ( t ) | ) d t means the sum of the actual power energy curtailment of all the DGs during period T. ∫ 0 T | P DG max | dt − ∫ 0 T min ( | P DG ge ( t ) | , | P L ( t ) | ) d t means the sum of the power energy curtailment of all the DGs assuming that the power outputs of all the DGs keep their maximum value (installed capacity) during period T. The value range of E _Ω in Eq. 4 is [−100%, 0%). Particularly, when | P DG ge ( t ) | = | P DG max | , ∀ t ∈ [ 0 , T ] , E _Ω = −100%.

According to Eqs. 1-4, it can be seen that “E _Ω ≥ 0” is equivalent to “matched,” while “E _Ω < 0” is equivalent to “unmatched”.

The DG output power is characterized by intermittency and uncertainty due to the uncertain weather conditions. Therefore, it is necessary to take uncertain weather conditions into account in the DG-load matching degree, which makes it to be calculated by an integral expression. Generally, the calculation speed can be improved by simplifying the integral. For example, the DG output curve can be simplified by assuming the DG output power is constant during a certain time. The result is relatively conservative, but the calculation speed is improved. In practical application, the speed and accuracy of calculation can be reasonably balanced according to the requirement of planning issues.

In distribution network planning, the DG-load matching degree can be applied to the primary capacity planning for the DG integrated in distribution networks. On the basis of the typical load curve of the distribution network, DG installed capacity can be obtained with the goal of maximum DG-load matching degree. The result is the maximum DG installed capacity such that the output power of the DG can be totally accommodated. When DGs are connected to the distribution network, the DG locations can be optimized by taking them as variables of the DG planning model.

The accommodation ratio is used in this work to assess the DG accommodation capability of a distribution network. The accommodation ratio, represented by λ _DG, is defined as the proportion of the actual output power energy to the maximum available output power energy of all the DGs in a distribution network during period T, as is shown in the following equation: λ DG = W DG W DG ge = ∫ 0 T | P DG ( t ) | d t ∫ 0 T | P DG ge ( t ) | d t   × 100 % ,                     (5) where W DG ge is the sum of the maximum available output power energy of all the DGs during period T. W _DG is the sum of the actual output power energy of all the DGs during period T, which is also called accommodation power energy of the DG. | P DG ( t ) | is the actual output power of all the DGs at the moment t. λ _DG ∈[0,1]. The DG accommodation capability of a distribution network will be better if λ _DG is larger.

Load power and network operational constraints are the key influences of DG accommodation capability. To analyze the influences separately, the accommodation ratio is divided into two subindices, including the DG-load accommodation ratio and DG-network-load accommodation ratio.

The DG-load accommodation ratio, represented by λ _GL, is equal to the accommodation ratio without any network operational constraints. The definition of the DG-load accommodation ratio is consistent with the matching degree, which reflects the overall DG-load power balance. The concept of DG-load accommodation ratio is similar to the “substation capacity-load ratio” (Xiao et al., 2018a), which is a classical index for distribution network planning, and also neglects the network operational constraints.

The DG-network-load accommodation ratio, represented by λ _GNL, is equal to the accommodation ratio considering network operational constraints. The definition of the DG-network-load accommodation ratio reflects the effects of network operational constraints on the DG accommodation capability, which is effective supplementary to the DG-load accommodation ratio.

The DG-load accommodation ratio (λ _GL) and DG-network-load accommodation ratio (λ _GNL) are depicted in Figure 1. The area between the blue dot line (λ _DG = 100%) and red curve (λ _GL) is the power energy curtailment of the DG due to the load power limit. The area between the red curve (λ _GL) and green curve (λ _GNL) is the power energy curtailment of the DG due to the network operational constraints limit.

The DG-network-load accommodation ratio curve will tend to the DG-load accommodation ratio curve if proper planning or optimization measures are taken, such as regulating voltage and expanding feeder capacity. Particularly, λ _GNL = λ _GL if the network operational constraints are completely eliminated.

The proposed accommodation ratio is similar to the existing DG penetration (Anderson et al., 2009). The DG penetration can be divided into power penetration, capacity penetration, and energy penetration. The power penetration and capacity penetration are the power ratios of DG to load (Anderson et al., 2009); the energy penetration (Moghaddam et al., 2018), which is most similar to the concept of accommodation ratio, is the energy ratio of DG to load. The accommodation ratio and energy penetration both utilize the power energy of the DG and load to analyze the DG accommodation capability. The difference is that the accommodation ratio, which reflects the utilization of DG, is the proportion of the actual output power energy to the maximum available output power energy of the DG, while the energy penetration, which reflects the supply capability of the DG in the distribution network, is the proportion of the maximum available output power energy of the DG to load consumption. Compared with energy penetration, the accommodation ratio can separately reflect the effects of load power and network operational constraints on the DG accommodation capability directly.

This work is mainly focused on the urban distribution network, which is featured as high load density and short power supply range. Thus, the following assumptions are used:

(1) DC power flow (Purchala et al., 2005) is used. First, the network loss can be included in the power flow of the feeder outlets (Xiao et al., 2011) because the feeders are usually short in length and the network loss ratio is small in the urban power grid. Second, the voltage constraints can be neglected because the system and the DGs are all capable of regulating the bus voltage; thus, the voltage can be kept within the security limits (Shi et al., 2016).

(1) Objective function

The objective is to maximize the sum of the actual output power energy of all the DGs during period T, max W DG = ∫ 0 T ∑ k ∈ G | P DG , k | d t,                       (10) where | P DG , k | is the actual output power of the DG installed on node k at the moment t. G is the set of all the DG nodes.

(2) Network operational constraints

The network operational constraints include power flow constraints and security constraints. The power which flows out of bus, such as load consumption power, is noted as positive, while the power which injects into of bus, such as DG output power, is noted as negative. The power which flows from a substation to a feeder terminal is noted as positive. The power which flows from a feeder to the connected SOP is noted as positive. Due to the assumptions in Assumptions, the constraints are simplified as follows.

The branch flow P _Bi,j can be expressed as the algebraic sum of the net power of downstream nodes and the power injected into the downstream SOPs. The power flow equations are formulated as follows: ± | P B i , j | = ∑ k , m ∈ Ω ( B i , j ) ( ± | P k | ± | Δ P i , m | )   ,                       (11) ± | P k | = { | P L , k | ,                     k ∈ L − | P DG , k | ,         k ∈ G     ,                                 (12) ∑ F i ∈ Ω ( SOP m ) ± | Δ P i , m |     = 0   ,                                                                       (13) where B _i,j is the branch j of feeder F _i . | P B i , j | is the power flow of B _i,j . ± indicates the direction of power. Ω ( B i , j ) represents the set of downstream nodes and SOPs of B _i,j . L and G represent the sets of all the load and DG nodes, respectively. | P k | is the net power of node k. | P L , k | is the load power of node k. | Δ P i , m | is the power flow between F _i and SOP _m . Ω ( SOP m ) represents the set of feeders connected to SOP _m . Equation 11 is the power flow calculation of the branch. Equation 12 is the node power equation. Equation 13 is the equilibrium of active power from each terminal of the SOP.

Due to the assumptions in Assumptions, the voltage constraints are neglected. The security constraints of the FDN with DG installed are mainly thermal capacity constraints, including the feeder capacity constraints, node power constraints, DG output constraints, SOP capacity constraints, and the reverse power flow constraints. | P B i , j | ≤ C B i , j ,               ∀ B i , j ∈ B                       (14) { 0 ≤ | P L,k | ≤ | P k max | ,          k ∈ L 0 ≤ | P DG,k | ≤ | P k max | ,       k ∈ G               (15) 0 ≤ | P DG , k | ≤ | P DG , k ge | ,         k ∈ G                  (16) | Δ P i , m | ≤ C SOP m ,                 ∀ SOP m ∈ S               (17) ± | P B i , 1 | ≥ 0 ,                       ∀ B i , 1 ∈ B                                          (18) where C B i , j represents the capacity of B _i,j . B is the set of all the branches. | P k max | is the maximum permitted power of node k. | P DG , k ge | is the maximum available output power of the DG installed on node k. C SOP m is the capacity of the SOP _m terminal. S is the set of all the SOPs. | P B i , 1 | is the power flow of the outlet of F _i . Equation 14 is the constraint of branch capacity. Equation 15 is the constraint of node power. Equation 16 is the constraint of DG output. Equation 17 is the constraint of SOP capacity. Equation 18 is the constraint of reverse power flow.

The model in Simulation Model of Sequential Production of the FDN is linear, which can be solved by the linear programming software after identical deformation (Xiao et al., 2011). The detailed deformation is shown in Supplementary Material 1. In this paper, LINGO is used to solve the model to obtain W _DG. W _DG is applied to Eq. 5, and the DG-network-load accommodation ratio is obtained.

The case grid of the FDN is shown in Figure 3. For comparison, the case grid of a rigid distribution network (RDN) with the same topology is used, and the FDN is divided into two local networks corresponding to the RDN. The total daily power energy of the load is 597.8 MWh for both the FDN and rigid distribution network, and the details are shown in Supplementary Tables S1 and S2. For both the FDN and rigid distribution network, each DG capacity is 8 MW and the distribution transformer capacity is 10 MVA. For the FDN, each terminal of the SOP is 10 MVA.

However, considering the uncertainty of DG output power caused by weather conditions, three typical scenarios are designed to represent the different matching degrees, as is shown in Table 1.

In Figure 10, the results of the FDN and rigid distribution network (RDN) in three scenarios are summarized, including DG-load matching degrees (MDs, represented by E _Ω), DG-load accommodation ratios (GLARs, represented by λ _GL), and DG-network-load accommodation ratios (GNLARs, represented by λ _GNL).

According to the conclusion of Scenarios 1–3 and Figure 10, it can be seen that

(1) the DG-load accommodation ratio of a distribution network is determined by the matching degree of each local network. Particularly, since the FDN is overall connective, the DG-load accommodation ratio of the FDN is only determined by the overall matching degree.

To remove the bottlenecks of DG accommodation in the FDN, the following measures are proposed:

(5) If the locating of DGs is under discussion, it is recommended that the DGs should be integrated to the areas with enough load to ensure the output power of DGs can be accepted instead of exporting.

For the cases in this work, the specific measures are as follows:

Measure 1: change the location of DGs, enlarge the capacity of feeders, and increase the capacity of the SOP terminal to 10.5 MVA, as is shown in Table 5.

Measure 2: based on measure 1, adjust the incremental capacity. The capacity of branch B_1,5 is increase by 0.9 MVA, B_1,7 0.4 MVA, B_1,8 4.8 MVA, B_2,4 0.7 MVA, and B_2,7 8.8 MVA. The capacity of the rest of the branches is increased as shown in Table 5. The details are shown in Supplementary Table S3.

The cases with overall higher capacity of DGs are also studied to verify the conclusion mentioned above, as is shown in Supplementary Material 4.