Truncated Strategy Based Dynamic Network Pricing for Energy Storage

Xiaohe Yan; Hongcai Zhang; Chenghong Gu; Nian Liu; Furong Li; Yonghua Song

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Truncated Strategy Based Dynamic Network Pricing for Energy Storage PDF

- ORCID：
Xiaohe Yan (Member, IEEE)
✉
- ORCID：
Hongcai Zhang (Member, IEEE)
✉
- ORCID：
Chenghong Gu (Member, IEEE)
✉
- ORCID：
Nian Liu (Member, IEEE)
✉
- ORCID：
Furong Li (Senior Member, IEEE)
✉
- ORCID：
Yonghua Song (Fellow, IEEE)
✉

1. State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (North China Electric Power University), Beijing, China； 2. Department of Electronic and Electrical Engineering, University of Bath, Bath, U.K.； 3. State Key lab of Internet of Things for Smart City, University of Macau, Macau, China

Updated：2023-03-25

DOI：10.35833/MPCE.2021.000631

Abstract

With the increasing penetration of local renewable energy and flexible demand, the system demand is more unpredictable and causes network overloading, resulting in costly system investment. Although the energy storage (ES) helps reduce the system peak power flow, the incentive for ES operation is not sufficient to reflect its value on the system investment deferral resulting from its operation. This paper designs a dynamic pricing signal for ES based on the truncated strategy under robust operation corresponding to the network charge reduction. Firstly, the operation strategy is designed for ES to reduce the total network investment cost considering the uncertainties of flexible load and renewable energy. These nodal uncertainties are converted into branch power flow uncertainties by the cumulant and Gram-Charlier expansion strategy. Then, a time of use (ToU) pricing scheme is designed to guide the ES operation reflecting its impact on network investment based on the long-run investment cost (LRIC) pricing scheme. The proposed ToU LRIC method allocates the investment costs averagely to network users over the potential curtailment periods, which connects the ES operation with network investment. The curtailment amount and the distribution of power flow are assessed by the truncated strategy considering the impact of uncertainties. As demonstrated in a Grid Supply Point (GSP) distribution network in the UK, the network charges at the peak time reduce more than 20% with ES operation. The proposed method is cost-reflective and ensures the fairness and efficiency of the pricing signal for ES.

Keywords

Energy storage; network investment; renewable energy; robust optimization; uncertainty

I. Introduction

WITH the ambitious target to reduce CO₂ emissions, the global renewable energy capacity is predicted to rise sharply to 43% of total installed capacity over the next five years [

1]. However, the uncertainty of renewable energy threatens power systems due to the aging of the pre-planned network with limited capacity. Since it is more costly to reinforce the network, the wind power is curtailed around 1.3 GWh at the peak time in 2015 in the UK, costing more than 90 million euros [2].

The energy storage (ES) can help the network accommodate more renewable energy by shifting the peak over time and mitigating the uncertainties, which potentially defers the network investment [

3], [4]. However, ESs are mainly scheduled without consideration of the uncertainties in the power system [5], [6]. To handle the uncertainties of renewable energy, the robust optimization [7]-[11] is widely used. A robust optimization of second-order cone programming is proposed in [9] to reduce the impact from the uncertain load and distributed generation, which are qualified by different uncertainty sets. There are four typical uncertainty sets in the robust optimization, i.e., interval uncertainty, ellipsoidal uncertainty, budgeted uncertainty, and norm uncertainty. A budgeted uncertainty set is used in [12], [13], where the wind and load uncertainties is considered as budgeted sets in the security-constrained unit commitment. The correlation between load and wind uncertainties is considered in the robust model [12]. To accommodate uncertain renewable energies in the security-constraint unit commitment problem, the max-min problem is converted into a mixed-integer programming problem by a binary expansion [13]. Although the load and generation uncertainties are considered in the operation method for ES [14], [15], the value of ES operation on network investment is not quantified.

In the severe case, there will be more curtailment in the system, bringing forward system reinforcement and increasing operation costs. The ES operation can directly impact the system investment decisions. Therefore, it is significant to set accurate pricing signals to incentivise the efficient use of existing networks. In distribution networks, the long-run incremental cost (LRIC), i.e., the charging for network cost, is widely used in the UK [

16], which sets the pricing signal based on the unused capacity of networks at the peak loading time annually to recover the network investment. The network charges will be reduced resulting from the system investment deferral and investment cost reduction.

Although ES can enhance the capacity with a reduced investment cost of the system, it cannot receive sufficient incentives from the system operator. Therefore, it is significant to design effective and efficient network charges to promote ES. Currently, the pricing schemes are designed only for the traditional generation and load, which purely withdraw or inject power from/into power systems. Since the ES has both features of load and generation, the existing LRIC pricing scheme based on the impact of nodes on annual peak power flow of system is inappropriate to capture the ES operation in hours [

17]. Thus, a time of use (ToU) pricing is required [18], which should reflect the investment cost savings resulting from the ES operation. In the existing research, the ES conducts the energy arbitrage in response to ToU tariffs, generating pricing signals to guide the ES operation [19]. The ToU pricing scheme can provide a fairer price to the users. With an appropriate pricing scheme, the economic incentives will be awarded to ES if it can help release more spare capacity to accommodate load and renewable energy.

The contributions of this paper are concluded as follows.

1) The traditional flat-rate single-point LRIC is extrapolated into a ToU LRIC to ensure the fairness of the pricing scheme to network users by respecting time-variant impact on systems.

2) The truncated strategy based dynamic incentives guiding ES operation is designed to reduce the total network charges. Thus, the existing spare capacity of the system can effectively accommodate increasing renewable energy and loads.

3) A robust operation strategy is designed for ES to minimize the investment cost under uncertainties.

The rest of this paper is organized as follows. Section II designs the ES operation model to reduce the system LRIC. Section III designs a TOU LRIC for the renewable energy and loads. They are demonstrated on a local GSP distribution network in Section IV. Section V draws conclusions.

II. ES Operation Modelling

In this section, the ES is operated to reduce the system network charges, incentivise the efficient use of existing systems, and recover the investment from network users. The ES operation is designed to minimize the peak branch power flow under uncertainties, which is quantified by system LRIC reduction [

18].

A. Network Pricing Scheme

LRIC is a forward network pricing scheme, which is widely used in the UK. To accommodate more generation and demands and allocate the incurred network investment costs to users appropriately, the LRIC evaluates the network investment costs and then charges customers according to their incremental impact on system peak power flow via an incremental approach. This method reflects both the distance and utilization levels of each customer. In the LRIC scheme, the future investment cost is only related to the peak power flow and unused capacity of the system. Then, the LRIC is allocated to network users according to their contributions to the annual change of system peak power flow. The network investment horizon is calculated in (1) and (2) and the present value of the network is evaluated in (3).

C_{l} = P_{l} {(1 + r)}^{n_{l}}

(1)

n_{l} = \frac{l o g C_{l} - l o g P_{l}}{l o g (1 + r)}

(2)

P V_{l} = \frac{A S_{l}}{{(1 + d)}^{n_{l}}}

(3)

where $C_{l}$ is the capacity of branch $l$ ; $P_{l}$ is the peak power flow of branch l; $n_{l}$ is the number of years that power flow grows from $P_{l}$ to $C_{l}$ with a given load growth rate $r$ ; $P V_{l}$ is the present value of the asset with discount rate $d$ ; and $A S_{l}$ is the asset cost.

The incremental cost is conducted for the customer to evaluate its contribution to the change in the present value of the network. The investment horizon and present value are updated with nodal energy injection in (4) and (5), respectively. The change in the present value of branches $Δ P V_{l}$ is calculated in (6) and the network charge from branch $l$ $L R I C_{l}$ is shown in (7). The sensitivity factor between nodal power injection and change of branch power flow [

21]-[23] is shown in (8).

n_{l, n e w} = \frac{l o g C_{l} - l o g (P_{l} + Δ P_{n, l})}{l o g (1 + r)}

(4)

P V_{l, n e w} = \frac{A S_{l}}{{(1 + d)}^{n_{l, n e w}}}

(5)

Δ P V_{l} = P V_{l, n e w} - P V_{l}

(6)

L R I C_{l} = \frac{\sum_{l} Δ I C_{l}}{Δ P I_{n}} = \frac{\sum_{l} Δ P V_{l} \cdot a_{f}}{Δ P I_{n}}

(7)

M_{n, l} = \frac{Δ P_{n, l}}{Δ P I_{n}}

(8)

where $n_{l, n e w}$ and $P V_{l, n e w}$ are the new reinforcement year and new present value of asset, respectively, which are affected by the nodal injection at node $n$ $Δ P I_{n}$ ; $Δ P_{n, l}$ is the power flow change along the circuit due to the nodal injection $Δ P I_{n}$ ; $Δ I C_{l}$ is the incremental cost of branch $l$ over its lifespan due to the nodal injection; $a_{f}$ is the annuity factor; and $M_{n, l}$ is the sensitivity factor of branch power flow in the DistFlow model.

B. Design of Robust Operation Model

The objective of ES operation is to reduce the peak power flow to increase the penetration of renewable energy, which is reflected as the minimization of system LRIC. The objective function and constraints are listed as (9) and (10)-(12), respectively, which are derived from the traditional LRIC scheme. In (9), the total LRIC of the system is the summation of the nodal LRIC from $N$ nodes. In (10), the nodal LRIC of node $n$ $L R I C_{n}$ is the summation of its network charges from L branches.

\underset{S o C_{2,3, \dots, 24}, P_{c, t}, P_{d, t}}{m i n}

\underset{S o C_{1}, P F_{l, t}}{m a x}

\sum_{n = 1}^{N} L R I C_{n}

(9)

s.t.

L R I C_{n} = \sum_{l = 1}^{L} L R I C_{l}

(10)

∆ P V_{l} = \frac{A S_{l}}{P_{l}} \frac{l o g (1 + d)}{l o g (1 + r)} {(\frac{P_{l}}{C_{l}})}^{\frac{l o g (1 + d)}{l o g (1 + r)}} ∆ P_{n, l} a_{f}

(11)

P_{l} \geq m a x (P F_{l, t}^{e}) p f_{l, t}^{e} \in U_{1}

(12)

where $S o C_{1}$ is the daily start-up SoC of ES; $S o C_{t}$ ( $t = 2,$ $3, \dots, 24$ ) is the SoC of ES at time $t$ ; $P_{c, t}$ and $P_{d, t}$ are the energy amounts for ES charging and discharging at time $t$ , respectively; $P F_{l, t}$ is the original power flow of branch $l$ at time $t$ without ES operation; $P F_{l, t}^{e}$ is the power flow of branch $l$ at time $t$ with ES operation; and $U_{1}$ is the uncertainty set of probabilistic power flow calculated based on load and generation uncertainties by the cumulant and Gram-Charlier expansion strategy [

24].

The constraints for branch power flow, SoC, and ES operation (i.e., charging or discharging) are listed as follows.

1)　Constraints for Branch Power Flow

P F_{l, t}^{e} = P F_{l, t} + M_{n, l, t} K_{n} (\frac{P_{c, t}}{η_{c, t}} - P_{d, t} η_{d, t})

(13)

0 \leq η_{c, t} \leq 1

(14)

0 \leq η_{d, t} \leq 1

(15)

\{\begin{matrix} \prod_{n = 1}^{N} K_{n} = 0 K_{n} \in {0,1} \\ \sum_{n = 1}^{N} K_{n} = 1 K_{n} \in {0,1} \end{matrix}

(16)

where $η_{c, t}$ and $η_{d, t}$ are the charging and discharging efficiencies of ES at time $t$ , respectively; $M_{n, l, t}$ is the sensitivity factor of branch power flow at time $t$ ; and $K_{n}$ is an integer indicating the location of ES.

2)　Constraints for SoC

S o C_{t} = S o C_{t - 1} + \frac{P_{c, t - 1} - P_{d, t - 1}}{C_{e s}} S o C_{1} \in U_{2}

(17)

S o C_{m i n} \leq S o C_{t} \leq S o C_{m a x}

(18)

where $C_{e s}$ is the ES capacity; $S o C_{m i n}$ and $S o C_{m a x}$ are the lower and upper limits of $S o C_{t}$ , respectively; and $U_{2}$ is the uncertainty set of $S o C_{1}$ .

3)　Constraints for ES Operation

P_{c, t} \leq O_{r a t e}

(19)

P_{d, t} \leq O_{r a t e}

(20)

P_{c, t} \leq B C_{e s}

B \in {0,1}

(21)

P_{d, t} \leq (1 - B) C_{e s}

B \in {0,1}

(22)

where $O_{r a t e}$ is the rate limit of ES operation; and $B \in {0,1}$ is the binary integer, which ensures that there is no conflict between the charging and discharging processes.

Since the probabilistic power flow during the potential curtailment periods has a significant impact on the ES operation, the conservativeness should be considered when choosing the uncertainty set. Since the uncertainty range during non-potential curtailment periods is of relatively low importance, a budgeted uncertainty set is selected, which can lower the boundaries to achieve the extreme value during these periods. Since $S o C_{1}$ is decided by the end SoC of the previous day, it is designed as an interval set. Equations (25) and (26) give the uncertainty sets for the branch power flows and $S o C_{1}$ , respectively.

U_{1} = \{P F_{l, t}^{e} |P F_{l, t}^{e} = {\bar{P F}}_{l, t}^{\begin{matrix} e \end{matrix}} + ξ_{l, t} {\hat{P F}}_{l, t}^{\begin{matrix} e \end{matrix}}, - 1 \leq ξ_{l, t} \leq 1, \sum_{t = 1}^{24} ξ_{l, t} \leq Γ_{l}\}

(25)

U_{2} = \{S o C_{1} | S o C_{1} \in [{\bar{S o C}}_{1} - ε_{1}, {\bar{S o C}}_{1} + ε_{1}]\}

(26)

where ${\bar{P F}}_{l, t}$ is the forecasting value of power flow derived from the predicted load of branch $l$ at time $t$ ; ${\hat{P F}}_{l, t}$ is the power flow derived from the load uncertainty of branch $l$ at time $t$ ; $ξ_{l, t}$ and $Γ_{l}$ are the parameters representing the desired conservation level of the ES operator; ${\bar{S o C}}_{1}$ is the forecasting value of the initial SoC; and $ε_{1}$ is the deviation.

To speed up the solution time, (11) should be linearized. Derived from (11), $Δ P V_{l}$ can be described as:

Δ P V_{l} = A S_{l} \cdot \frac{l o g (1 + d)}{l o g (1 + r)} Δ P_{n, l} a_{f} {(\frac{P_{l}}{C_{l}})}^{\frac{l o g (1 + d)}{l o g (1 + r)} - 1}

(27)

Because of the change of $(P_{l} / C_{l})^{\frac{l o g (1 + d)}{l o g (1 + r)} - 1}$ with respect to the change of $P_{l}$ $Δ P_{l}$ is small (5% of the forecasting value), the exponent can be linearized to accelerate the calculation, as shown in Fig. 1.

Fig. 1 Linearization of $(P_{l} / C_{l})^{\frac{l o g (1 + d)}{l o g (1 + r)} - 1}$ .

Therefore, $Δ P V_{l}$ can be further derived as:

Δ P V_{l} = A S_{l} \cdot \frac{l o g (1 + d)}{l o g (1 + r)} Δ P_{n, l} a_{f} J_{l} \frac{P_{l}}{C_{l}}

(28)

where $J_{l}$ is the linearization factor.

C. Design of ToU LRIC Pricing Scheme

To capture the influence of ES on the network investment cost and guide its operation, a dynamic pricing scheme named ToU LRIC is designed based on LRIC. The ToU LRIC is designed for each branch to allocate the LRICs of branches from a single peak time to potential curtailment periods. Then, the ToU LRICs of branches are aggregated to the node based on the nodal contribution to the curtailment periods.

Originally, the LRICs are allocated to network users according to their contributions to annual peak power flow. However, although the peak value is the key driver for the reinforcement, the investment can be deferred if the curtailment of load or generation is low. In year $n i v$ , the annual investment cost ( $A I C$ ) equals annuity curtailment cost ( $A C C$ ), as shown in (29) and (30), which means the curtailment cost will be higher in the following year and the system should be reinforced.

A C C_{l} = C C_{l, n i v}

(29)

A I C_{l} = \frac{A S_{l}}{{(1 + d)}^{n i v}}

(30)

where $C C_{l, n i v}$ is the curtailment cost of branch l in year $n i v$ .

It is assumed that the annual power flow follows the normal distribution, and the variance of power flow and increase rate of load are constant. The probability distribution functions (PDFs) of annual power flows in the current year and year $n i v$ are shown in Fig. 2, where D is the potential curtailment level; $D p$ and $D p'$ are the peak power flows in the current year and year niv, respectively; C is the branch capacity; and $Δ d$ and $Δ d'$ are the loal level differences in the current year and year niv, respectively.

Fig. 2 PDFs of annual power flows in current year and year $n i v$ .

The curtailment can be calculated by the PDF of the annual power flow in year $n i v$ , as shown in the shadow area in Fig. 2. As a truncated distribution, the uncertainties are calculated by the truncated strategy in (31)-(33). The truncated shadow area can be calculated by (31) based on the integration of the PDF of power flow in (32). The LRIC should be allocated to these curtailments (the shadow truncated area) according to the corresponding curtailment levels in year $n i v$ .

C C_{l, n i v} = \int_{C_{l}}^{D p_{l}^{'}} [P F_{t, l} \cdot f_{t, i n v} (P F_{t, l})] d (P F_{t, l}) \cdot U C_{t, l} = g_{l} (P F_{t, l}) \cdot U C_{t, l}

(31)

f_{t, i n v} (P F_{t, l}) = \frac{1}{\sqrt[]{2 π σ_{l}^{2}}} e^{- \frac{(P F_{t, l} - {μ_{p}}_{l})^{2}}{2 σ_{l}^{2}}}

(32)

D p_{l}^{'} = \frac{A s_{l}}{U C_{t, l}} g_{l}^{- 1} (P F_{t, l})

(33)

where $D p_{l}^{'}$ is the peak power flow of branch e in year $n i v$ ; $U C_{t, l}$ is the unit curtailment cost of branch $l$ ; $C C_{l, n i v}$ is the total curtailment cost of branch $l$ ; $g_{l} (P F_{t, l})$ is the curtailment amount of branch $l$ ; $f_{t, i n v} (P F_{t, l})$ is the distribution function of $P F_{t, l}$ ; and $μ_{p, l}$ and $σ_{l}$ are the mean and variance of the distribution of the annual power flow of branch $l$ in year $n i v$ .

As shown in Fig. 2, the load level difference $∆ d'$ between the branch capacity $C$ and peak power flow $D p'$ in year niv can be determined in (34). Then, $Δ d'$ is used to calculate the difference $Δ d$ between peak power flow $D p$ and the potential curtailment level $D$ in the current year.

The power flow above the potential curtailment level will shoulder the investment cost in the current year. Therefore, the load level difference $Δ d_{l}$ between $D p_{l}^{'}$ and $C_{l}$ of branch $l$ can be derived as:

∆ d_{l} = ∆ d_{l}^{'} \frac{μ_{l}}{μ_{p, l}} = (D p_{l}^{'} - C_{l}) \frac{μ_{l}}{μ_{p, l}}

(34)

where $μ_{l}$ is the mean of the distribution of the annual power flow of branch $l$ in the current year.

Thus, in the current year, the potential curtailment level $D_{l}$ can be determined by applying the difference $∆ d_{l}$ from the current peak power flow $D p_{l}$ as (35). The power flow above the potential curtailment level is the potential curtailment in the current year.

D_{l} = D p_{l} - ∆ d_{l}

(35)

Therefore, the original LRIC price can be increased to the ToU LRIC of branch by extrapolating the allocation of the current LRIC from a single peak time to the potential curtailment periods. Therefore, by aggregating ToU LRIC of branch, the nodal ToU LRIC of node n can be calculated as:

L R I C_{n, t} = \sum_{l = 1}^{L} L R I C_{l} \frac{m a x (P F_{t, l} - D_{l}, 0)}{\sum_{t = 1}^{365} m a x (P F_{t, l} - D_{l}, 0)}

(36)

The ToU incentives for ES are determined by the difference of nodal ToU LRICs under original power flows and new power flows with ES operation. The ToU LRIC difference resulting from ES operation is the ToU incentive signal for ES, which is denoted as ToU LRICe.

D. Whole Process of Proposed Method

The flow chart in Fig. 3 shows the process of setting the ToU LRIC pricing signal to network users and ES. With the system data, the probabilistic power flow is determined by the combined cumulant and Gram-Charlier expansion strategy [

25]. The original network charge and the time to reinforcement horizon are calculated based on the traditional LRIC pricing scheme as a benchmark for that with ES operation. The ES is operated to reduce the system LRIC via robust optimization under uncertainties. With the assumption of constant load growth rate, the system curtailment amount in year

n i v

is analyzed when

A I C

equals

A C C

. Based on the truncated strategy, the peak difference between the peak power flow and branch capacity in year

n i v

is used to determine the potential curtailment amount in the current year. The curtailment of each branch is allocated to the nodes based on their contributions. Then, the current LRIC is allocated to different periods according to the nodal curtailments over time. By reducing the network charges with the proposed ES operation, the difference between the new LRIC (LRICe) and the original LRIC of the system should be calculated, which is allocated to ES over time as ToU pricing signals based on its charging or discharging amount.

Fig. 3 Flow chart of whole process of proposed method.

III. Case Studies

The proposed pricing models with ES operation are demonstrated in a GSP distribution network in the UK, as shown in Fig. 4 [

26]. It assumes that the system asset lifespan and the annuity factor are 40 years and 0.0831, respectively [16]. A discount rate of 5.6% and a typical load growth of 2% are chosen. The photovoltaic (PV) generation (G1) is located at busbar 1005 with 5% uncertainty of the peak output (25 MW) and the wind generation (G2) is located at busbar 1013. The asset cost highly depends on the voltage level, which means the branch with a higher voltage level normally has a higher cost. For example, the branches 2 and 3 cost

£ 1.8 \times 10^{6}

, almost 5 times as much as branches 16 and 17.

Fig. 4 GSP area test system.

To simplify the analysis, the following assumptions are adopted: ① the loss of ES is 10%; ② the minimum and maximum SoC levels are 0.2 and 0.8, respectively; ③ the maximum charging and discharging rates are 2 MW/h with the capacity of 6 MWh; ④ the start-up SoC is $0.2 \pm 0.05$ ; ⑤ the uncertainty set of the load is assumed to be between $\pm 5 %$ of the predicted value.

Branches 11 and 24 have the largest capacity (50 MW), accommodating the renewable generation connected at busbars 1005 and 1013. The capacity of branch 23 is 6.5 MW, which connects busbars 1006 and 1007. The detailed branch data and parameters can be achieved from [

27]. The hourly output of PV generation

P_{p v}

is estimated by fitting the historical hourly solar irradiance data into Beta distribution.

A. Deterministic and Robust Models

The ES operation is to minimize the system LRIC, which is determined by the deterministic and robust models. The difference between the results of these two models is analyzed and compared. In Fig. 5, the SoC of ES with the deterministic and robust models is applied and displayed.

Fig. 5 SoC of ES with deterministic and robust models.

With the deterministic model, ES charges from 00:00 to 11:00 and 15:00 to 16:00. The maximum charging rate is 1.0 MW/h at 00:00. The maximum discharging rate is 1.44 MW/h at 20:00. Correspondingly, the SoC achieves the maximum value of 0.8 at 12:00 and 18:00.

With the robust model, the charging and discharging periods are similar to those with the deterministic model. The maximum charging rate is 1.14 MW/h at 00:00. The discharging period is from 18:00 to 22:00 with the maximum discharging rate of 1.13 MW/h at 20:00. The start-up SoC is 0.25, which is a severe condition for ES operation. The SoC reaches 0.8 at 18:00 with the robust model. Because of the conservativeness of the robust model, the charging or discharging amount for ES is flatter.

Figure 6 shows the power flow of branch 3 with deterministic and robust models in the severe scenario. The grey area in Fig. 6 is the probabilistic power flow of branch 3 under load and PV uncertainties. At 19:00, the original power flow range without ES operation is between 21.59 MW and 24.82 MW. While with the ES operation, the peak power flow reduces to 23.64 MW at 18:30 with the robust model, which is higher than that with the deterministic model. However, a new peak of 24.02 MW occurs at 20:00 with the deterministic model, which indicates that the deterministic model is not an optimal solution in the severe scenario.

Fig. 6 Power flow of branch 3 with deterministic and robust models in severe scenario.

Since the impact of ES is small on branch 23, the difference of the power flows with and without the ES operation is small. The robust and deterministic models can reduce the peak power flows to 7.64 MW and 7.66 MW at 21:00, respectively.

The differences in the impact of the robust and deterministic models on the reduction of peak power flow of different branches are shown in Table I. The robust model can reduce more peak power flows of branches 2 and 3 because of their expensive investment costs. Although the robust model performs worse than the deterministic model on branches 16 and 17, the difference between these two models is relatively small and the investment costs of these two branches are relatively low.

TABLE I Reduction Difference of Peak Power Flow of Different Branches

Branch No.	Reduction difference (MW)
2	0.21
3	0.28
16	-0.01
17	-0.01
23	0.02

Note: the positive values mean the reduction of the peak power flow with the robust model is higher than that with the deterministic model and the negative ones mean the opposite.

B. Time to Reinforcement Deferral and Nodal LRIC Change

Table II lists investment horizon of different benches in the following three scenarios: ① without ES operation; ② with ES operation deterministic model; and ③ with ES operation robust model. The ES operated with the deterministic model could defer network investment on the majority of the branches. Although the investment horizon is forwarded 0.1 year on branch 17 due to the severe case, the robust model has better performances on other branches.

TABLE II Investment Horizon of Branches in Different Scenarios

Branch No.	Time horizon (year)
Branch No.	Scenario ①	Scenario ②	Scenario ③
2	13.5	14.2	14.6
3	12.0	12.6	13.2
16	8.1	8.1	8.1
17	12.3	12.3	12.2
23	22.6	22.7	22.8

Figure 7 shows the contribution of LRICs of different branches to loads at busbars 1006 and 1007, which are denoted as D1006 and D1007, respectively. The LRICs of these branches are more fluctuating for busbar 1006. The LRICs of branches 2 and 3 have much higher contributions to the total LRIC at busbar 1006, which are $4072 £ / M W$ and $3812 £ / M W$ , respectively. The LRICs of branches 16 and 17 have a much smaller contribution to LRIC of busbar 1006, which are $125 £ / M W$ and $109 £ / M W$ , respectively, because of their cheap asset costs. Since D1006 leads to a reversed power flow of branch 23, the LRIC of branch 23 is negative, i.e., -107 £/MW. The LRICs of these branches for busbar 1007 is relatively flatter, with the highest value of 2259 £/MW from branch 16 and the lowest value of $599$ £/MW from branch 4.

Fig.7 LRICs of branches to load at different busbars.

The total LRICs of D1006 and D1007 and the generation at busbar 1011 (denoted as G1011) are listed in Table III.

TABLE III LRIC of Load and Generation at Different Locations

Scenario	LRIC ( $£ / M W$ )
Scenario	D1006	D1007	G1011
①	9783.0	8502.6	-8503.5
②	9535.9	8423.1	-8424.0
③	9411.0	8383.7	-8384.6

Note: the positive values mean the loads should pay the use of network charges and the negative ones mean the peak power flow can be reduced and the generation will get paid.

C. ToU LRIC

To extrapolate the LRIC allocation from a single peak time to the ToU level, the potential curtailment level of each branch is shown in Table IV. The potential curtailment levels of branches 16 and 17 are 1.6 MW and 1.3 MW less than the peak power flow, respectively. These values are highly related to the unused capacity of branch and the fluctuation degree of power flow.

TABLE IV Potential Curtailment Level of Each Branch

Branch No.	Curtailment (MW)	Branch No.	Curtailment (MW)
1	9.6	13	8.0
2	18.9	14	10.4
3	19.7	15	10.3
4	5.6	16	8.7
5	7.6	17	8.1
6	2.9	18	3.4
7	2.2	19	3.4
8	3.4	20	3.6
9	3.4	21	3.6
10	3.2	22	7.8
11	24.1	23	6.3
12	8.7	24	8.1

The ToU LRICs of the nodes are aggregated from the ToU LRIC of the branches based on their contributions to the system peak power flow.

Figure 8 shows the ToU LRIC of branch 3 in a year, which is allocated based on the potential curtailment level. It can be observed that the ToU LRIC only exists in two main periods and it is zero on most days of the year. Compared with the traditional network charge from branch 3 ( $9411 £ / M W$ ), the maximum ToU LRIC of this branch is $3.15 £ / M W$ at 18:00 on day 19. Day 34 has the longest ToU LRIC signals, starting from 09:00 to 22:00. Most days of the year are not allocated network charges because their power flow levels are below the threshold of the potential curtailment level.

Fig. 8 ToU LRIC of branch 3 in a year.

Figure 9 shows the ToU LRIC of branch 3 on day 34. Without ES operation, the ToU LRIC starts from 09:00 to 24:00 with the peak value of $3.50 £ / M W$ at 18:00. With the robust model, the peak value of ToU LRIC decreases to $2.81 £ / M W$ and the total network charges decreases from $£ 39.36$ to $£ 37.82$ .

Fig. 9 ToU LRIC of branch 3 on day 34.

The ToU LRICs of D1006 without and with ES operation are aggregated from branches, as shown in Fig.10(a) and (b), respectively. In Fig. 10(a), the ToU LRIC exists in two main periods, which are 10:00-12:00 and 17:00-23:00 on day 2 to day 85 and 18:00-22:00 on day 286 to day 355. The peak network charge of ToU LRIC is $8.6 £ / M W$ at 20:00 on day 19, which is reduced to $6.5 £ / M W$ and mitigated with the ES operation, as shown in Fig.10(b). With the ES operation, the ToU LRIC of D1006 is smaller from 18:00 to 23:00, where the ToU LRIC is flatter. Although the ToU LRIC from 10:00 to 12:00 is increasing, the peak LRIC and the total LRIC at busbar 1006 are reduced. Therefore, the ToU LRIC is fairer for the customers and it is more efficient to cover the network investment cost with the ES operation.

Fig. 10 ToU LRICs of D1006. (a) Without ES operation. (b) With ES operation.

D. Dynamic Incentives for ES Operation

The original LRIC cannot reflect the ES operation due to the single peak time. The ToU LRIC improves this situation by extrapolating the single time point to multiple periods, which shows the dynamic incentives over its daily operation period. The incentives for ES operation are determined by the difference between the original ToU LRIC without ES operation and the ToU LRICe with ES operation at the node of ES location. On day 34, the incentives for ES operation are shown in Fig. 11. The positive and negative values are the incentives for ES discharging and ES charging, respectively. The incentive period for the ES discharging is 18:00-22:00, and the incentive periods for the ES charging are 09:00-13:00, 16:00-17:00, and 23:00. The maximum incentive rate of the ES discharging and ES charging are $1.85 £ / M W$ at 20:00 and $1.0 £ / M W$ at 17:00, respectively.

Fig. 11 Incentives for ES operation on day 34.

IV. Conclusion

In this paper, a dynamic pricing scheme is designed for ES and network users based on LRIC. With the proposed robust optimization method, the time to reinforcement can be deferred by ES operation. Through extensive demonstration, the following conclusions are obtained.

1) The truncated strategy can accurately reflect branch curtailment and can efficiently quantify the curtailment amount under uncertainties.

2) The ToU LRIC is fairer for network users to share the investment cost and capture its impact on network investment.

3) The network charges at peak time is reduced by more than 20% with the ES operation, which is more efficient to allocate the network investment cost to network users.

4) The robust optimization based ES operation can produce a better performance to reduce the system LRIC under severe uncertainty conditions. It is beneficial for system operators to defer system investments, whilst accommodating more uncertain renewable energies. It also benefits load and renewable generation to have lower network charges.

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