2012 3rd IEEE PES Innovative Smart Grid Technologies Europe (ISGT Europe), Berlin
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Coordinated Electric Vehicle Charging Strategy for Optimal Operation of Distribution Network Kaiqiao Zhan, Zechun Hu, Member, IEEE, Yonghua Song, Fellow, IEEE, Zhuowei Luo, Student Member, IEEE, Zhiwei Xu, Student Member, IEEE and Long Jia
Abstract--Because of global warming and shortage of fossil fuels, much attention has been paid to plug-in electric vehicles (PEVs) worldwide in recent years. With possibly millions of PEVs on road in the future, it is essential for power network operators, especially distribution network operators, to implement coordinated PEV charging. In this paper, an optimal charging method is proposed for PEVs to maximize profits of distribution companies. By introducing voltage constraints, the optimization method guarantees both security and economics of distribution network operation. To reduce computational difficulties, an iterative approach is presented and linear programming models are built and solved in each iteration with updated nodal voltages. The computing speed of the method can meet the need of real-time operation. Using the 12.66kV, 33-bus distribution network as the test system, simulation results show merits of the coordinated charging method and effectiveness of the proposed voltage constraints. Index Terms--Plug-in electric vehicle (PEV), Coordinated charging, Distribution network, Voltage constraints, Optimal operation.
B
I. INTRODUCTION
ECAUSE of advantages in reducing greenhouse gas emissions and dependence on petroleum [1], the development of PEVs has captured world’s attention. It can be predicted that charging of large-scale PEVs will pose significant impacts on distribution network operation. Uncoordinated charging may cause problems like branch overloading [2], voltage drop [3], harmonic problems [4] and so forth. In [5], the impact caused by the charging load on a typical low voltage distribution grid in UK is studied under different penetration and aggregation level scenarios. In [4], harmonic problems caused by the charging load are studied and some suggestions are made to PEV owners. In [6], the relationship between losses and PEV penetration in micro grids is studied by using the Monte Carlo simulation method. In [7], the impact of the charging load on transformer lives is discussed in details. Though the charging load of PEVs may cause problems to the network, on the other hand, it can also bring benefits to This work is supported by the National High Technology Research and Development Program of China (2011AA05A110). Kaiqiao Zhan, Zechun Hu, Yonghua Song, Zhuowei Luo, Zhiwei Xu and Long Jia are with the Department of Electrical Engineering, Tsinghua University, Beijing, P.R.C. (e-mail:
[email protected],
[email protected],
[email protected],
[email protected],
[email protected],
[email protected].)
economic and secure operation of power systems by optimizing the charging processes. Thus, coordinated charging strategies for PEVs have become a research focus recently. In [8], a method is proposed to reduce power losses by scheduling the charging process of each PEV in the distribution network. Reference [9] proves that coordinated charging can improve the power quality, mitigate voltage imbalance and reduce power losses. In [10], the relationships between losses, the load factor and variation of load power are analyzed, and then a coordinated charging strategy is proposed to reduce the losses and smooth variation of the aggregated load. In [11], PEVs are optimally scheduled to charge at corresponding nodes which will bring minimum impact on system losses by analyzing the sensitivities. In published work on PEV coordinated charging [8-11], the losses are often chosen as the optimization targets to guarantee economics of distribution network operation. However, it may not help the distribution network achieve maximum profits if the electricity purchase/retail price varies over time. In this paper, an optimization method is proposed to maximize profits of the distribution company instead of losses. Voltage constraints are considered in the model to guarantee secure operation. By updating nodal voltages iteratively, a linear programming model is solved in each iteration instead of solving a complex nonlinear non-convex programming model, making the computing speed meet the needs of on-line applications. The coordinated charging method is simulated on the 33-bus distribution system and several conclusions are drawn. This paper is organized as follows. Section II describes the proposed formulation to maximize profits of distribution companies. The detailed derivation of voltage constraints and the iterative process are represented and analyzed. Section III shows the results tested on the 33-bus distribution system. In Section IV, some conclusions are drawn based on the simulation results. II. MATHEMATICAL FORMULATION FOR COORDINATED CHARGING A. Assumptions and Feature Descriptions To build a simple model, several reasonable assumptions and features of the model are listed as follows: (1) The residential distribution grid has a radial or tree topology. (2) The root node of the distribution system is regarded as a slack node with known voltage magnitude. Without loss of
978-1-4673-2597-4/12/$31.00 ©2012 IEEE
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generality, the voltage angle of the slack node can be set equal to zero. (3) It is assumed that the distribution company is not only the distribution network operator but also the electricity retailer. As an intermediary, the distribution company purchases electricity from generation companies or other suppliers at purchase prices and sells it to customers at retail prices. (4) Customers charge their PEVs in private garages, i.e. PEVs have fixed charge locations. Thus the network operator can only control the charging time and charging power of PEVs. (5) Customers require that their PEVs should be fully charged within the charging periods they set. (6) The charging power is continuously adjustable. B. Optimization Model The electricity consumption in the distribution system can be divided into three parts, which are the normal resident load, the charging load and losses. The sum of them equals the injected power into the root node from the outside network. With the injected power and the electricity purchase price curve, the cost of the distribution company can be obtained. With the normal resident load, the charging load and the electricity retail price curve, the revenue of the distribution company can be obtained. The difference between the revenue and the cost is the profit we try to maximize. Based on the above analysis, an optimization model can be built as follows: t max
nmax
t =0
n =1
max ∑ ( β t ∑（ Pn ,t +Pn ,t） − α tV0, t I injected , t )Δt 0
a
(1)
subject to: tmax
∀n : μ ∑ Pn ,t ⋅ Δt = K n Cmax
(2)
∀t , ∀n : 0 ≤ Pn,t ≤ Kn Pmax
(3)
t =1
1− λ
∀t , ∀n : Qn ,t =
2
λ
Pn,t
U na,t
∀t , ∀n : I na,t =
Pn ,t + U na,t 2 + U nb,t 2 a U U n ,t ⋅ Pn0,t + U nb,t ⋅ Qn0,t Q + n ,t U na,t 2 + U nb,t 2 U na,t 2 + U nb,t 2 b n ,t
∀t , ∀n : I nb,t =
(4)
U
b n ,t
Pn ,t − + U nb,t 2 b U U n ,t ⋅ Pn0,t − U na,t ⋅ Qn0,t Q + n ,t U na,t 2 + U nb,t 2 U na,t 2 + U nb,t 2 U
a n ,t
a ∀t : I injected ,t =
(5)
a 2 n ,t
nmax
∑ I na,t
(6) (7)
n =1
where: a I injected ,t is the real part of the current injected into the root node from the outside network at time interval t . V0,t is the voltage magnitude of the root node at time interval
t. αt
is the purchase price of electricity at time interval t .
βt
is the retail price of electricity at time interval t .
Δt
is the time interval. In this paper its value is 15 minutes.
Pn ,t + jQn ,t is the charging power at node n at time interval t . Pn0,t + jQn0,t is the normal resident load power at node n at
time interval t . It is predicted based on the historical load data each day. μ is the efficiency of the charger.
K n is the number of PEVs charging at node n . Cmax is the maximum capacity of a PEV. Pmax is the maximum allowable charging power.
λ
is the power factor of charging power.
U na,t
+ jU nb,t is the voltage of node n at time interval t .
I na,t + jI nb,t is the injected current of node n at time interval t .
Constraints (2) describe the customers’ charging demand. Constraints (3) suggest that the charging power should not exceed the maximum power. Constraints (4) describe the relationship between the active and reactive charging power. Constraints (5) and (6) describe the relationship between the injected power and the injected current at node n . In fact, they are transformations of the following equation: I na,t − jI nb,t = In*, t =
Sn ,t P + Pn0,t + jQn , t + jQn0,t = n ,t U n ,t U na,t + jU nb,t
(8)
Constraints (7) are obtained according to the Kirchhoff’s circuit laws. The sum of currents injected into each node in the distribution equals the current injected into the root node from the outside network ignoring the line-to-ground capacitance. Only (5) and (6) are nonlinear among above constraints. To simplify the problem, an iterative approach is used in this paper. The nodal voltages are supposed to be known in each iteration. Thus (5) and (6) can be regarded as linear constraints and the optimization problem becomes a linear programming problem which is rather easy to solve. After solving the linear programming problem, the power flow is calculated with the optimal solution, and then the nodal voltages are updated. Repeat this process until convergence and the optimal charging power of each PEV is obtained. C. Voltage Constraints Uncoordinated charging of large-scale PEVs may cause serious voltage drop problems in distribution networks [3]. Thus constraints of voltage magnitude limits should be considered in the model. To keep the model linear, voltage constraints are linearized in the following. Let’s consider the voltage constraints of node k :
U min ≤ U k ,t ≤ U max
(9)
It is supposed that node k can be reached along path l via nodes 1, ... , k − 1 starting from the root node 0. The branch connecting node k and node k − 1 is marked as branch k . Because the load flow is not calculated in the proposed formulation, the following equations are derived in order to calculate nodal voltage magnitudes approximately. Thus, constraint (9) can be taken into account and linearized. Through a series of derivation, we can get following equations:
U k = U ka + jU kb = U k −1 + ( Rk + jX k )( I ka + jI kb ) (10)
3 a b a ⎪⎧ Rk I k − X k I k =ΔU k ⎨ b a b ⎪⎩ Rk I k + X k I k =ΔU k U a ΔU a +U kb−1ΔU kb U k ≈ U k −1 + k −1 k U
(11) (12)
k
Δ U k =
U ka−1ΔU ka +U kb−1ΔU kb U
(13)
k
U k
(m)
≈ U 0
k
(m)
+ ∑ Δ U n n =1
(14)
( m −1)
Thus the linear voltage constraints (14) are obtained. The detailed proof is given in the appendix.
flow goes back to step 3. Step 6: Output the optimal results. III. SIMULATIONS AND ANALYSIS A. Results of Coordinated and Uncoordinated Charging The proposed method is tested on the 12.66kV, 33-bus distribution system. Fig. 2 shows the single line diagram. The maximum load of the system in a day is (3715+j2300) kVA. The node 0 connects the distribution network and the outside network. It is a reference node and the other nodes are all PQ nodes.
D. Flow Chart of the Proposed Solution Method The flow chart of the proposed solution method is shown in Fig. 1. Step 1
input required infomation
Step 2
set k = 0 the initial value of charging power is Pn0,t + jQn0,t
Step 3
calculate the node voltage Vnk, t+1 with Pnk, t + jQnk,t (flow calculation)
Step 4
calculate Pnk,t+1 + jQnk,+1 with the Vnk, t+1 t (optimization method)
Step 5
Step 6
Pnk,t+1 − Pnk, t ≤ε ?
Fig. 2. Single line diagram of the 33-bus distribution system.
k =k + 1
No
Yes output the optimal results
Fig. 1. Flow chart of the proposed solution method for PEVs’ optimal charging.
The following are the details of each step. Step 1: Input the required information. The information includes the topology of the distribution network, parameters of each branch, the predicted load curve, PEVs’ charging places and so forth. Step 2: Initialize the settings. Set the iteration index to zero. Set values of charging power to given values. Step 3: Calculate the power flow. Using the charging power optimized in the previous step, we can obtain the nodal voltages by power flow calculation. Step 4: Solve the linear programming problem. Using the nodal voltages obtained in power flow calculation, the charging power can be optimized with the proposed formulation. Step 5: If the difference of the charging power obtained from two successive iterations is below the given level, the solution method converges and the flow goes to step 6. Otherwise, the
To calculate maximum profits the distribution company may obtain, an extreme scenario is chosen and relevant information is given as below. (1) PEVs are parked in fixed places like private garages and available for the network operator to dispatch all the day. (2) The initial SOC of each PEV is 0.10. PEVs should be fully charged by the end of charging processes. (3) The maximum charging power is 3kW. The battery capacity is 32kWh. The charging efficiency is 0.98. (4) There are 240 PEVs evenly distributed in nodes of the distribution network. Supposing the average normal electricity consumption of a household is about 6kW, thus there are about 619 households in this region. Supposing each household owns one vehicle on average, the PEV penetration is about 38.8%. (5) Choose a typical day load curve of north China in winter as the resident load curve, which is shown in Fig. 3. From the curve we know that there are two load peaks separately at about 11:00 and 19:00. (6) It’s assumed that the purchase price varies over time while the retail price stays the same because of the contract provisions. This assumption is not necessary for the model but makes it easier to analyze. (7) The tolerance of convergence is set to 1e-5. To compare with the coordinated charging scenario, an uncoordinated charging scenario is simulated. In this scenario, once PEV owners get home from work, PEVs are connected to the power network to charge. Based on the PEV owners’ behavior, most of PEVs start charging from 17:30 to 20:30. The charging load peak overlaps the resident load peak. According to probability distributions of some parameters (like the SOC, charging periods and so forth) [12], Monte Carlo simulations are carried out for 10000 times and the average profit is calculated. The comparisons between coordinated and uncoordinated charging are shown in Table I.
4
Fig. 4. Hypothetical electricity purchase price curve to test the voltage constraints.
Fig. 3. Day load curve of the resident load.
From Table I, one can see that profits under the coordinated charging scenario are 10.7% higher than those under the uncoordinated charging scenario. However, the total losses (The expression is
tmax
∑ (V0,t I t =0
a injected , t
nmax
− ∑ Pn , t )Δt .) are almost the n =1
same. The voltage profile is much worse for the uncoordinated charging scenario. B. Tests of Voltage Constraints To test whether the voltage constraints work as expected, the electricity purchase price curve is artificially modified to be contrary to the actual situation. As is shown in Fig. 4, the hypothetical curve has low values between 19:00 to 21:00, when the normal resident load is generally rather high. Fig. 5 shows the simulation results of scenarios with and without voltage constraints. Without the voltage constraints the PEVs tend to charge when the electricity purchase price is low to get more profits, while with the voltage constraints the optimization method may sacrifice some profits to satisfy the security constraints. Detailed comparisons of related parameters are shown in Table II. It indicates that voltage constraints can improve voltage conditions without much loss of profits. C. Parameter Analysis In this part, parameters which may affect profits, like the electricity purchase price, PEV penetration and so forth are analyzed. Fig. 6 shows the relationship between the average electricity purchase price and the profits. From this figure, it can be seen that profits decrease as the purchase price grows. The sensitivity is about -70.03 MWh/day. Table III shows the relationship between profits and the penetration level of PEVs. One can see that profits of the charging load are almost proportional to the total number of PEVs. The cost of losses has little growth with the increase of PEV number. It indicates that by using the proposed optimization method, the distribution network can accommodate more PEVs without violating voltage limits and obtain more profits.
Fig. 5. PEV optimal charging power with and without voltage constraints.
Fig. 6. Relationship between the average electricity purchase price and profits of the distribution company.
5 TABLE I COMPARISONS BETWEEN COORDINATED AND UNCOORDINATED CHARGING Charging mode
Profits of the charging load ($/day)
Minimum node voltage (p.u.)
losses of the distribution network (MWh)
Coordinated charging
174.20
0.9300
2.55
Uncoordinated charging
157.35
0.9217
2.57
TABLE II SIMULATION RESULTS OF SCENARIOS WITH AND WITHOUT VOLTAGE CONSTRAINTS Different models
Profits ($/day)
Minimum voltage
losses of the distribution network (MWh)
With voltage constraints
1955.25
0.9300
2.58
Without voltage constraints
1960.58
0.9189
2.60
Number of PEVs 160 240 320
TABLE III PROFIT ANALYSIS OF DIFFERENT PEV PENETRATION LEVELS Profits of the resident load ($/day) Profits of the charging load ($/day) 1764.00 153.87 1764.00 230.62 1764.00 305.13
Penetration 25.8% 38.8% 51.7%
cost of losses ($/day) 53.61 56.42 59.65
Let: IV. CONCLUSIONS The PEVs coordinated charging within a distribution network is studied in this paper. An optimal charging formulation is built to maximize profits of the distribution network company while satisfying charging demand and network security constraints. The formulation is solved iteratively and a linear programming problem is built in each iteration. Thus the proposed method can solve the problem efficiently. Simulation results indicated that coordinated charging can help distribution companies improve their profitability and the distribution voltage profile. The voltage constraints considered in this paper can improve the voltage level effectively. Some parameters like the electricity purchase price and the PEV penetration level can significantly affect the profits obtained from coordinated charging.
⎧⎪ Rk I ka − X k I kb =ΔU ka ⎨ b a b ⎪⎩ Rk I k + X k I k =ΔU k and then we have:
U k = (U ka−1 + ΔU ka )2 + (U kb−1 + ΔU kb )2 Because the line voltage drop is much smaller than the nodal voltage, we can expand the above equation in the Taylor series and get: a
U min ≤ U k ,t ≤ U max
a
U min ≤ U
+U
b 2 k ,t
≤ U max
U k −1
b k
a a a b ⎪⎧U k = U k −1 + Rk I k − X k I k ⎨ b b b a ⎪⎩U k = U k −1 + Rk I k + X k I k
a k
b
( m)
≈ U k −1
( m −1)
+Δ U k
( m −1)
( m −1)
( m)
≈ U 0
k
(m)
+ ∑ Δ U n n =1
is known and Δ U n
( m −1)
( m −1)
can be expressed
linearly. VI. REFERENCES [1] [2]
U k = U + jU = U k −1 + ( Rk + jX k )( I + jI ) a k
b
where, m represents the current iteration number. To take advantage of more distribution information, we can expand the above expression along the path l back. Then the desired expression is finally obtained.
U k
The constraints are nonlinear constraints. To keep the model linear, they should be linearized. Without loss of generality, we can suppose that node k can be reached along path l via nodes 1, ... , k starting from the root node 0. The branch connecting node k and node k − 1 is marked as branch k . Thus the following equations are obtained.
a
and then we have:
It can also be represented as: a 2 k ,t
b
U ΔU +U k −1ΔU k Δ U k = k −1 k U k U k
The voltage constraint is given below for node k at time interval t .
b
Let:
V. APPENDIX A. Derivation of Voltage Constraints
a
U ΔU +U k −1ΔU k U k ≈ U k −1 + k −1 k U k
b k
[3]
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6 J. C. Gomez and M. M. Morcos, "Impact of EV bbattery chargers on the power quality of distribution systems," IEEE Tranns. on Power Delivery, vol. 18, pp. 975-981, Mar. 2003. [5] P. Papadopoulos, L. M. Cipcigan, N. Jenkins, and I. Grau, "Distribution networks with electric vehicles," in Proc. 2009 IEE EE Universities Power Engineering Conf., pp. 1-5. [6] F. J. Soares, J. A. P. Lopes, and P. M. R. Almeeida, "A Monte Carlo method to evaluate electric vehicles impacts in disttribution networks," in Proc. 2010 IEEE Innovative Technologies for an Efficient and Reliable Electricity Supply Conf., pp. 365-372. [7] L. Kelly, A. Rowe, and P. Wild, "Analyzing thhe impacts of plug-in electric vehicles on distribution networks in Britishh Columbia," in Proc. 2009 IEEE Electrical Power Energy Conf., pp. 1-66. [8] K. Clement-Nyns, E. Haesen, and J. Driesen, "Thhe impact of charging plug-in hybrid electric vehicles on a residential distribution grid," IEEE Trans. on Power Systems, vol. 25, pp. 371-380, Febb. 2010. [9] M. Singh, I. Kar, and P. Kumar, "Influence of EV on grid power quality and optimizing the charging schedule to mitigate vvoltage imbalance and reduce power loss," in Proc. 2010 IEEE Power Ellectronics and Motion Control Conf., pp. 196-203. [10] E. Sortomme, M. M. Hindi, S. D. J. MacPhersonn, and S. S. Venkata, "Coordinated charging of plug-in hybrid electricc vehicle to minimize distribution system losses," IEEE Trans. on Smartt Grid, vol. 2, pp. 198205, Mar. 2010. [11] S. Deilami, A. S. Masoum, P. S. Moses, and M. A A. S. Masoum, "Realtime coordination of plug-in electric vehicle chargging in smart girds to minimize power losses and improve voltage proffile," IEEE Trans. on Smart Grid, vol. 2, pp. 456-467, Mar. 2011. [12] Z. W. Luo, Y. H. Song, Z. C. Hu, Z. W. Xu, X. Y Yang, and K. Q. Zhan, "Forecasting charging load of plug-in electric vvehicles in China," in Proc. 2011 IEEE Power and Energy Society Generral Meeting, pp. 1-8. [4]
VII. BIOGRAPHIES Kaiqiao Zhan was born inn March, 1989. He received his Bachelor's degrree in 2010 and is pursuing his Ph.D. degree inn the Department of Electrical Engineering at Tsinghua University (THU), Beijing, China. He woorks in the Smart Grid Laboratory (SGOOL) Operation and Optimization L as a research assistant. Kaiqiaao's research interests include electric vehicles aand power systems modeling and operations. Zechun Hu was born in Nanjing, China. He received the B.S. degree andd Ph.D. degree from Xi’an Jiao Tong University, Shhaanxi, China, in 2000 and 2006, respectively. He woorked in Shanghai Jiao Tong University after graduation and also worked in University of Bath as a research officer from 2009 to 2010. He joined the Depaartment of Electrical Engineering at Tsinghua Univversity in 2010 where he is now an associate professoor. His major research interests include optimal plaanning, operation of power systems and electric vehhicles. Yonghua Song was born inn January 1964. He received his BEng and P PhD from Chengdu University of Science and Teechnology, and China Electric Power Research Instittute in 1984 and 1989 respectively. He was a Postdoctoral Fellow at Tsinghua University from Junee 1989 to March 1991. He then held various positionss at Bristol University, Bath University and John Mooores University from 1991 to 1996. In January 19997, he was appointed Professor of Power Systems at Brunel University where he was Pro-Vice Chaancellor for Graduate Studies from August 2004. In 2002, he was awarrded DSc by Brunel University for his original achievements in power systeem research. In 2004, he was elected Fellow of the Royal Academy of Engineeering (UK). In January 2007, he took up a Pro-Vice Chancellorship and Profeessorship of Electrical Engineering at the University of Liverpool. In 2008, he was elected Fellow of the Institute of Electrical and Electronics Engineers (U USA). He returned to
Tsinghua University in February 2009 as a Professor at the Department of Electrical Engineering. In April 2009, he wass appointed Assistant President of the University and Deputy Director of the Laaboratory of Low-Carbon Energy. In June 2009, he was elected Vice-President of Chinese Society for Electrical Engineering (CSEE) and appointed Chairm man of the International Affairs Committee of the CSEE. His research areass include Smart Grid, electricity economics, and operation and control of poweer systems. Zhuowei Luo was born in May 1984. He received his M.S. degree in 2008 at Hunan University, Changsha, China. He H is pursuing his Ph.D. degree in the Department of Electrical Engineering at Tsinghua University y (THU), Beijing, China. He is a member of the team t working on Smart Grid Operation and Optim mization Laboratory (SGOOL) at Tsinghua Universitty. His fields of interest include electric vehicles an nd power systems modeling and operations. Zhiwei Xu was born in August 1989. He is currently a Ph.D. candidate in Department of Electric Engineering at Tsin nghua University. He works in the Smart Grid Operatiion and Optimization Laboratory (SGOOL) as a reseearch assistant. Zhiwei's research interests include eleectric vehicles and power systems modeling and operaations.
Long Jia was born n in Oct 1988. He received his Bachelor's degree in n 2011 and is pursuing his Ph.D. degree in the Depaartment of Electrical Engineering at Tsinghua Universsity (THU), Beijing, China. He is a member of the team working on Smart Grid Operation and Optim mization Laboratory (SGOOL) at Tsinghua Universitty. He is also an IEEE student member. His fields of interest include power system modelling and operaations and electric vehicles.
