Proceedings of the 45th ISCIE International Symposium on Stochastic Systems Theory and Its Applications Okinawa, Nov. 1-2, 2013
Optimal Power Flow Problem with Energy Storage, Voltage and Reactive Power Control Ashkan Zeinalzadeh Hawaii Natural Energy Institute, University of Hawaii at Manoa E-mail:
[email protected] Reza Ghorbani Dept. of Mechanical Engineering, University of Hawaii at Manoa E-mail:
[email protected] Ehsan Reihani Dept. of Mechanical Engineering, University of Hawaii at Manoa E-mail:
[email protected] Abstract The integration of renewable energy into an electric grid introduces new challenges to achieve optimal operating point for the power flow problem, because of randomness in generation and the distance between generation and consumption. These challenges can be overcome by using storage with appropriate capacity and efficient strategy to charge and discharge. In this work, we introduce a model of the power flow problem with storage so that it can be used to inject active and reactive power into the grid. We formulate an optimal power flow problem for a distribution grid with storage as a multi-period control problem based on the deterministic and stochastic nature of load and generation. We propose a multi-objective optimization function, in which the operating objectives are as follows: i) injecting reactive power into the power system for decreasing distribution loss and voltage deviation; and ii) maximizing renewable energy integration with minimum curtailment.
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voltage regulation across the grid and maximize RE penetration over a time horizon. We consider a distribution grid with a single generator and a finite number of customers with solar PV systems and battery storage. Customers simultaneously decide on • how to charge/discharge the battery • the amount of injected active and reactive power to the distribution system in order to maximize their expected profit over multipleperiods of time. We consider a mix of deterministic and stochastic load and PV models for customers. The fundamental challenges for our model are as follows: • Controling the state of charge of the storage devices • Dealing with stochastic renewable energy sources • Dealing with stochastic load • System-wide voltage constraints
Introduction
The Optimal Power Flow (OPF) problem at the transmission level aims to find an optimal operating point of a power system that minimizes generation cost and transmission loss subject to certain constraints on power and voltage variables. OPF can be defined at the distribution level by an optimal load flow problem with maximization of renewable energy integration over a time horizon. Dynamics of renewable energies (RE) are stochastic. Therefore, classical static OPF methods cannot be applied if we are interested in making the grid reliable and highly efficient, with high penetration of renewable energies. In this work, we present a distributed multi-period OPF for a distribution grid system with solar photo-voltaic (PV) power generators equipped with smart inverters and energy storages. Smart inverters can control active/reactive power based on voltage fluctuations. This work considers a distributed optimization of smart inverters on a high RE penetrated grid with energy storage. The goal is to preserve the
• Highly distributed control optimization. We introduce a decentralized problem in which optimization will be done in a distributed manner such that each PV generator maximizes power injection to the grid while minimizing a local cost function. In our model, customers need not know about each others’ decisions except the implicit information in observed voltage at PCC. However, through the history of voltage at PCC each controller realizes the deterministic or stochastic nature of the load-generation for the time horizon. The voltage magnitude at PCC is affected by the decisions of all customers and the grid generator. We will study the grid efficiency for a scenario in which customers do not communicate to each other. The proposed methods provide an innovative way to integrate renewable energy into the grid and provide better management of storage to improve the penetration level and voltage regulation. The proposed model allows both supply-side and demandside elasticity. The key merit of the work is to optimize
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and control the power grid for improving penetration of renewable resources and higher supply and demand elasticity. This approach allows the distribution network to operate efficiently and satisfy grid objectives optimally. This work will address the following questions: How can the mix of solar PV systems and energy storage be optimized to fulfill the load? What are the effects of stochastic load and solar PV models? What are the major factors that differentiate between the uses of distributed storage versus single high-power storage? Unlike this work, most of the available literature has simplified the grid model by ignoring the effect of reactive power injection on voltage by consumers. 1.1 Storage It is predicted that fossil fuel will be substituted by renewable energy, because it is more environmentally sustainable and widely accessible. We probably never get rid of all fossil fuel, but fossil fuel is currently responsible for 70 percent of all energy generation, including transportation and electrical generation. We need to decrease it from 70 percent to 10 percent. The grid has been operating for over 100 years without energy storage. The whole grid is like a giant inventory system. The energy is generated and transmitted simultaneously to consumers. In case of renewable energy the available energy must be consumed or stored; otherwise it will be wasted. There is a huge market for energy storage, but so far there is no solid and economically feasible solution for it. Energy storage is expensive and there is a need to add value to it before it makes sense to use it widely. Energy storage can reduce the amount of the peak power, but we can also reduce the peak by adding a natural gas generator to the grid. If the price of natural gas goes higher, using energy storage would be more competitive. It would be important to demonstrate the benefit of energy storage beyond reducing the peak, e.g., increasing transmission capacity for a growing and dense area is expensive. One way to avoid that cost would be to use distributed generation and storage at a substation and supply the energy locally. We also need to figure out how to combine renewable energy and storage to deliver the energy at the same price or cheaper than conventional energy. There is a need to create a new control architecture to incorporate the storage such that the RE integration to the grid becomes more affordable, reliable and environmentally sustainable. There are a number of potential applications for energy storage in a grid, e.g., mitigating fluctuations, reducing peak, improving load following, providing efficient spinning reserve, minimizing losses, decreasing the need for expanding the transmission and generation capacities, and regulating frequency, voltage and power factor. After all, energy storage is a long-term opportunity and there is a need to facilitate the development of real energy storage options.
1.1.1
Survey
Integration of energy storage into the grid is a subject of ongoing study [1]-[4]. The integration of energy storage into the grid leads to a finite-horizon optimal control problem that enables optimization of power allocation over time. [1] consider a single generator and a single load with deterministic demand profile and power generation. They assume a quadratic generation cost and strictly increasing battery cost. They study the effect of storage on multi-stage optimal generation policy. They establish that the optimal policy is to charge the battery initially, and then discharge the battery to supply generation in final stages. Unlike this work, they ignore the reactive power. [2] investigate the impact of large-scale integration of energy storage on generation costs and peak reductions. They formulate a finite-time OPF problem with simple charge/discharge dynamics for energy storage, using a procedure based on a convex semi-definite program. Unlike [1], they consider the constraints on storage capacity and charging/discharging rate. Their model allows them to study the impacts of energy storage on cost and peak generation versus changes in power rate, capacity and distribution over the network. Unlike this work, they neglect the uncertainties due to fluctuations in demand and intermittency in generation. [3] consider a network with deterministic demand profile and nondecreasing convex generation cost. They optimally place, size and control the energy storage for the network. They establish that under optimal policy, zero storage capacity is allocated to generator buses that connect to the grid via single links. In this work, it is assumed that demand is stochastic. 1.2 Reactive Power and Voltage Regulating Voltage violations in distribution systems can occur for a number of events, e.g., sudden high or low load, cable outage, dropped generation and capacitor bank outages. Transmission systems are traditionally more responsible than distribution systems to regulate voltage. We assume distribution system provide a nominal level of voltage regulation, and consumers with PV generators elevate the level of voltage regulation. Consumers can behave as intelligent agents sensing local voltage and making decisions regarding reactive power. Using reactive power to reduce the loss during peak-load time operations becomes more economically effective in power systems, [5] and [6]. Growth in decentralized PV penetration level in a low-voltage distribution system can lead to voltage violations in times of high solar radiation and light load time. PV generators have good potential to provide reactive power. Reactive power from decentralized PV power generation can be used to minimize voltage violations along the feeder. The influence of the reactive power for voltage regulation in networks with a larger impedance angle is stronger than in networks with a smaller impedance angle. Therefore, it is important that reactive power be injected at the proper locations, sizes and times. Otherwise, the reactive
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power can cause more losses and voltage violations. During light-load times, losses are lower, but over-voltage can occur and injecting reactive power can cause more losses. An optimal strategy must improve voltage levels while reducing active power losses and minimizing the cost of the reactive power source. The ability to inject active and reactive power without voltage violation depends on all local consumers’ decisions. Injecting reactive power locally using smart inverters has multiple benefits: The generators with lower reactive power support capability have lower cost per generated MW and are cheaper; to maximize the limit of active power that can be transferred through a line, reactive power must be minimized; reactive load can cause more voltage drop than the same size of active load; reactive losses can be reduced by generating reactive power locally; the robustness of the system in response to voltage violation is location-dependent, because the reactive power source is unevenly distributed; reactive power generated from capacitors is proportional to the square of voltage, and the transient response of capacitors can be problematic; injecting reactive power into the grid to respond to voltage variation using smart inverters is more effective than capacitors and tap-changing transformers, which are mostly static; excessive use of capacitors can cause imbalance in reactive power distribution and voltage collapse; transferring reactive power from a line can increase active power losses; reactive power losses increase with square of the flow; distributed reactive resources can reduce the reactive losses resulted from increased active power and therefore relieve congestion; supporting reactive power locally can reduce the losses through the feeder and improve local power equality.
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Model
Consider a radial distribution network with the set of customers C = {1, · · · , n}; the set of PV generators G = {1, · · · , n}; the set of batteries B = {1, ..., n}; the set of distribution lines L = {(0, 1), (1, 2), ..., (i, i + 1), ..., (n − 1, n)}. The following notations are used in this work.
Fig. 1: Distribution model
• Pi,t : Active power loss of ith line at time t • Qi,t : Reactive power loss of ith line at time t • Vi,t : Voltage magnitude at customer i at time t
• PDi,t : Active power requested by customer i • QDi,t : Reactive power requested by customer i • Pgi,t : Active power generated by inverter i • Qgi,t : Reactive power generated by inverter i • Pt : Generator active power • Qt : Generator reactive power • Ppvi,t : Power generated by ith PV • Si,t : Apparent power injected into node i • Pkl,t : Active power transferred from node k to l • Ii,t : Current flowing through customer i • bi,t : Battery power level • ri,t : Battery charge or discharge rate • Ri : Line i’s resistance • Xi : Line i’s inductance • ω : Angular frequency. Define the following vectors: Pg = {Pgi,t |∀i ∈ G, ∀t}, Qg = {Qgi,t |∀i ∈ G, ∀t}, Ppv = {Ppvi,t |∀i ∈ G, ∀t}, PD = {PDi,t |∀i ∈ C, ∀t}, QD = {QDi,t |∀i ∈ C, ∀t}, We assume Ppv , PD and QD are stochastic. Each customer is equipped with a PV generator, smart inverter and battery. By using a smart inverter, customers can decide on the amount of active power (Pg ), reactive power (Qg ) based on PV power (Ppv ) and power factor limitation. 2.1 Centralized problem Dynamics of renewable energies are stochastic; therefore, classical static OPF methods cannot be efficient and reliable if we are interested in high penetration of renewable energies. We are interested in describing a supplementary controller that can promote efficiency and voltage reliability. In the centralized problem, we assume that power flow constraints (1)-(8) are satisfied by the operator while a central controller decides locally on charging and discharging strategies for batteries and how much active and reactive power each customer can generate. There is no communication among consumers, but each consumer communicates with the central controller. The apparent power of ith customer is given as Si,t = Pi,t +jQi,t = (PDi,t −Pgi,t )+j(QDi,t −Qgi,t ). (1)
• θi,t : Voltage angle at customer i at time t
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The magnitude of the current flowing through customer i can be achieved from Si,t . |Ii,t | = Vi,t
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Active and reactive power loss for the line (i, i+1) are given as Pi ,t ≈ (|Ii,t | + · · · + |IN,t |)2 Ri , (3) Qi ,t ≈ (|Ii,t | + · · · + |IN,t |)2 ωXi .
(4)
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The voltage magnitude Vi,t at customer i is bounded by given constraints as Vimin ≤ Vi,t ≤ Vimax .
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The battery energy level bi,t evolves according to 0 ≤ bi,t = bi,t−1 − ri,t .
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If Ppvi,t < Pgi,t then battery discharge rate is ri,t = Pgi,t − Ppvi,t ,
(10)
otherwise battery charge rate is ri,t = Ppvi,t − Pgi,t .
(11)
Let fi be a mapping from energy to power for ith battery and fi (bi,t ) be the available power from ith battery at time t. The active and reactive power generated by ith inverter must satisfy (12) and (13) Pgi,t ≤ Ppvi,t + fi (bi,t ),
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Qgi,t ≤ tan(cos−1 (P F i ))Pgi,t .
(13)
(12) means that the power export from ith inverter cannot be more than the sum of PV and battery power. (13) is the power factor constraint for customer i. Customers can have different lower limits on their power factors based on their location in the distribution grid. It is desired that customers with bigger impedance angles be able to inject more reactive power for voltage regulation. The voltage loss at customer i is approximated as follows:
V − V1,t ≈
N R1 Pn,t cos(θn ) + X1 Qn,t sin(θn ) , (14) Vn,t n=1
for i = 2, ... V −Vi,t ≈ V −Vi−1,t +
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(15) The objective is to increase the penetration while decreasing distribution loss and voltage violations. The loss is subtracted from objective function (16) in order to increase the penetration without increasing the loss in the system. We assume Pgi,t and Qgi,t are controllable parameters which can be achieved from (16). The objective function is given as max
Pg,t ,Qg,t
N N N α Pgi,t −β Pi ,t −γ |V −Vi,t | , (16) i=1
i=1
i=1
with respect to (9)-(15). In this work, it is assumed that α, β and γ are convex weights and are constant. These values could be time variant, and each consumer can have different weights. Finding optimal weights is another interesting problem which can be considered as a future work. 2.2 Decentralized problem Similar to the centralized problem, it is assumed that power flow constraints (1)-(8) are satisfied by the operator. Under decentralized controller each consumer tries to maximize its own penetration while minimizing its own loss and voltage violation. Unlike the centralized problem, each consumer decides on the charging and discharging strategy for its own battery and the amount of active and reactive power injected into the grid. There is no communication among consumers, and they are able to observe their own voltage. Consumers are aware of their own line impedance. Note that consumers ignore the effect of their own decisions on others’ objective functions. Consumer i objective function is given as follows, max αPgi,t − βRi |Ii,t |2 − γ|V − Vi,t | (17) Pgi,t ,Qgi,t
with respect to (9)-(15). 2.3 Simulation To simulate the centralized and decentralized controller a radial distribution grid of 7 customers in Hawaii is considered. The average of 132 days of PV generations and loads are used as PV generations and load models in simulation. The simulation graphs are given in figures 2-9 for centralized and decentralized controller. The daily load and PV generation, customer voltage, batteries voltage, line loss, grid loss, active power, reactive power and power factor are given in figures 2 - 9 respectively. In the simulation, it is assumed that α = 0.03, β = 0.05 and γ = 0.92 for both controllers. 2.4 Conclusion The goal in this work is to understand the impact of distributed reactive power generation on voltage variation and optimal power generation. A centralized controller flattens the demand more than decentralized controllers during peak
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time because the battery usage is more efficient and battery reserves last for a longer period of time. Decentralized controllers have higher active power penetration from PV generators into grid, which causes more power loss during off-peak time, but the centralized controller charges the batteries more during off-peak time; subsequently, the centralized controller can regulate voltage in peak time by providing active and reactive support through battery storage. Future work can be summarized as follows: Study a mix of centralized and decentralized controllers; develop a learn-
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ing process to find optimal weighings in the objective function of the optimization problem; derive an optimal dynamic strategy for charging and discharging the battery by reactive power support; study the transient effect of injecting reactive power; study how to make the load at distribution level deterministic despite of all uncertainty and the stochastic nature of local generation; consider different kinds of correlation among demand distributions to study penetration of PV generation into the grid.
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Acknowledgements This work is partially supported by NSF award 1310709.
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