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Optimization Modeling in a Smart Grid Damian Lampl, Md Chowdhury, Pranav Dass, Kendall E. Nygard1 Dept. of Computer Science North Dakota State University Fargo, ND, USA {Damian.Lampl, MD.Chowdhury,Pranav.Dass,Kendall.Nygard}@ndsu.edu 1. Corresponding Author Vahid Khiabani Dept. of Construction Management and Operations Management MSUM Moorhead Moorhead, Minnesota, USA [email protected]

Abstract—Communication and control in a fully realized Smart Electrical Grid involves heterogeneous wired and wireless networks working cooperatively, supporting data streams among many types of sensors. We address the selfhealing problem, in which the goal is to intelligently automate corrective actions when a disruption to Grid operation occurs. Such actions include redirecting electricity flows along alternative pathways, and selectively tripping breakers. A primary objective of a self-healing method is to prevent cascading failures. Motivated by the need for corrective actions in self-healing to produce efficient and reliable grid operations, we formulated and developed an optimization model that generates sets of high performance electricity flows in an arbitrary Grid configuration. The model is a Capacitated Transshipment Problem (CTP) that we solve using a very fast and customized algorithm. The versatility of the model in supporting multiple performance metrics and the speed achieved in generating sets of optimal electricity flows makes the model useful in evaluating self-healing strategies. Index Terms — smart electrical grid, self-healing capacitated transshipment problem, linear programming, network flow optimization I. INTRODUCTION A Smart Grid is an electrical generation and distribution system that is fully networked, instrumented, and automated [2]. From a communication network perspective, there are three distinct levels. At the most distributed level, within a demand site such as a home, a wireless network is typically

used to interconnect appliances and various other devices and systems. Intelligent control is called for to regulate consumption of energy for such things as heating water and living spaces. At a second level, smart meters receive information from the low level network, and are in turn themselves networked within neighborhoods. Other devices are also in the neighborhood network with the smart meters and form the distribution system. Wireless networking is typical within a neighborhood. Finally, a wide area network (WAN) interconnects utility owned and operated equipment and systems, such as distribution substations, power plants, and long-haul transmission lines. Multitudes of sensing devices, such as Phasor Measurement Units (PMUs) that report detailed waveform information, are deployed throughout the grid. Selfhealing functionality relies heavily on streaming sensing data to drive models and analytics aimed at choosing effective actions for maintaining safe, efficient, and reliable grid performance. An electrical grid experiences faults caused by numerous factors such as failures of generators or routers; or power lines damaged by weather events or vandals. Faults can propagate through the connected networks of an electrical grid and result in remote butterfly effects. The effects can be cascading failure and consumer power outages over wide areas. It is not possible to prevent such faults [3], but their effects can be minimized by isolating fault sources with sensor information and taking corrective actions. Corrective actions taken by power companies traditionally are mostly focused on scheduling and dispatching crews and equipment to make repairs and replace devices or connections in the grid infrastructure. However, human decision making and actions often cannot be fast enough to avoid significant downtimes for

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consumers, providing a basic motivation for intelligent automation in a Smart Grid. One strategy for mitigating the effects of malfunctions in the grid is to dynamically reroute power to physically avoid trouble spots. However, rerouting power can itself be a source of problems, as power lines that are overloaded or nearly so can result in cascading failures over wide areas. Thus, control decisions and actions to reroute power must be done with full consideration of possible ramifications distributed in the grid infrastructure. The software tool that we have developed serves the purpose of rapidly determining optimal distribution patterns and dispatches of power along available channels, including the reporting of metrics that evaluate costs and quality of service. Another important consideration in optimizing grid operations is the emerging deployment of microgrids. A microgrid is a local energy generation system, powered by small-scale generators, batteries, or alternative sources like solar panels. A microgrid is coupled with a primary grid, and can be disconnected as needed so that a local area can function as an island during an emergency, or to cut costs. Thus, microgrids provide a decentralized control function that can help maintain quality of service. Our self-healing model supports the use of microgrids. The mathematical model that we have developed is a linear programming optimization model with a special structure that can be conceptualized as an abstract network with nodes and arcs. As described in the literature, the model is a Capacitated Transshipment Problem (CTP). One type of parameter for the model pertains to known data on grid topology such as locations of sites where power is generated or demanded and interconnection nodes. Another type of parameter pertains to the capabilities of grid devices to do useful work, such as capacities of transmission lines to carry power and of power plants to generate electricity. The output of the model is the values of variables that specify dispatching decisions, flows of power, and performance metrics. Under conditions of normal operation or of disruption, data from distributed sensors are streamed to populate the model and trigger computational devices within the Grid to solve the model. Our customized model solver is fast and modest in terms of computational resources, so it can be preinstalled on computational devices distributed in the Smart Grid. General linear programming solvers could be applied to the model accurately, but would have the disadvantage of requiring unacceptably long computation times. This remainder of paper is organized as follows. Section III provides a brief overview of linear programming modeling. In section IV, the CTP formulation is presented and is applied to the Smart Grid. The algorithmic process for solving the model is detailed in section V. Section VI provides the results and analysis, followed by the conclusion in section VII. II. OBJECTIVE Representing the Smart Grid network using a CTP model allows multiple different cost and network flow related problems to be easily solved. To make a Smart Grid self-

healing, whenever a critical failure is detected, the CTP solver can be used to find an optimal and inherently feasible redirected path for redistributing energy throughout the grid, resulting in minimizing customer outages. Apart from the self-healing aspect of the Smart Grid, the CTP solver offers other key benefits such as its ubiquitous availability to any machine or mobile device connected to the internet, regardless of the operating system. Since the CTP Solver is able to connect to a database as well as read XML files, it could be easily integrated with other Smart Grid systems such as failure notification solutions, providing automatic optimal electric flow rerouting based on the supplied network topology of available nodes and arcs. Since arc capacities are taken into consideration, the cascading failure dynamic could possibly be avoided by ensuring network flow is feasibly rerouted. The CTP solver incorporates an object-oriented approach, thereby ensuring ease of use and maintainability for its users. This further allows the developers to quickly determine the application areas that need updates and implement them in a timely and efficient manner. The CTP solver automates its processes so that the user does not need to learn a new application-specific language or syntax to follow them. The CTP solver involves use of bidirectional arcs in its design, thus allowing the network flow in both directions between a node pair, resulting in effectively limiting the network file size and memory requirements of a dataset containing all bidirectional arcs. In this work, we have developed the mathematical models based on the design goals of the CTP solver we have already discussed in order to determine the optimal network flow of a given Smart Grid network. III. LINEAR PROGRAMMING MODELS Linear Programming models are formulated to maximize or minimize an objective function that is devised to measure performance of a solution. Linear constraints in the form of equations or inequalities are supported. Linear programming is an exact model, in that once solved, the solution is guaranteed to be the very best (genuinely optimal) as measured by the objective function. In some applications heuristic models are applied as an alternative, but such models do not guarantee optimality. The three basic steps given below are followed when formulating a linear programming model. 1. Determination of the decision variables 2. Formulating the objective function 3. Formulating the constraints The decision variables are the quantities that the model seeks to calculate, providing the solution to the problem. The objective function is the expression that the modeler wishes to optimize, and the constraints are limitation requirements. The general form of a linear programming model is given below in Figure 1 [4].

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Parameters

Parameters

C = [cj] = Vector of costs or value measures per unit of decision variable value A = [aij] = matrix of technological coefficients that measure the rate at which variable xi consumes resource j. b = [bi] = Vector of coefficients that measure constraint limitations of resource

c = [cij] = Measures of the costs or values per unit of flow through arcs indexed by tail and head nodes i and j u = [uij] = Vector of flow capacities on arcs l = [lij] = Vector of lower bounds for flow on arcs b = [bi] = Vector of supplies and demands at nodes indexed by i. Positive values for supplies, negative for demands

Variables Z = Objective function that measures the value of a solution x = [xi] = Vector of decision variables Formulation

Variables Z = Objective function that measures the value of a solution x = [xi] = Vector of optimal flows Formulation

Optimize z = c1x1 + c2x2 + . . . + cnxn Subject To:

Minimize z = Σcijxij

a1,1x1 + a1,2x2 + . . . + a1,nxn {≤, =, ≥} b1 a2,1x1 + a2,2x2 + . . . + a2,nxn {≤, =, ≥} b2 . . . am,1x1 + am,2x2 + . . . + am,nxn {≤, =, ≥} bm x1,x2, . . . xn ≥ 0

Subject To:

Figure 1: Linear Programming Model General Form When instantiated to model electricity distribution in the Smart Grid, we think of the decision variables as representing flows of power, and resource constraints as representing capacity limitations on devices and power lines. IV. THE CAPACITATED TRANSSHIPMENT MODEL The CTP is conceptualized as a network problem with supply and demand nodes, transshipment nodes, and connective arcs. The basic concept is to find an optimal set of flows that transfers units from supply nodes through the network to meet requirements at the demand nodes, conserving flow at transshipment points, and without violating capacity constraints. The CTP is presented in algebraic form in Figure 2.

(1) (2) (3) (4)

xji xij xij xij

≥ ≤ ≥

xij + bi = 0 0 uij lij

for for for for

all all all all

arcs arcs arcs arcs

i,j i,j i,j i,j

Figure 2: CTP Standard Form The objective function is to minimize the total of all arc flows multiplied by their costs. Constraint (1) ensures flow balance at every node by ensuring that total flow out of a node is the same as the total flow in, adjusted for supplies or demands at the node itself. These constraints also ensure that supply units are fully distributed from all supply nodes to all demand nodes, creating flow balance for the entire network. Constraint set (2) ensures that all arcs have a non-negative unit flow. Constraint (3) ensures that no arc capacities (upper bounds) are violated. Constraint set (4) ensures that no arc lower bounds are violated. In a self-healing application to the Smart Grid, a candidate grid configuration, even one that reflects serious disruptions or damage, can be optimized. This then supports a best possible means of running the grid under adverse conditions. The special CTP formulation allows for a customized solver with highly desirable characteristics to be developed as detailed in the following section. A standard Smart Grid test problem is the IEEE 14-Bus System, illustrated in Figure 3. The corresponding flow network configuration that can be modeled as a CTP is illustrated in Figure 4.

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1.

Initialization An XML file or local database is populated with sensor readings and pre-established topological information. The initialization step reads the data and creates a candidate solution basis tree, as illustrated shown in Figure 5. From an artificial root node, a directed arc is connected to each actual network node using penalty values for the arc costs, which will force them out of the basis early. The absolute values of supplies (or negative demands) at the actual nodes are used to set initial values of the arc flows from the artificial node. In the algorithm, these artificial arcs are forced from the basis tree one by one due to their large penalty costs, leaving only actual network arcs in the final, optimal solution.

Figure 3: IEEE 14-Bus Test System Diagram

Figure 5: Example Initial Basis Tree Figure 4: IEEE 14-Bus Test System Network Representation V. MODEL SOLVER There are currently a number of solvers that have been developed and are available for producing optimal solutions to general linear programming problems. However, we needed a solver that would scale extremely well and produce solutions in near real time. Our custom solver software was written as an ASP.NET C# application using a simplex algorithm modified to exploit the special structure of the model. The powerful characteristic of the CTP that we exploit is that any linear programming basis corresponds to a spanning tree of the network representation. This enables simplex basis changes to be carried out in all integer arithmetic on graphical tree structures, greatly expediting the computations when compared with working inverses of basis matrices. Following the general scheme for applying the simplex method, we carried out the following five steps: 1. 2. 3. 4. 5.

Initialization Reduced Cost Calculation Cycle Creation Basis Update Repeat Steps 2-4 Until Optimality

Node potentials are also calculated for the initial basis tree and used to determine the best candidate arc not already in the basis tree, to replace a basic arc. The node potentials are the dual variables in linear programming terms, represented algorithmically as the sum of the arc costs following the path from any given node back to the root node in the basis tree. 2.

Reduced Cost Calculation The reduced cost is the per unit rate at which the objective function would change if a given non-basic arc were inserted into the basis tree. If the evaluation metric is a cost that should be minimized, the best reduced cost belongs to the arc that will potentially lower the total network cost by the greatest per unit amount. For any given non-basic arc, the reduced cost is calculated by subtracting the node potential of the arc's tail node and its cost from the node potential of its head node. In effect, this evaluates an alternative pathway for power to flow. If no candidate arc is found to reduce the total cost of the network, then the solution is optimal. Otherwise, the arc with the best reduced cost is chosen to enter the basis tree. At each step, both upper and lower bound on arc flows must be evaluated in order to maintain a feasible solution. 3.

Cycle Creation By definition, the basis tree is a connected graph with no cycles. This means there is a path between any two nodes, but

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not a path from any node to itself. When a non-basic arc is added to the basis tree, a cycle is created and an arc must be removed to preserve the basis tree's acyclic property. An example cycle is shown in Figure 6. Using the depth of the entering arc's nodes in the basis tree, the cycle is created by following the back path from each entering arc's node to the root node of the basis tree. The node depth allows the two back paths to be traversed in pairs during the same iteration, starting at the deepest node in the cycle and working up the basis tree until the two back paths meet at the same parent node or the root node is reached, either of which completes the cycle.

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5.

Repeat Steps 2-4 Until Optimal At Step 2, the reduced cost information for all non-basic arcs is calculated. If there are no arcs that can improve the solution, optimality is guaranteed. For the example, the optimal basis tree is illustrated in Figure 7.

Figure 7: Example Optimal Solution VI. RESULTS AND ANALYSIS

Figure 6: Example Cycle As each arc is added to the cycle, its maximum feasible flow change is calculated based on the arc's direction in relation to the cycle created by the entering arc. This value is the largest flow that could be added or subtracted from a sameor opposite-cycle direction arc, respectively, without violating the arc's flow capacity or lower bound. Using this flow value, the algorithm adjusts the flow solution in the new basis tree to move the current solution incrementally toward the optimal solution. The solution adjustment respects upper and lower bounds at every step, ensuring that feasible solutions are found at every increment. These feasible solutions can be evaluated using auxiliary criteria that might be imposed when the Smart Grid experiences equipment failures. Robustness of the solution is one such auxiliary criterion. 4.

Basis Update

Once a new arc enters the basis tree and one arc leaves, a new basis tree is determined. To fully specify the new tree, the node potentials and depth values are updated. Once the basis tree has been updated, it is ready to be used for the next iteration if the optimal solution has not yet been reached.

The key question is whether the customized solver can compute solutions fast enough to function in a self-healing system. For all of our test problems an end to end process was followed, starting with a populated data set of parameters and network topology, following all steps to generate a full optimal solution, and reporting the results. Several abstract networks with known solution obtained from the literature were tested to ensure that the solver accurately obtained the optimal solution [7]. To test the method on example power grid networks, test problems for the IEEE 14-Bus System, IEEE 30-Bus System, IEEE 57-Bus System, and IEEE 118-Bus System were downloaded from the University of Washington Electrical Engineering website. Various objective function evaluation metrics were evaluated for each problem, to generate realistic Smart Grid scenarios. The base case utilized simply used physical distances between nodes, to essentially determine overall shortest paths for electricity flow to follow. All computation tests were performed on an Intel Core 2 Quad 2.67GHz processor with 4GB DDR2 800 RAM, running on Windows Vista Ultimate x64. A local virtual directory was created for the solver, it using IIS and running the .NET 4.0 framework. Each suite of test problems was run multiple times with parameter changes, and average computation times recorded. The 14, 30, and 57 bus systems solved in well under .1 seconds and the 118 bus system solved in less than .2 seconds. Although we have not yet tested the solution algorithm on truly large problems, our computational experience thus far suggests that the algorithm will scale well. In any case, the procedure is clearly of potential value for

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dynamic power dispatching and allocation in smaller power grid components, such as microgrids. Another advantage is the ease of setting new parameters for the solver within a selfhealing system. More specifically, when devices and lines modeled by nodes and arcs in a functioning Smart Grid system fail or malfunction, an updated network topology can easily be provided to the CTP solver. This produces the capability of quickly finding high performance ways to redistribute power throughout the network, meeting electricity demands with minimal interruption of service. VII. CONCLUSION In this research, a customized CTP solver was developed as a tool for formulating and analyzing the performance of a Smart Grid network. Multiple evaluation metrics are supported, allowing a diverse set of problems to be studied using the same solver. Solutions are generated with little required computation time, opening potential for use in self-healing Smart Grid situations which inherently demand near real-time results. These solutions are guaranteed to be optimal, ensuring the best possible flow of electricity throughout the network according to the provided parameters. As sensors in the electrical grid become more sophisticated and high bandwidth communication networks are in place, the model provides the potential to receive the data streams and generate operational grid actions through the computational

efficiencies of linear programming to minimize the effect of infrastructure failures. REFERENCES [1] Kaplan, S. M. (2009). Smart Grid. Electrical Power Transmission: Background and Policy Issues. The Capital.Net, Government Series. pp. 1-42. [2] Solanki, J., S. Khushalani, and N. N. Schulz, “A Multiagent Solution to Distribution Systems Restoration,” IEEE Transactions on Power Systems, vol. 22, no. 3, pp. 1026–1034, 2007. [3] Nygard, K. E., S. Ghosn, M. Chowdhury, D. Loegering, R. Mcculloch And P. Ranganathan, "Optimization Models For Energy Reallocation In A Smart Grid," In IEEE Infocom 2011 Workshop On M2mcn, 2011 [4] Ignizio, J. P., Linear Programming, Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1994. [5] Bazaraa, M., J. Jarvis, and H. Sherali, Linear Programming and Network Flows, 4th Edition, Hoboken, New Jersey: John Wiley & Sons, Inc.,2009. [6] Brown, G. G. and G. H. Bradley, "Design and Implementation of Large Scale Primal Transshipment Algorithms," Management Science, vol. 24, no. 1, pp. 134, 1977.