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Global efforts aim at utilizing the available information technology in order to enhance the power grid with resilience properties, precise accounting and new services. Towards this end, we provide methods to support an adaptive power grid and study

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Multi-Agent Distributed Method for Dynamic Power System Rebalancing
Giorgos Georgiadis, Christopher Saunders,
Member, IEEE
, Marina Papatriantafilou
Abstract
—
Global efforts aim at utilizing the available information technology in order to enhance the power grid with resilience properties, precise accounting and new services. Towards this end, we provide methods to support an adaptive power grid and study the relative changes in supply and/or demand while in a quasi-steady state. Acknowledging that the absolute amount of the power demand/supply is not always the limiting factor, the proposed algorithm focuses on the deviations in the generated and/or demanded resources, as well as the location of these changes in the grid. Through our experimental evaluation study we show that the algorithm leads to a faster reconvergence to a new steady state operating point when compared to convergence from initial conditions after changes complete. We also show that changes can be resolved in a distributed and local manner, needing only limited and fast communication with nodes that are far from the locus of change.
Index Terms
—
Smart grid, multi-agent system, resource allocation, distributed algorithms
I.
I
NTRODUCTION
Recent modernization efforts on an international level aim to enhance the power grid with resilience properties, precise accounting and new services through the use of information technologies, with the resulting grid commonly referred to as
smart grid
. The motivating force behind this transition is the willingness to incorporate more efficiently into grid operations an increasing amount of renewable energy sources (RES) (e.g. photovoltaics and wind generation) and distributed energy resources (DER) (e.g. electric car fleets), both of which bring benefits as well as challenges. Coupling intermittent power generation from RES with the high mobility and power demand of electric vehicles may lead to grid instability but also holds the promise of high RES utilization. One way of addressing these concerns is the deployment of demand response (DR) programs between end consumers and an intermediate entity (usually a distribution system operator or an aggregator). Under these programs, the end consumer grants the right to the signatory entity of curtailing part of its power consumption at times of need, while getting a discounted electricity bill in return. In order to operate a DR program successfully, a utility needs to apply increased coordination and control of both generation and consumption assets. In a way similar to the optimal power flow (OPF) problem, the total power generation in the grid must be adapted to the total power demand and vice versa. Indeed, well established OPF methods [1]-[4] can be adapted to work with this resource allocation problem in order to i.e. increase generation as needed or curtail demand if generation capacity is inadequate. The problem considered within this paper assumes that initially the grid is in a steady state, where a suitable power flow solution has been found and implemented which takes into account any necessary economic and security constraints. From this initial steady state, deviations from the scheduled generation and demand profile can lead to system frequency errors, line over-currents, and other possible system security concerns. In these cases, it may be too time consuming to delay actions while the system executes additional OPF calculations and eventually reschedules resources. During these brief periods, the limiting factor on generator output adjustments is not the maximum production capabilities of the generator (i.e. as would be the case in OPF calculations), but the ability to ramp production up/down in order to accommodate demand. Similarly, the absolute amount of the power demand is not a limiting factor either in this scenario, but rather the difference in demanded power, as well as the location of these changes in the grid. In this work we are looking at the relative changes in supply and/or demand while in a quasi-steady state, and the proposed methodology should seek to address certain key issues, such as how the supply and demand changes propagate through the grid, if there is a suitable distributed multi-agent algorithm for addressing the quasi-steady state dynamics, and if the algorithm investigated has acceptable reconvergence times to a new stable steady state operating point. A multi-agent control system which is an extension of [5]-[6] was chosen for this work in order to propagate the supply/demand changes through the grid, and to then efficiently rebalance the power flow using a distributed algorithm which requires only knowledge of its local region in the grid. We show that the proposed method is capable of handling these slow dynamics in power grids, both on the supply and on the demand side, leading to reduced reconvergence times as compared to the same method, when used for the initial resource allocation that led to the steady state. We also show that it is possible to move beyond simple proportional allocation according to hard constraints (such as line thermal capacities) into sophisticated resource allocation according to individual node preferences (e.g. temporary abundance of generated power in the case of RES), while honoring the same hard constraints. Finally we show that nodes converge at different speeds, meaning that some nodes
C. Saunders is with the Department of Electronic and Computer Engineering, Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom (email: cssaunders@ieee.org). G. Georgiadis, M. Papatriantafilou, are with the Department of Computer Science and Engineering, Chalmers University of Technology, 41296 Gothenburg, Sweden (emails: georgiog@chalmers.se; ptrianta@chalmers.se)
will achieve a steady state long before others. This is a property that can be exploited by system designers for applications such as prior identification of critical nodes based on reconvergence rates in worst case scenarios. The remainder of this paper is organized as follows. In Section 2 we introduce the problem under consideration and describe our approach. Section 3 contains our experimental study along with extensive comments, while Section 4 discusses our results in depth. We present our concluding remarks in Section 5. II.
P
ROBLEM
F
ORMULATION AND
A
LGORITHMIC
D
ESCRIPTION
Following [6], we consider the resource allocation problem of splitting a desired amount of a specific resource among nodes
i V
connecting to each other through edges
e E
into a network
( , )
G V E
, according to the capabilities of each node. These capabilities come in the form of pairs of values
min max
( , )
i i
for each node
i
, representing the minimum and maximum amount of the desired quantity respectively (e.g. active power) that the node is willing and capable of contributing. In [6]-[7] Dominguez-Garcia et al describe a mechanism to split a global quantity
d
to each node proportionally to its capabilities, by using two helper quantities
i
and
i
for each node
i
. Through these quantities, the amount of contributing resources can be calculated recursively using the following formula
min max min
ii i i ii
k k k
(1) for each node
i
, where
i
x
is the initial requested resource by node
i
. The helper quantities update their values using the formulas
and
1 1
i ii i
k P k k P k
with initial values
minmax min
and
0 0
i i ii i i
x
where
P
is a matrix based on the adjacency matrix of the network
G
(cf [6]-[7]). The desired values of the contributing quantity
i
are obtained in the steady state of the recursion. In practice, this formula is calculated in each node using information obtained asynchronously through interaction with its neighbors, as part of a multi-agent system. However, instead of considering a resource amount (e.g. active power) to be allocated among the grid's nodes, we focus on resource
changes
, or deviations from an already existing steady state. In the example of active power as a resource to be allocated, the changes express the ramp up and ramp down capabilities of different nodes, to contribute with more or less active power if needed respectively, and can be positive or negative respectively. In effect, the resource to be allocated with the above mentioned multi-agent method in our scenario is the changes themselves. This yields the changes described below with regard to the srcinal method of Dominguez-Garcia et al [6]. Since the considered changes must necessarily refer to a specific time period and our algorithm runs periodically, in the experiments considered here we assume a time period of 5 minutes for both. This means that all metrics used by the algorithm (for example the values of
min
and
max
) must be converted and expressed in the proper derivative units, e.g. as energy in MWh if the considered primary resource is active power in MW. Note that this choice is not a hard restriction, since any appropriate time period for the considered application can be chosen. The values of
min
and
max
are determined by the physical capabilities of nodes as before and therefore cannot change easily. However, when applied to the algorithm, we treat these values as the lower and upper bounds respectively of the resources that can be contributed by a particular node for the considered time period (i.e. 5 minutes in our case), and the values used in practice can lie anywhere within this range. For example, a node may decide at a given point in time that it will reduce its contribution to 90% of the resources currently allocated in the steady state solution, or conversely increase to 110% contribution, provided that the absolute value is less than the physically-determined upper bound. Throughout the remainder of this manuscript, the symbols
min
and
max
will therefore refer to the flexible nodal output limits and will be utilized for algorithm control, and do not refer to the physical lower and upper bounds. Along with the ramp up or down in generation, the nodes may pose load demands which are also expressed in relative values to the steady state. For example, a node may request 10MW of additional energy for the next 5 minutes, or declare that is going to need 10MW less than in the proceeding 5 minute timeslot, in which case the value of 10MW is given with a negative sign. Continuing from eq. (1), we incorporate the above changes into our algorithm in the following way. Instead of using the lower and upper bounds of individual node resource contribution as
min
i
and
max
i
in eq. (1), we use the adapted values for the considered time period (i.e. 5 minutes) for initialization purposes only. Later on, these values of
min
i
and
max
i
can change to any value within the srcinal range, according to the generation capabilities of specific nodes. On the other hand, the amount of load demand can also change according to individual node wishes, and this is expressed as a new initial requested resource
i
x
in eq. (1). In addition to the new value of
i
x
(in the case of demand changes) or
min
i
and
max
i
(in the case of generation
changes) used in eq. (1), we update
i
and
i
with the following additive terms in order to repair the previous solution:
'max' max min' min
and
i i i ii i i i i i
k x x k k k
where the new values of the quantities
max min
, ,
i i i
x
are denoted with
' max' min'
, ,
i i i
x
respectively. Note that these terms depend only on the current values of
i
and
i
, the previous constraints or initial values and the changes themselves, and can be calculated easily without additional information or communication between nodes. The algorithm is able to converge to a new solution after a number of iterations. We repeat this process periodically (in our case, every 5 minutes) for the demand and generation changes that have occurred since the end of the last period. III.
E
XPERIMENTAL
S
TUDY
A.
Test System
For our experiments we used the topology of the IEEE 39-bus system [8] (Fig. 1), as well as the relevant data regarding the
generation capabilities of its nodes. As mentioned in the previous section, we are considering a periodically reoccurring time horizon of 5 minutes for system rebalancing, and all data have been converted to the corresponding units and refer to this time period. Note that only 10 out of the 39 nodes (nodes 1 to 10) of the 39-bus system possess generational capabilities (i.e. are of interest for our method
’s
convergence) and are presented in the following results. All presented results are average values of 50 experiments.
B.
Benchmark
We structure our experiments along two scenarios, one each to study changes in demand and generation respectively, as follows: 1.
For the
demand change
scenario, we identified two groups of nodes (nodes 34,35,36,37,38,27 and nodes 12,13,15,18,22,26,28) that lie at the core or the edge of the system respectively (with edge/core nodes being closer to/further away from nodes that possess generating capabilities respectively), as well as an additional third group comprising of all nodes. By creating a random change in the load demand of all group nodes at the same time, we aim to study the effect of demand changes at different parts of the grid. 2.
For the
generation change
scenario, we focus on all generating nodes (nodes 1 to 10) and study two events: first all generating nodes reduce the upper limit of their contribution randomly up to -20% of its previous value, and then they all increase it randomly also up to +20%. By changing the generating capabilities of all nodes at the same time we aim to study the effects of a sudden loss of generating capacity due to an unforeseen event. On a separate instance we repeat the experiment with the same goals but we reduce and subsequently increase generation by -40% and +40% respectively.
Fig. 1 IEEE 39-bus system
In all of the above cases, nodes that are participating in a demand or generation change are initially sending messages to neighboring nodes in order to negotiate a new steady point. These nodes in turn propagate the change, wait for replies and eventually answer back to the srcinating nodes, generating in this way bidirectional message exchanges throughout the network
1
. For the rest of the paper, in order to improve the presentation of our results, we are assuming synchronous communication rounds, where all nodes may transmit or receive at most one message per round in synchrony. However, this is not a prerequisite for the correct operation of our method: nodes can communicate either synchronously or asynchronously. We study the time needed (measured in synchronous communication rounds) for the system to reconverge to a steady state after a change, while we use as a reference point the time needed to converge initially using the same method. Note that the particular calculation of the reference point is not a prerequisite to apply our method: any number of resource allocation techniques (e.g. OPF methods) can be used to achieve the initial steady state that we use as a basis. As mentioned above, we present our results only for the generating nodes since these are the ones that are of interest
1
Note that we do not consider link or node failures, i.e. messages do not get lost and nodes disconnect gracefully or their absence can be detected by other means (for example
special periodic “alive” messages).
Fig. 2 Comparison of reconvergence times for demand changes in specific node groups Fig. 3 Comparison of reconvergence times for generation
changes up to ±
20% in all relevant nodes Fig. 4 Comparison of reconvergence times for generation
changes up to ±
40% in all relevant nodes
for the method. Our results can be found in figures Fig. 2, Fig. 3 and Fig. 4.
C.
Simulation results
Overall, we note the variation in reconvergence speed among different nodes across all experiments. Indeed, nodes such as e.g. nodes 7 and 9 need consistently less time to converge and reconverge, and in some cases (for example the generation change experiments) the savings from achieving reconvergence against converging to a steady state anew are higher than those of other nodes. This can be easily explained if we see that the specific nodes can be found at the edge of the system, having fewer neighbors at distances 2 and 3 than other generating nodes, and therefore having relatively limited reach within the grid (fewer exchanged messages needed to communicate their preferences). Regarding demand changes, we see that the above mentioned positioning of the nodes in the grid plays an equally important role. Demand changes in the core group of nodes lead to high reconvergence times, similar to the ones needed for initial convergence, whereas changes in the edge group lead to significant reductions, with changes to all nodes being somewhere in between. This can be explained as before due to the position of the disturbance's focus in relation to the rest of the grid: a change at the core, where nodes are closer together, leads to many communication messages being exchanged back and forth before a solution can be found collectively. In contrast, a change at the edge of the grid leads to fewer messages and earlier reconvergence since nodes need to negotiate a solution with fewer neighbors. For the generation change experiments, shown in figures Fig. 3 and Fig. 4, we focus on the effect of sudden and
massive changes in generating capacity in the system. It is easy to see that the reduction and increase events have slightly different effect in reconvergence speed but both lead to significant savings when contrasted with the initial convergence speed. What is particularly evident is the
difference between the ±
20% and
±
40% scenarios: the higher variation in generation capacity changes, the more time is needed for the system to reconverge. This is to be expected, since greater (and massive) loss of generating capabilities leads to more difficult rebalancing of the remaining capacity in order to cover the unchanged demand. Depending on the applica
tion at hand, variation up to ±
20% may be reasonable to assume, however it is important to be taken into account during design and system wide deployment of resource allocation algorithms. IV.
D
ISCUSSION
The method described above can easily be used in conjunction with established OPF methods, using their steady state solution as a starting point, and utilizing the proposed approach to adapt the system to new scenarios that are likely to appear in the future. Allowing for changing demand and generation constraints, we are able to address the variability of renewable generation technologies such as photovoltaic panels and wind generators, the dynamicity in load demand due to
e.g. intermittent charging of electric vehicles in a city level or varying electricity prices, as well as enhancing system reliability and security through autonomous adaptation to unexpected outages. It is also possible, through the utilization of this method, to express a willingness on behalf of the various actors involved in the grid operations to allocate a resource differently than by simply using the principle of proportionality. This is different from the method of Dominguez-Garcia et al [6], which allocates the desired resource proportionally among nodes with respect to their capabilities. Since these capabilities are largely determined by hard parameters of the grid (e.g. generating capabilities of nodes and distribution line capacities) which do not change often, the resulting allocation is deterministic and fixed. Using the method described here, it is possible to adapt the resource allocation schedule rapidly to adapt to temporary changes in the grid and rebalance to a point which is collectively decided by all grid nodes. Regarding the time performance of the proposed method, note that larger systems are expected to need more time to converge to a stable solution than smaller systems. However, convergence time does not depend on the physical size of the grid but rather on its longest communication path [9] (along which the necessary messages are being exchanged) and, by extension, on the inte
rconnection density of the grid’s various
elements (i.e. generators, loads). Furthermore, the proposed method utilizes locality to achieve fast reconvergence, without having to execute the full calculations needed to converge initially. For example, a demand or generation change in only a small, closely packed set of nodes creates an intense
communication “wave”, with back
-and-forth exchange of messages, which centers at the locus of change and dissipates with increasing distance from that center (i.e. nodes further away affect less the srcinating nodes, and need fewer communication messages). This is particularly important, since it implies that these changes will not necessarily extend to the whole grid but resolve as locally as possible. In fact, the experiments show differential reconvergence among nodes, with nodes which are further away from the locus of a particular change needing less time to reach a steady state. This property can be exploited by design engineers for a range of applications, from estimating average times of reconvergence for a given grid to identifying critical nodes based on reconvergence rates in worst case scenarios. V.
C
ONCLUSIONS
The proposed method focuses on deviations in power grids that are at a steady state between required load demand and supplied generation capacity. While these deviations may often require manual intervention for rebalancing operations, in the proposed algorithm they are treated in a distributed and local way, by letting the deviating nodes communicate their new preferences to immediate neighbors and beyond, while trying to repair the existing solution. The presented experimental study uses the IEEE 39-bus system as a test case and shows that the proposed algorithm achieves significant reduction in communication rounds needed for reconvergence when compared to convergence from initial conditions (after changes complete). We also show that it is possible to move beyond simple proportional allocation according to hard constraints, such as line thermal capacities, into sophisticated resource allocation according to individual node preferences (e.g. temporary abundance of generated power in the case of RES), while honoring the same hard constraints. Finally, we show that the gains in reconvergence time differ among nodes according to their relative location to the deviating nodes in the grid, with some nodes achieving a steady state long before others. This opens new research questions for showing guarantees depending on graph-properties of the grid and can lead to interesting applications such as prior identification of critical nodes based on reconvergence rates in worst case scenarios. R
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